Tootfinder

Sparsity, Stress-Independence and Globally Linked Pairs in Graph Rigidity Theory
D\'aniel Garamv\"olgyi, Bill Jackson, Tibor Jord\'an
https://arxiv.org/abs/2509.03150

Sparsity, Stress-Independence and Globally Linked Pairs in Graph Rigidity Theory
A graph is $\mathcal{R}_d$-independent (resp. $\mathcal{R}_d$-connected) if its $d$-dimensional generic rigidity matroid is free (resp. connected). A result of Maxwell from 1867 implies that every $\mathcal{R}_d$-independent graph satisfies the sparsity condition $|E(H)|\leq d|V(H)|-\binom{d+1}{2}$ for all subgraphs $H$ with at least $d+1$ vertices. Several other families of graphs $G$ arising naturally in rigidity theory, such as minimally globally $d$-rigid graphs, are kno…

This https://arxiv.org/abs/2505.19945 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Signed Angle Rigid Graphs for Network Localization and Formation Control
Graph rigidity theory studies the capability of a graph embedded in the Euclidean space to constrain its global geometric shape via local constraints among nodes and edges, and has been widely exploited in network localization and formation control. In recent years, the traditional rigidity theory has been extended by considering new types of local constraints such as bearing, angle, ratio of distance, etc. Among them, the signed angle constraint received extensive attention since it is practic…

Graphs with tree decompositions of small graphs and realizing them as the Reeb graphs of real algebraic functions
Naoki Kitazawa
https://arxiv.org/abs/2508.00729 https://…

Graphs with tree decompositions of small graphs and realizing them as the Reeb graphs of real algebraic functions
We have been interested in graphs and realizing them as Reeb graphs of explicit real algebraic functions. The Reeb graph of a differentiable function is the quotient space of the manifold of the domain, regarded as the space consisting of all components of preimages of all single points. Reeb graphs have been fundamental and strong tools in geometry of manifolds since the birth of theory of Morse functions, in the former half of the 20th century. We can easily see that the Reeb graph of the n…

Oriented Diameter of Mixed Graphs with Given Maximum Undirected Degree
Ran An, Hengzhe Li, Jianbing Liu, Gaoxing Sun
https://arxiv.org/abs/2507.02277 https…

Oriented Diameter of Mixed Graphs with Given Maximum Undirected Degree
In 2018, Dankelmann, Gao, and Surmacs [J. Graph Theory, 88(1): 5--17, 2018] established sharp bounds on the oriented diameter of a bridgeless undirected graph and a bridgeless undirected bipartite graph in terms of vertex degree. In this paper, we extend these results to \emph{mixed graphs}, which contain both directed and undirected edges. Let the \emph{undirected degree} $d^*_G(x)$ of a vertex $x \in V(G)$ be the number of its incident undirected edges in a mixed graph $G$ of order $n$, and…

In our last #ISE2025 lecture last week, we were discussing what makes a node "important" in a knowledge graph. A simple heuristics can be borrowed from graph theory or communication theory: Degree Centrality
Interestingly, in Wikidata In-degree centrality states Jane Austen to be to most "important" female author, while Out-degree centrality claims J.K. Rowling as being…

Kauffman bracket polynomials, perfect matchings and cluster variables
Weiqing Tian
https://arxiv.org/abs/2508.02535 https://arxiv.org/pdf/2508.02535…

Kauffman bracket polynomials, perfect matchings and cluster variables
We introduce a class of links whose bracket polynomials admit an expansion over perfect matchings of a plane bipartite graph. This class includes 2-bridge links, pretzel links, and Montesinos links. Our first main result (Theorem A) provides a partial answer to a question posed by Kauffman concerning the connection between spanning tree expansions of the Jones polynomial and the Clock Theorem. Building on Theorem A, we apply our framework to cluster theory and prove in Theorem B that the bracke…

Sampling Theory of Jointly Bandlimited Time-vertex Graph Signals
Hang Sheng, Hui Feng, Junhao Yu, Feng Ji, Bo Hu
https://arxiv.org/abs/2508.21412 https://a…

Sampling Theory of Jointly Bandlimited Time-vertex Graph Signals
Time-vertex graph signal (TVGS) models describe time-varying data with irregular structures. The bandlimitedness in the joint time-vertex Fourier spectral domain reflects smoothness in both temporal and graph topology. In this paper, we study the critical sampling of three types of TVGS including continuous-time signals, infinite-length sequences, and finite-length sequences in the time domain for each vertex on the graph. For a jointly bandlimited TVGS, we prove a lower bound on sampling densi…

This https://arxiv.org/abs/2502.07975 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csGT_…

Sink equilibria and the attractors of learning in games
Characterizing the limit behavior -- that is, the attractors -- of learning dynamics is one of the most fundamental open questions in game theory. In recent work in this front, it was conjectured that the attractors of the replicator dynamic are in one-to-one correspondence with the sink equilibria of the game -- the sink strongly connected components of a game's preference graph -- , and it was established that they do stand in at least one-to-many correspondence with them. We make threefold p…

In our last #ISE2025 lecture last week, we were discussing what makes a node "important" in a knowledge graph. A simple heuristics can be borrowed from graph theory or communication theory: Degree Centrality
Interestingly, in Wikidata In-degree centrality states Jane Austen to be to most "important" female author, while Out-degree centrality claims J.K. Rowling as being…

The principle of least action for random graphs
Ioannis Kleftogiannis, Ilias Amanatidis
https://arxiv.org/abs/2507.01468 https://arxi…

The principle of least action for random graphs
We study the statistical properties of the physical action $S$ for random graphs, by treating the number of neighbors at each vertex of the graph (degree), as a scalar field. For each configuration (run) of the graph we calculate the Lagrangian of the degree field by using a lattice quantum field theory(LQFT) approach. Then the corresponding action is calculated by integrating the Lagrangian over all the vertices of the graph. We implement an evolution mechanism for the graph by removing one ed…

Unified boson sampling
Luca Bianchi, Carlo Marconi, Laura Ares, Davide Bacco, Jan Sperling
https://arxiv.org/abs/2509.02058 https://arxiv.org/pdf/2509.0205…

Unified boson sampling
Boson sampling is a key candidate for demonstrating quantum advantage, and has already yielded significant advances in quantum simulation, machine learning, and graph theory. In this work, a unification and extension of distinct forms of boson sampling is developed. The devised protocol merges discrete-variable scattershot boson sampling with continuous-variable Gaussian boson sampling. Thereby, it is rendered possible to harness the complexity of more interesting states, such as squeezed photo…

Bounce-Averaged Theory In Arbitrary Multi-Well Plasmas: Solution Domains and the Graph Structure of their Connections
I. E. Ochs
https://arxiv.org/abs/2507.00778

Bounce-Averaged Theory In Arbitrary Multi-Well Plasmas: Solution Domains and the Graph Structure of their Connections
Bounce-averaged theories provide a framework for simulating relatively slow processes, such as collisional transport and quasilinear diffusion, by averaging these processes over the fast periodic motions of a particle on a closed orbit. This procedure dramatically increases the characteristic timescale and reduces the dimensionality of the modeled system. The natural coordinates for such calculations are the constants of motion (COM) of the fast particle motion, which by definition do not chang…

Reinforcement learning for graph theory, Parallelizing Wagner's approach
Alix Bouffard, Jane Breen
https://arxiv.org/abs/2509.01607 https://arxiv.org/p…

Reinforcement learning for graph theory, Parallelizing Wagner's approach
Our work applies reinforcement learning to construct counterexamples concerning conjectured bounds on the spectral radius of the Laplacian matrix of a graph. We expand upon the re-implementation of Wagner's approach by Stevanovic et al. with the ability to train numerous unique models simultaneously and a novel redefining of the action space to adjust the influence of the current local optimum on the learning process.

Historical Contingencies Steer the Topology of Randomly Assembled Graphs
Cole Mathis, Harrison B. Smith
https://arxiv.org/abs/2507.00335 https://

Historical Contingencies Steer the Topology of Randomly Assembled Graphs
Graphs are used to represent and analyze data in domains as diverse as physics, biology, chemistry, planetary science, and the social sciences. Across domains, random graph models relate generative processes to expected graph properties, and allow for sampling from distinct ensembles. Here we introduce a new random graph model, inspired by assembly theory, and characterize the graphs it generates. We show that graphs generated using our method represent a diverse ensemble, characterized by a br…

Optimal Dual Frame Pairs: A Synergy with Graph Theory
Shankhadeep Mondal, Ram Narayan Mohapatra
https://arxiv.org/abs/2507.23249 https://arxiv.org/pdf/2507…

Optimal Dual Frame Pairs: A Synergy with Graph Theory
This paper investigates the optimization of dual frame pairs in the context of erasure problems in data transmission, using a graph theoretical approach. Frames are essential for mitigating errors and signal loss due to their redundancy properties. We address the use of spectral radius and operator norm for error measurements, presenting conditions for the optimality of dual pairs for one and two erasures. Our study shows that a tight frame generated by connected graphs and its canonical dual p…

Computational complexity of covering regular trees
Jan Bok, Ji\v{r}\'i Fiala, Nikola Jedli\v{c}kov\'a, Jan Kratochv\'il
https://arxiv.org/abs/2507.00564

Computational complexity of covering regular trees
A graph covering projection, also referred to as a locally bijective homomorphism, is a mapping between the vertices and edges of two graphs that preserves incidences and is a local bijection. This concept originates in topological graph theory but has also found applications in combinatorics and theoretical computer science. In this paper we consider undirected graphs in the most general setting -- graphs may contain multiple edges, loops, and semi-edges. This is in line with recent trends in …

Weakly branch actions: first-order theory, rigidity and Boston's conjecture
Jorge Fari\~na-Asategui
https://arxiv.org/abs/2507.22507 https://arxiv.org/…

Weakly branch actions: first-order theory, rigidity and Boston's conjecture
We disprove a well-known conjecture of Boston (2000), which claims that a just-infinite pro-$p$ group is branch if and only if it admits a positive-dimensional embedding in the group of $p$-adic automorphisms. This is obtained as a result of a comprehensive study of the rigidity of branch actions. Firstly, we generalize the notion of the structure graph, introduced by Wilson in 2000, to weakly branch groups and use it to prove several results on the first-order theory of weakly branch groups,…

Graph theory inspired anomaly detection at the LHC
Jack Y. Araz, Dimitrios Athanasakos, Mateusz Ploskon, Felix Ringer
https://arxiv.org/abs/2506.19920 http…

Graph theory inspired anomaly detection at the LHC
Designing model-independent anomaly detection algorithms for analyzing LHC data remains a central challenge in the search for new physics, due to the high dimensionality of collider events. In this work, we develop a graph autoencoder as an unsupervised, model-agnostic tool for anomaly detection, using the LHC Olympics dataset as a benchmark. By representing jet constituents as a graph, we introduce a method to systematically control the information available to the model through sparse graph c…

A Tactical Behaviour Recognition Framework Based on Causal Multimodal Reasoning: A Study on Covert Audio-Video Analysis Combining GAN Structure Enhancement and Phonetic Accent Modelling
Wei Meng
https://arxiv.org/abs/2507.21100

A Tactical Behaviour Recognition Framework Based on Causal Multimodal Reasoning: A Study on Covert Audio-Video Analysis Combining GAN Structure Enhancement and Phonetic Accent Modelling
This paper introduces TACTIC-GRAPHS, a system that combines spectral graph theory and multimodal graph neural reasoning for semantic understanding and threat detection in tactical video under high noise and weak structure. The framework incorporates spectral embedding, temporal causal edge modeling, and discriminative path inference across heterogeneous modalities. A semantic-aware keyframe extraction method fuses visual, acoustic, and action cues to construct temporal graphs. Using graph atten…

Replaced article(s) found for cs.IT. https://arxiv.org/list/cs.IT/new
[1/1]:
- Graph Reconstruction from Noisy Random Subgraphs
Andrew McGregor, Rik Sengupta
http…

Tree decompositions with small width, spread, order and degree
David R. Wood
https://arxiv.org/abs/2509.01140 https://arxiv.org/pdf/2509.01140

Tree decompositions with small width, spread, order and degree
Tree-decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. The main property of tree-decompositions is the width (the maximum size of a bag $-1$). We show that every graph has a tree-decomposition with near-optimal width, plus several additional properties of interest. In particular every graph $G$ with treewidth at most $k$ has a tree-decomposition with width at most $72k+1$, where each vertex $v$ appears in at most $\text{deg}_G(v)+1$ bags, the num…

Decorated Clusters and Geometrical Frustration in Cluster Spin Glass: A Random Graph Approach
S. G. Magalhaes, F. M. Zimmer, R. Erichsen Jr
https://arxiv.org/abs/2507.22690 http…

Decorated Clusters and Geometrical Frustration in Cluster Spin Glass: A Random Graph Approach
We develop a theory to investigate how geometrically frustrated clusters that become decorated affect the Cluster Spin Glass phase. The cluster structure is assumed to be a tetrahedron composed of Ising spins with z-anisotropy placed at its vertices that interact antiferromagnetically. We consider the probability $1-p_J$ of finding an impurity at a vertex of the tetrahedron that interacts ferromagnetically with the remaining elements inside the tetrahedron. An intercluster disorder is added as …

Sectional Kolmogorov N-widths for parameter-dependent function spaces: A general framework with application to parametrized Friedrichs' systems
Christian Engwer, Mario Ohlberger, Lukas Renelt
https://arxiv.org/abs/2507.00678

Sectional Kolmogorov N-widths for parameter-dependent function spaces: A general framework with application to parametrized Friedrichs' systems
We investigate parametrized variational problems where for each parameter the solution may originate from a different parameter-dependent function space. Our main motivation is the theory of Friedrichs' systems, a large abstract class of linear PDE-problems whose solutions are sought in operator- (and thus parameter-)dependent graph spaces. Other applications include function spaces on parametrized domains or discretizations involving data-dependent stabilizers. Concerning the set of all parame…

from my link log —
Unlocking sudoku's secrets with graph theory and abstract algebra.
https://chalkdustmagazine.com/features/unlocking-sudokus-secrets/
saved 2025-04-14

Unlocking sudoku's secrets - Chalkdust
Sara Logsdon looks to graph theory and abstract algebra for help on the puzzle page

Potential Theory and the Boundary of Combinatorial Graphs
Stefan Steinerberger
https://arxiv.org/abs/2507.20833 https://arxiv.org/pdf/2507.20833

Potential Theory and the Boundary of Combinatorial Graphs
Let $G=(V,E)$ be a finite, connected graph. We investigate a notion of boundary $\partial G \subseteq V$ and argue that it is well behaved from the point of view of potential theory. This is done by proving a number of discrete analogous of classical results for compact domains $Ω\subset \mathbb{R}^d$. These include (1) an analogue of Pólya's result that a random walk in $Ω$ typically hits the boundary $\partial Ω$ within $\lesssim \mbox{diam}(Ω)^2$ units of time, (2) an analogue of the Fa…

Tangling and Untangling Trees on Point-sets
Giuseppe Di Battista, Giuseppe Liotta, Maurizio Patrignani, Antonios Symvonis, Ioannis G. Tollis
https://arxiv.org/abs/2508.18535 htt…

Tangling and Untangling Trees on Point-sets
We study a question that lies at the intersection of classical research subjects in Topological Graph Theory and Graph Drawing: Computing a drawing of a graph with a prescribed number of crossings on a given set $S$ of points, while ensuring that its curve complexity (i.e., maximum number of bends per edge) is bounded by a constant. We focus on trees: Let $T$ be a tree, $\vartheta(T)$ be its thrackle number, and $χ$ be any integer in the interval $[0,\vartheta(T)]$. In the tangling phase we co…

Homomorphism Indistinguishability and Game Comonads for Restricted Conjunction and Requantification
Georg Schindling
https://arxiv.org/abs/2506.19746 https…

Homomorphism Indistinguishability and Game Comonads for Restricted Conjunction and Requantification
The notion of homomorphism indistinguishability offers a combinatorial framework for characterizing equivalence relations of graphs, in particular equivalences in counting logics within finite model theory. That is, for certain graph classes, two structures agree on all homomorphism counts from the class if and only if they satisfy the same sentences in a corresponding logic. This perspective often reveals connections between the combinatorial properties of graph classes and the syntactic struc…

The theory of reachability in trace-pushdown systems
Dietrich Kuske
https://arxiv.org/abs/2507.15733 https://arxiv.org/pdf/2507.15733

The theory of reachability in trace-pushdown systems
We consider pushdown systems that store, instead of a single word, a Mazurkiewicz trace on its stack. These systems are special cases of valence automata over graph monoids and subsume multi-stack systems. We identify a class of such systems that allow to decide the first-order theory of their configuration graph with reachability. This result complements results by D'Osualdo, Meyer, and Zetzsche (namely the decidability for arbitrary pushdown systems under a severe restriction on the depende…

Edge Clique Partition and Cover Beyond Independence
Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, Kirill Simonov
https://arxiv.org/abs/2506.21216 https:…

Edge Clique Partition and Cover Beyond Independence
Covering and partitioning the edges of a graph into cliques are classical problems at the intersection of combinatorial optimization and graph theory, having been studied through a range of algorithmic and complexity-theoretic lenses. Despite the well-known fixed-parameter tractability of these problems when parameterized by the total number of cliques, such a parameterization often fails to be meaningful for sparse graphs. In many real-world instances, on the other hand, the minimum number of …

Higher-rank graphs and the graded $K$-theory of Kumjian-Pask Algebras
Roozbeh Hazrat, Promit Mukherjee, David Pask, Sujit Kumar Sardar
https://arxiv.org/abs/2507.19879 https://

Higher-rank graphs and the graded $K$-theory of Kumjian-Pask Algebras
This paper lays out the foundations of graded $K$-theory for Leavitt algebras associated with higher-rank graphs, also known as Kumjian-Pask algebras, establishing it as a potential tool for their classification. For a row-finite $k$-graph $Λ$ without sources, we show that there exists a $\mathbb{Z}[\mathbb{Z}^k]$-module isomorphism between the graded zeroth (integral) homology $H_0^{gr}(\mathcal{G}_Λ)$ of the infinite path groupoid $\mathcal{G}_Λ$ and the graded Grothendieck group $K_0^{g…

Probabilistic Graphical Models: A Concise Tutorial
Jacqueline Maasch, Willie Neiswanger, Stefano Ermon, Volodymyr Kuleshov
https://arxiv.org/abs/2507.17116

Probabilistic Graphical Models: A Concise Tutorial
Probabilistic graphical modeling is a branch of machine learning that uses probability distributions to describe the world, make predictions, and support decision-making under uncertainty. Underlying this modeling framework is an elegant body of theory that bridges two mathematical traditions: probability and graph theory. This framework provides compact yet expressive representations of joint probability distributions, yielding powerful generative models for probabilistic reasoning. This tut…

Understanding the rift between update rules in Evolutionary Graph Theory: The intrinsic death rate drives star graphs from amplifying to suppressing natural selection
Max Dew, Christopher E. Overton
https://arxiv.org/abs/2506.15528

Understanding the rift between update rules in Evolutionary Graph Theory: The intrinsic death rate drives star graphs from amplifying to suppressing natural selection
Evolutionary graph theory is the study of evolutionary dynamics in structured populations. A well-known problem in evolutionary graph theory is that the spread of mutation (measured by fixation probability) is impacted by the graph type chosen and the update rule. For example, the star graph is an amplifier of natural selection under the birth-death with fitness on birth (Bd) update rule but a suppressor of natural selection under the death-birth with fitness on birth (dB) update rule. A contin…

Stereotype graph: A mathematical framework of category stereotypes via graph theory
Yijia Yan
https://arxiv.org/abs/2506.12533 https://

Stereotype graph: A mathematical framework of category stereotypes via graph theory
In social psychology and cognitive science, there has been much interest in studying category stereotypes. However, we still lack a consensual mathematical definition or framework, which is necessary for us to hold a deeper understanding of stereotypes in human cognition. In this paper, we use graph theory to portray category stereotypes in human cognition, based on pairs of labels having special relations. By using methods and conclusions in graph theory (including algebraic graph theory and v…

$C_4$-free subgraphs of high degree with geometric applications
Zach Hunter, Aleksa Milojevi\'c, Istvan Tomon, Benny Sudakov
https://arxiv.org/abs/2506.23942

$C_4$-free subgraphs of high degree with geometric applications
The Zarankiewicz problem, a cornerstone problem in extremal graph theory, asks for the maximum number of edges in an $n$-vertex graph that does not contain the complete bipartite graph $K_{s,s}$. While the problem remains widely open in the case of general graphs, the past two decades have seen significant progress on this problem for various restricted graph classes -- particularly those arising from geometric settings -- leading to a deeper understanding of their structure. In this paper, w…

Physics-Informed EvolveGCN: Satellite Prediction for Multi Agent Systems
Timothy Jacob Huber, Madhur Tiwari, Camilo A. Riano-Rios
https://arxiv.org/abs/2507.22279 https://

Physics-Informed EvolveGCN: Satellite Prediction for Multi Agent Systems
In the rapidly evolving domain of autonomous systems, interaction among agents within a shared environment is both inevitable and essential for enhancing overall system capabilities. A key requirement in such multi-agent systems is the ability of each agent to reliably predict the future positions of its nearest neighbors. Traditionally, graphs and graph theory have served as effective tools for modeling inter agent communication and relationships. While this approach is widely used, the presen…

Notes on the cohomology of partial Hessenberg varieties
Tatsuya Horiguchi, Mikiya Masuda, Takashi Sato, Haozhi Zeng
https://arxiv.org/abs/2507.23259 https://

Notes on the cohomology of partial Hessenberg varieties
Hessenberg varieties are a family of subvarieties of full flag varieties. This family contains well-known varieties such as Springer fibers, Peterson varieties, and permutohedral varieties. It was introduced by De Mari-Procesi-Shayman in 1992 and has been actively studied in this decade. In particular, unexpected relations to hyperplane arrangements and the Stanley-Stembridge conjecture in graph theory have been discovered. Hessenberg varieties can be defined in partial flag varieties. In this …

FIRE-GNN: Force-informed, Relaxed Equivariance Graph Neural Network for Rapid and Accurate Prediction of Surface Properties
Circe Hsu, Claire Schlesinger, Karan Mudaliar, Jordan Leung, Robin Walters, Peter Schindler
https://arxiv.org/abs/2508.16012

FIRE-GNN: Force-informed, Relaxed Equivariance Graph Neural Network for Rapid and Accurate Prediction of Surface Properties
The work function and cleavage energy of a surface are critical properties that determine the viability of materials in electronic emission applications, semiconductor devices, and heterogeneous catalysis. While first principles calculations are accurate in predicting these properties, their computational expense combined with the vast search space of surfaces make a comprehensive screening approach with density functional theory (DFT) infeasible. Here, we introduce FIRE-GNN (Force-Informed, Re…

Hodge-de Rham Theory on Higher-Dimensional Level-L Sierpinski Gaskets
Sze-Man Ngai, Shui-Hong Zhou
https://arxiv.org/abs/2508.12319 https://arxiv.org/pdf/2…

Hodge-de Rham Theory on Higher-Dimensional Level-L Sierpinski Gaskets
This paper extends the Hodge-de Rham theory of Aaron \textit{et al.} [Commun. Pure Appl. Anal. {\bf 13} (2014)] to higher-dimensional level-$l$ Sierpinski gaskets $SG_{\ell}^{n},$ providing a framework for analyzing differential forms and Laplacians on these fractal structures. We construct a sequence of graphs approximating $SG_{\ell}^{n}$ and define $k$-forms, de Rham derivatives, and their duals on these graphs. We prove that the extension of a $1$-form on a generation-$m$ graph to a $1$-for…

Computer-assisted graph theory: a survey
Jorik Jooken
https://arxiv.org/abs/2508.20825 https://arxiv.org/pdf/2508.20825

Computer-assisted graph theory: a survey
Computers and algorithms play an ever-increasing role in obtaining new results in graph theory. In this survey, we present a broad range of techniques used in computer-assisted graph theory, including the exhaustive generation of all pairwise non-isomorphic graphs within a given class, the use of searchable databases containing graphs and invariants as well as other established and emerging algorithmic paradigms. We cover approaches based on mixed integer linear programming, semidefinite progra…

Treedepth Inapproximability and Exponential ETH Lower Bound
\'Edouard Bonnet, Daniel Neuen, Marek Soko{\l}owski
https://arxiv.org/abs/2507.13818 https:…

Treedepth Inapproximability and Exponential ETH Lower Bound
Treedepth is a central parameter to algorithmic graph theory. The current state-of-the-art in computing and approximating treedepth consists of a $2^{O(k^2)} n$-time exact algorithm and a polynomial-time $O(\text{OPT} \log^{3/2} \text{OPT})$-approximation algorithm, where the former algorithm returns an elimination forest of height $k$ (witnessing that treedepth is at most $k$) for the $n$-vertex input graph $G$, or correctly reports that $G$ has treedepth larger than $k$, and $\text{OPT}$ is t…

Entanglement Classification in the Graph States: The generalization to $n$-Qubits States using the Entanglement Matrix
Sameer Sharma
https://arxiv.org/abs/2507.11458

Entanglement Classification in the Graph States: The generalization to $n$-Qubits States using the Entanglement Matrix
Graph states represent a significant class of multi-partite entangled quantum states with applications in quantum error correction, quantum communication, and quantum computation. In this work, we introduce a novel formalism called the Entanglement Matrix for quantifying and classifying entanglement in n-qubit graph states. Leveraging concepts from graph theory and quantum information, we develop a systematic approach to analyze entanglement by identifying primary and secondary midpoints in gra…

The Redundancy of Full Nodes in Bitcoin: A Network-Theoretic Demonstration of Miner-Centric Propagation Topologies
Dr Craig S Wright
https://arxiv.org/abs/2506.14197

The Redundancy of Full Nodes in Bitcoin: A Network-Theoretic Demonstration of Miner-Centric Propagation Topologies
This paper formally examines the network structure of Bitcoin CORE (BTC) and Bitcoin Satoshi Vision (BSV) using complex graph theory to demonstrate that home-hosted full nodes are incapable of participating in or influencing the propagation topology. Leveraging established models such as scale-free networks and small-world connectivity, we demonstrate that the propagation graph is dominated by a densely interconnected miner clique, while full nodes reside on the periphery, excluded from all tra…

Machine Learning for Evolutionary Graph Theory
Guoli Yang, Matteo Cavaliere, Mingtao Zhang, Giovanni Masala, Adam Miles, Mengzhu Wang
https://arxiv.org/abs/2507.08363

Machine Learning for Evolutionary Graph Theory
The stability of communities - whether biological, social, economic, technological or ecological depends on the balance between cooperation and cheating. While cooperation strengthens communities, selfish individuals, or "cheaters," exploit collective benefits without contributing. If cheaters become too prevalent, they can trigger the collapse of cooperation and of the community, often in an abrupt manner. A key challenge is determining whether the risk of such a collapse can be detected in ad…

On the coefficients of Coxeter polynomials of trees and bipartite quivers
Niv Harel, Sefi Ladkani
https://arxiv.org/abs/2509.02375 https://arxiv.org/pdf/25…

On the coefficients of Coxeter polynomials of trees and bipartite quivers
We apply spectral graph theory and a theorem of A'Campo to express the first and second coefficients of the Coxeter polynomials associated with certain bipartite quivers in terms of the degrees of the vertices in their underlying graphs. As a consequence, we provide a new proof of a result by Happel, expressing the second coefficient of the Coxeter polynomial of a tree in terms of its vertex degrees.

Replaced article(s) found for math.GR. https://arxiv.org/list/math.GR/new
[1/1]:
- Aspects of the commuting graph
V. Arvind, Xuanlong Ma, Peter J. Cameron, Natalia V. Maslova

Supercharacters of finite abelian groups and applications to spectra of $U$-unitary Cayley graphs
Tung T. Nguyen, Nguyen Duy T\^an
https://arxiv.org/abs/2508.10348 https://

Supercharacters of finite abelian groups and applications to spectra of $U$-unitary Cayley graphs
We define super-Cayley graphs over a finite abelian group $G$. Using the theory of supercharacters on $G$, we explain how their spectra can be realized as a super-Fourier transform of a superclass characteristic function. Consequently, we show that a super-Cayley graph is determined by its spectrum once an indexing on the underlying group $G$ is fixed. This generalizes a theorem by Sander-Sander, which investigates the case where $G$ is a cyclic group. We then use our theory to define and study…

HEIMDALL: a grapH-based sEIsMic Detector And Locator for microseismicity
Matteo Bagagli, Francesco Grigoli, Davide Bacciu
https://arxiv.org/abs/2507.10850 …

HEIMDALL: a grapH-based sEIsMic Detector And Locator for microseismicity
In this work, we present a new deep-learning model for microseismicity monitoring that utilizes continuous spatiotemporal relationships between seismic station recordings, forming an end-to-end pipeline for seismic catalog creation. It employs graph theory and state-of-the-art graph neural network architectures to perform phase picking, association, and event location simultaneously over rolling windows, making it suitable for both playback and near-real-time monitoring. As part of the global s…

How Complex is a Complex Network? Insights from Linear Systems Theory
Giacomo Baggio, Marco Fabris
https://arxiv.org/abs/2507.06389 https://

How Complex is a Complex Network? Insights from Linear Systems Theory
This paper leverages linear systems theory to propose a principled measure of complexity for network systems. We focus on a network of first-order scalar linear systems interconnected through a directed graph. By locally filtering out the effect of nodal dynamics in the interconnected system, we propose a new quantitative index of network complexity rooted in the notion of McMillan degree of a linear system. First, we show that network systems with the same interconnection structure share the s…

Sabotage the Mantel Theorem
Natalie Behague, Debsoumya Chakraborti, Xizhi Liu
https://arxiv.org/abs/2506.23794 https://arxiv.org/pdf/…

Sabotage the Mantel Theorem
One of the earliest results in extremal graph theory, Mantel's theorem, states that the maximum number of edges in a triangle-free graph $G$ on $n$ vertices is $\lfloor n^2/4 \rfloor$. We investigate how this extremal bound is affected when $G$ is additionally required to contain a prescribed graph $\mathbb{P}$ as a subgraph. We establish general upper and lower bounds for this problem, which are tight in the exponent for random triangle-free graphs and graphs generated by the triangle-free pro…

Graphon Signal Processing for Spiking and Biological Neural Networks
Takuma Sumi, Georgi S. Medvedev
https://arxiv.org/abs/2508.17246 https://arxiv.org/pdf…

Graphon Signal Processing for Spiking and Biological Neural Networks
Graph Signal Processing (GSP) extends classical signal processing to signals defined on graphs, enabling filtering, spectral analysis, and sampling of data generated by networks of various kinds. Graphon Signal Processing (GnSP) develops this framework further by employing the theory of graphons. Graphons are measurable functions on the unit square that represent graphs and limits of convergent graph sequences. The use of graphons provides stability of GSP methods to stochastic variability in n…

This https://arxiv.org/abs/2308.04512 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

An introduction to graph theory
This is a graduate-level introduction to graph theory, corresponding to a quarter-long course. It covers simple graphs, multigraphs as well as their directed analogues, and more restrictive classes such as tournaments, trees and arborescences. Among the features discussed are Eulerian circuits, Hamiltonian cycles, spanning trees, the matrix-tree and BEST theorems, proper colorings, Turan's theorem, bipartite matching and the Menger and Gallai--Milgram theorems. The basics of network flows are i…

Directed and Undirected Vertex Connectivity Problems are Equivalent for Dense Graphs
Yonggang Jiang, Sagnik Mukhopadhyay, Sorrachai Yingchareonthawornchai
https://arxiv.org/abs/2508.20305

Directed and Undirected Vertex Connectivity Problems are Equivalent for Dense Graphs
Vertex connectivity and its variants are among the most fundamental problems in graph theory, with decades of extensive study and numerous algorithmic advances. The directed variants of vertex connectivity are usually solved by manually extending fast algorithms for undirected graphs, which has required considerable effort. In this paper, we present a simple, black-box randomized reduction from directed to undirected vertex connectivity for dense graphs. As immediate corollaries, we largely s…

Counting Chemical Isomers with Multivariate Generating Functions
Rana Shojaei, Thilo Gross
https://arxiv.org/abs/2507.21713 https://arxiv.org/pdf/2507.2171…

Counting Chemical Isomers with Multivariate Generating Functions
Counting the number of isomers of a chemical molecule is one of the formative problems of graph theory. However, recent progress has been slow, and the problem has largely been ignored in modern network science. Here we provide an introduction to the mathematics of counting network structures and then use it to derive results for two new classes of molecules. In contrast to previously studied examples, these classes take additional chemical complexity into account and thus require the use of mu…

Recognizing Penny and Marble Graphs is Hard for Existential Theory of the Reals
Anna Lubiw, Marcus Schaefer
https://arxiv.org/abs/2508.10136 https://arxiv.…

Recognizing Penny and Marble Graphs is Hard for Existential Theory of the Reals
We show that the recognition problem for penny graphs (contact graphs of unit disks in the plane) is $\exists\mathbb{R}$-complete, that is, computationally as hard as the existential theory of the reals, even if a combinatorial plane embedding of the graph is given. The exact complexity of the penny graph recognition problem has been a long-standing open problem. We lift the penny graph result to three dimensions and show that the recognition problem for marble graphs (contact graphs of unit …

Nice vertices in cubic graphs
Wuxian Chen, Fuliang Lu, Heping Zhang
https://arxiv.org/abs/2508.21471 https://arxiv.org/pdf/2508.21471

Nice vertices in cubic graphs
A subgraph $G'$ of a graph $G$ is nice if $G-V(G')$ has a perfect matching. Nice subgraphs play a vital role in the theory of ear decomposition and matching minors of matching covered graphs. A vertex $u$ of a cubic graph is nice if $u$ and its neighbors induce a nice subgraph. D. Král et al. (2010) [9] showed that each vertex of a cubic brick is nice. It is natural to ask how many nice vertices a matching covered cubic graph has. In this paper, using some basic results of matching covered gra…

The Edge-Distinguishing Game
Nathaniel Benjamin, Elisa Benthem, Cooper Burkel, Marissa Chesser, Mike Janssen
https://arxiv.org/abs/2508.21185 https://arxiv…

The Edge-Distinguishing Game
In this paper, we introduce a graph coloring game called the Edge-Distinguishing Game (EDGe). The edge-distinguishing chromatic number of a graph is used to determine the moves each player can make. We determine which player has a winning strategy for particular graphs and graph families. Additionally, utilizing principles from game theory as well as previous work on a computational solution for the Game of Cycles.

Walking on Archimedean Lattices: Insights from Bloch Band Theory
Davidson Noby Joseph, Igor Boettcher
https://arxiv.org/abs/2507.12662 https://

Walking on Archimedean Lattices: Insights from Bloch Band Theory
Returning walks on a lattice are sequences of moves that start at a given lattice site and return to the same site after $n$ steps. Determining the total number of returning walks of a given length $n$ is a typical graph-theoretical problem with connections to lattice models in statistical and condensed matter physics. We derive analytical expressions for the returning walk numbers on the eleven two-dimensional Archimedean lattices by developing a connection to the theory of Bloch energy bands.…

Surjectivity of the Cannon--Thurston map in metric (graph) bundles
Rakesh Halder
https://arxiv.org/abs/2507.07076 https://arxiv.org/p…

Surjectivity of the Cannon--Thurston map in metric (graph) bundles
Metric (graph) bundles generalize the notion of fiber bundles to the context of geometric group theory and were introduced by Mj and Sardar. Suppose $X$ is a metric (graph) bundle over $B$ such that the fibers are (uniformly) hyperbolic, and the total space $X$ is also hyperbolic. In this generality, Mj--Sardar proved that the inclusion of a fiber into $X$ admits a continuous extension to the (Gromov) boundary. In this article, we prove that such a continuous extension map between boundaries …

Graph theory-based automated quantum algorithm for efficient querying of acyclic and multiloop causal configurations
Salvador A. Ochoa-Oregon, Juan P. Uribe-Ram\'irez, Roger J. Hern\'andez-Pinto, Selomit Ram\'irez-Uribe, Germ\'an Rodrigo
https://arxiv.org/abs/2508.04019

Graph theory-based automated quantum algorithm for efficient querying of acyclic and multiloop causal configurations
Quantum algorithms provide a promising framework in high-energy physics, in particular, for unraveling the causal configurations of multiloop Feynman diagrams by identifying Feynman propagators with qubits, a challenge analogous to querying directed acyclic graphs in graph theory. In this paper, we present the Minimum Clique-optimised quantum Algorithm (MCA), an automated quantum algorithm designed to efficiently query the causal structures within the Loop-Tree Duality. The MCA quantum algorith…

A generalization of Deterministic Finite Automata related to discharging
John M. Campbell
https://arxiv.org/abs/2506.14072 https://ar…

A generalization of Deterministic Finite Automata related to discharging
Deterministic Finite Automata (DFAs) are of central importance in automata theory. In view of how state diagrams for DFAs are defined using directed graphs, this leads us to introduce a generalization of DFAs related to a method widely used in graph theory referred to as the discharging method. Given a DFA $(Q, Σ, δ, q_{0}, F)$, the transition function $δ\colon Q \times Σ\to Q$ determines a directed path in the corresponding state diagram based on an input string $a_{1} a_{2} \cdots a_{n}$ …

Paired Disjunctive Domination Number of Middle Graphs
Hande Tuncel Golpek, Zeliha Kartal Yildiz, Aysun Aytac
https://arxiv.org/abs/2506.19529 https://

Paired Disjunctive Domination Number of Middle Graphs
The concept of domination in graphs plays a central role in understanding structural properties and applications in network theory. In this study, we focus on the paired disjunctive domination number in the context of middle graphs, a transformation that captures both adjacency and incidence relations of the original graph. We begin by investigating this parameter for middle graphs of several special graph classes, including path graphs, cycle graphs, wheel graphs, complete graphs, complete bip…

New allocation rule based on graph structures and their application to economic phenomena
Taiki Yamada, Taisuke Matsubae, Tomoya Akamatsu
https://arxiv.org/abs/2507.11808

New allocation rule based on graph structures and their application to economic phenomena
This study introduces the \emph{edge-based Shapley value}, a novel allocation rule within cooperative game theory, specifically tailored for networked systems, where value is generated through interactions represented by edges. Traditional allocation rules, such as the Shapley and Myerson values, evaluate player contributions based on node-level characteristics, or connected components. However, these approaches often fail to adequately capture the functional role of edges, which are crucial in…

Algorithm for computing the partition function of the Potts model for SP-graphs
Sofya Mukhamedzhanova, Bulat Sabirov, Amir Mukhamedzhanov
https://arxiv.org/abs/2507.22579 https:…

Algorithm for computing the partition function of the Potts model for SP-graphs
The q-state Potts model is a fundamental framework in statistical physics and graph theory, with its partition function encoding rich information about spin configurations. The multivariate Tutte polynomial (known as the partition function of the Potts model) can be defined on an arbitrary finite graph $G$ and encodes a lot of important combinatorial information about the graph. As a special case, it contains the familiar Tutte polynomial with two variables and, consequently, its specialization…

Lower Bounds on the Haraux Function
Patrick L. Combettes, Julien N. Mayrand
https://arxiv.org/abs/2508.15735 https://arxiv.org/pdf/2508.15735

Lower Bounds on the Haraux Function
The Haraux function is an important tool in monotone operator theory and its applications. One of its salient properties for maximally monotone operators is to be valued in $[0,+\infty]$ and to vanish only on the graph of the operator. Sharper lower bounds for this function were recently proposed in specific scenarios. We derive lower bounds in the general context of set-valued operators in reflexive Banach spaces. These bounds are new, even for maximally monotone operators acting on Euclidean …

Asymptotic Expansion of the Gaussian Integral Operators on Riemannian submanifolds of $\mathbb{R}^{n}$
Jia-Ming (Frank), Liou, Chi-Chien Lu
https://arxiv.org/abs/2506.13238

Asymptotic Expansion of the Gaussian Integral Operators on Riemannian submanifolds of $\mathbb{R}^{n}$
The Gaussian integral operator arises naturally as a local Euclidean approximation of the heat semigroup on a Riemannian manifold and plays a pivotal role in the analysis of graph Laplacians, particularly within the frameworks of manifold learning and spectral graph theory. In this paper, we study the asymptotic behavior of the Gaussian integral operator on a smooth Riemannian submanifold \( M \subset \mathbb{R}^n \), focusing on its expansion as \( \varepsilon \to 0^+ \). Under the assumption …

On the association scheme of perfect matchings and their designs
John Bamberg, Lukas Klawuhn
https://arxiv.org/abs/2507.00813 https://

On the association scheme of perfect matchings and their designs
We investigate generalisations of 1-factorisations and hyperfactorisations of the complete graph $K_{2n}$. We show that they are special subsets of the association scheme obtained from the Gelfand pair $(S_{2n},S_2 \wr S_n)$. This unifies and extends results by Cameron (1976) and gives rise to new existence and non-existence results. Our methods involve working in the group algebra $\mathbb{C}[S_{2n}]$ and using the representation theory of $S_{2n}$.

In Reverie Together: Ten Years of Mathematical Discovery with a Machine Collaborator
Randy Davila, Boris Brimkov, Ryan Pepper
https://arxiv.org/abs/2507.17780 https://

In Reverie Together: Ten Years of Mathematical Discovery with a Machine Collaborator
We present four open conjectures in graph theory generated by the automated conjecturing system \texttt{TxGraffiti}. Each conjecture is concise, grounded in natural graph invariants, and empirically validated across hundreds of graphs. Despite extensive effort, these statements remain unresolved--defying both proof and counterexample. They are not only mathematical challenges but creative expressions--born of symbolic pattern recognition and mathematician-defined heuristics, refined through yea…

Replaced article(s) found for math.CO. https://arxiv.org/list/math.CO/new
[1/1]:
- Applications of rational difference equations to spectral graph theory: expanded version
Elismar R. Oliveira, Vilmar Trevisan

On the Isoperimetric functions of a class of Artin Groups
Arye Juhasz
https://arxiv.org/abs/2507.16770 https://arxiv.org/pdf/2507.167…

On the Isoperimetric functions of a class of Artin Groups
We find a polynomial (n^6) isoperimetric function for Artin groups, the defining graph of which contains no edges labelled by 3. This in particular shows that even Artin groups have solvable word problem. We use small cancellation theory of relative extended presentations.

Reducible Iterated Graph Systems: multiscale-freeness and multifractals
Nero Ziyu Li, Frank Xin Hu, Thomas Britz
https://arxiv.org/abs/2506.18073 https://

Reducible Iterated Graph Systems: multiscale-freeness and multifractals
Iterated Graph Systems (IGS) aims to transplant ideas from fractal geometry into graph theory. Building on this framework, we extend Edge IGS from the primitive to the reducible setting. Within this broader context, we formulate rigorous definitions of multifractality and multiscale-freeness for graph fractals, and we establish conditions that are equivalent to the occurrence of these two phenomena. We further determine the corresponding fractal and degree spectra, proving that both are finite …

Metrics on Permutation Families Defined by a Restriction Graph
Danylo Tymoshenko, Leonhard Nagel
https://arxiv.org/abs/2507.10569 https://

Metrics on Permutation Families Defined by a Restriction Graph
Understanding the metric structure of permutation families is fundamental to combinatorics and has applications in social choice theory, bioinformatics, and coding theory. We study permutation families defined by restriction graphs--oriented graphs that constrain the relative order of elements in valid permutations. For any restriction graph $G$, we determine the maximum distance achievable by two permutations under the $\ell_\infty$-metric and provide an explicit algorithm that constructs opti…

An ${\mathfrak S}_3$-cover of $K_4$ and integral polyhedral graphs
Taizo Sadahiro
https://arxiv.org/abs/2508.18593 https://arxiv.org/pdf/2508.18593

An ${\mathfrak S}_3$-cover of $K_4$ and integral polyhedral graphs
We show that the star graph defined as the Cayley graph of Sn+1 generated by the star transpositions is an Sn -cover of the complete graph Kn+1 , which is known to have fine spectral properties. In the case n = 3, the star graph also has fine geometric properties: it em- beds into the honeycomb lattice and has a spectrum computable via both representation theory and an explicit Fourier formula. Inter- mediate covers correspond to the cube and truncated tetrahedron, offering a new interpretation…

Derangements in permutation groups with two orbits
Melissa Lee, Tomasz Popiel, Gabriel Verret
https://arxiv.org/abs/2506.11396 https://

Derangements in permutation groups with two orbits
A classical theorem of Jordan asserts that if a group $G$ acts transitively on a finite set of size at least $2$, then $G$ contains a derangement (a fixed-point free element). Generalisations of Jordan's theorem have been studied extensively, due in part to their applications in graph theory, number theory and topology. We address a generalisation conjectured recently by Ellis and Harper, which says that if $G$ has exactly two orbits and those orbits have equal length $n \geq 2$, then $G$ conta…

More on Nosal's spectral theorem: Books and $4$-cycles
Yongtao Li, Hong Liu, Shengtong Zhang
https://arxiv.org/abs/2508.14366 https://arxiv.org/pdf/250…

More on Nosal's spectral theorem: Books and $4$-cycles
Spectral graph theory studies how the eigenvalues of a graph relate to the structural properties of a graph. In this paper, we solve three open problems in spectral extremal graph theory which generalize the classical Turán-type supersaturation results. (a) We prove that every $m$-edge graph $G$ with the spectral radius $λ(G) > \sqrt{m}$ contains at least $\frac{1}{144} \sqrt{m}$ triangles sharing a common edge. This result confirms a conjecture of Nikiforov, and Li and Peng. Moreover, the …

Beyond Nash-Williams: Counterexamples to Clique Decomposition Thresholds for All Cliques Larger than Triangles
Michelle Delcourt, Cicely Henderson, Thomas Lesgourgues, Luke Postle
https://arxiv.org/abs/2508.20819

Beyond Nash-Williams: Counterexamples to Clique Decomposition Thresholds for All Cliques Larger than Triangles
A central open question in extremal design theory is Nash-Williams' Conjecture from 1970 that every $K_3$-divisible graph on $n$ vertices (for $n$ large enough) with minimum degree at least $3n/4$ has a $K_3$-decomposition. A folklore generalization of Nash-Williams' Conjecture extends this to all $q\ge 4$ by positing that every $K_q$-divisible graph on $n$ vertices (for $n$ large enough) with minimum degree at least $\left(1-\frac{1}{q+1}\right)n$ has a $K_q$-decomposition. We disprove this co…

On the subdirect product of graph bundles
Yanga Bavuma, Francesco G. Russo, Stefano Spessato
https://arxiv.org/abs/2507.14530 https://

On the subdirect product of graph bundles
The subdirect product of two finite groups $A$ and $B$ is defined as a subgroup of the direct product $A \times B$, which is a well-known notion in finite group theory. While it is clear that, under appropriate choices of sets of generators $S$, $S_A$ and $S_B$, the Cayley graph $Cay(A \times B, S)$ corresponds to the Cartesian product $Cay(A, S_A) \square Cay(B, S_B)$ of two graphs, there is no analogue at the level of graph product that reflects the notion of subdirect product of groups. This…

Shortest Paths in a Weighted Simplicial Complex
Sukrit Chakraborty, Prasanta Choudhury, Arindam Mukherjee
https://arxiv.org/abs/2506.12921 https://

Shortest Paths in a Weighted Simplicial Complex
Simplicial complexes are extensively studied in the field of algebraic topology. They have gained attention in recent time due to their applications in fields like theoretical distributed computing and simplicial neural networks. Graphs are mono-dimensional simplicial complex. Graph theory has application in topics like theoretical computer science, operations research, bioinformatics and social sciences. This makes it natural to try to adapt graph-theoretic results for simplicial complexes, wh…

Asymptotic structure. IV. A counterexample to the weak coarse Menger conjecture
Tung Nguyen, Alex Scott, Paul Seymour
https://arxiv.org/abs/2508.14332 https://

Asymptotic structure. IV. A counterexample to the weak coarse Menger conjecture
Coarse graph theory concerns finding 'coarse' analogues of graph theory theorems, replacing disjointness with being far apart. One of the most interesting open questions is to find a coarse analogue of Menger's theorem, which characterizes when there are $k$ vertex-disjoint paths between two given sets $S,T$ of vertices of a graph. We showed in an earlier paper that the most natural such analogue is false, but a weaker statement remained as a popular open question. Here we show that the weaker …

The exact Tur\'an number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs
Yuantian Yu, Shuchao Li
https://arxiv.org/abs/2508.07533 https://

The exact Turán number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs
Given a graph $H,$ we say that a graph is \textit{$H$-free} if it does not contain $H$ as a subgraph. The Turán number $\ex(n,H)$ of $H$ is the maximum number of edges in an $n$-vertex $H$-free graph, the set of all the corresponding extremal graphs is denoted by $\Ex(n, H)$. The study of Turán number of graphs is a central topic in extremal graph theory. A graph is \textit{color-critical} if it contains an edge whose deletion reduces its chromatic number. Simonovits showed that if $H$ is a c…

Counting homomorphisms in antiferromagnetic graphs via Lorentzian polynomials
Joonkyung Lee, Jaeseong Oh, Jaehyeon Seo
https://arxiv.org/abs/2506.13659 htt…

Counting homomorphisms in antiferromagnetic graphs via Lorentzian polynomials
An edge-weighted graph $G$, possibly with loops, is said to be antiferromagnetic if it has nonnegative weights and at most one positive eigenvalue, counting multiplicities. The number of graph homomorphisms from a graph $H$ to an antiferromagnetic graph $G$ generalises various important parameters in graph theory, including the number of independent sets and proper vertex-colourings, as well as their relaxations in statistical physics. We obtain homomorphism inequalities for various graphs $H…

Algorithmic methods of finite discrete structures. Topological graph drawing (part IV)
Sergey Kurapov, Maxim Davidovsky
https://arxiv.org/abs/2507.16759 ht…

Algorithmic methods of finite discrete structures. Topological graph drawing (part IV)
The chapter presents mathematical models intended for creating a topological drawing of a non-separable non-planar graph based on the methods of G. Ringel's vertex rotation theory. The induced system of cycles generates a topological drawing of a certain thickness. A method for determining the location of imaginary vertices by finding the intersection of connections on a plane is presented. A topological drawing of a maximum planar subgraph is used as a basis.

Inverse eigenvalue problem for discrete Schr\"odinger operators of a graph
Anzila Laikhuram, Jephian C. -H. Lin
https://arxiv.org/abs/2506.15430 https…

Inverse eigenvalue problem for discrete Schrödinger operators of a graph
A discrete Schrödinger operator of a graph $G$ is a real symmetric matrix whose $i,j$-entry, $i \neq j$, is negative if $\{i,j\}$ is an edge and zero if it is not an edge, while diagonal entries can be any real numbers. The discrete Schrödinger operators have been used to study vibration theory and the Colin de Verdière parameter. The inverse eigenvalue problem for discrete Schrödinger operators of a graph aims to characterize the possible spectra among discrete Schrödinger operators of a …

The wild number of an edge-colored graph
Katie Anders, Briana Foster-Greenwood, Rebecca Garcia, Naomi Krawzik
https://arxiv.org/abs/2508.06711 https://arxi…

The wild number of an edge-colored graph
We introduce the wild number of an edge-colored graph as a measure of how close an edge-colored graph is to having a spanning tree in every color. This combinatorial concept originates in the algebraic theory of generalized graph splines. After showing that determining the wild number of a graph is an NP-complete problem, we provide bounds on the wild number and find the exact wild number for trees, cycles, and families of graphs with restrictions on the edge-colorings. This article serves as a…

On g-Extra Connectivity of Corona-Type Graph Products
Arati Sharma, Satyam Guragain, Ravi Srivastava
https://arxiv.org/abs/2508.08853 https://arxiv.org/pdf…

On g-Extra Connectivity of Corona-Type Graph Products
Connectivity is one of the central ideas in graph theory, especially when it comes to building fault-tolerant networks. A cutset $S$ of $G$ is defined to be the set of vertices in $G$ whose removal disconnects the graph. An $R_g$ cutset of $G$ is a cutset whose removal disconnects the graph in such a way that each connected component has at least $g+1$ vertices. If $G$ has at least one $R_g$ cutset then the $g-$extra vertex connectivity (or the $g-$extra edge connectivity), denoted as $κ_g(G)$…

Algorithmic methods of finite discrete structures. Topological graph drawing (part III)
Sergey Kurapov, Maxim Davidovsky
https://arxiv.org/abs/2506.10936 h…

Algorithmic methods of finite discrete structures. Topological graph drawing (part III)
The manuscript considers mathematical models for creating a topological drawing of a graph based on the methods of G. Ringel's vertex rotation theory. An algorithm is presented for generating a topological drawing of a flat part of a graph based on the selection of a basis for the cycle subspace C(G) using the Monte Carlo method. A steepest descent method for constructing a topological drawing of a flat subgraph is described in the manuscript. The topological drawing of a graph is constructed u…

Tur\'an's theorem for Dowling geometries
Rutger Campbell, Donggyu Kim, Jorn van der Pol
https://arxiv.org/abs/2508.20843 https://arxiv.org/pdf/2508…

Turán's theorem for Dowling geometries
The Dowling geometry $Q_n(Γ)$, where $Γ$ is a finite group, is a matroid that generalizes the complete-graphic matroid $M(K_{n+1})$. We determine the maximum size of an $N$-free submatroid of $Q_n(Γ)$ for various choices of $N$, including subgeometries $Q_m(Γ')$, lines $U_{2,\ell}$, and graphic matroids $M(H)$. When the group $Γ$ is trivial and $N=M(K_t)$, this problem reduces to Turán's classical result in extremal graph theory. We show that when $Γ$ is nontrivial, a complex dependence …

Critical edge sets in vertex-critical graphs
Ema Skottova, Raphael Steiner
https://arxiv.org/abs/2508.08703 https://arxiv.org/pdf/2508.08703

Critical edge sets in vertex-critical graphs
Criticality is a fundamental notion in graph theory that has been studied continually since its introduction in the early 50s by Dirac. A graph is called $k$-vertex-critical ($k$-edge-critical) if it is $k$-chromatic but removing any vertex (edge) lowers the chromatic number to $k-1$. A set of edges in a graph is called critical if its removal reduces the chromatic number of the graph. In 1970, Dirac conjectured a rather strong distinction between the notions of vertex- and edge-criticality, …

Asymptotic normality of embedding distributions of some families of graphs
Yichao Chen, Wenjie Fang, Zhicheng Gao, Jinlian Zhang
https://arxiv.org/abs/2507.15751

Asymptotic normality of embedding distributions of some families of graphs
Computing the embedding distribution of a given graph is a fundamental question in topological graph theory. In this article, we extend our viewpoint to a sequence of graphs and consider their asymptotic embedding distributions, which are often the normal distribution. We establish the asymptotic normality of several families of graphs by developing adapted tools and frameworks. We expect that these tools and frameworks can be used on other families of graphs to establish the asymptotic normali…

The Hamilton cycle space of random graphs
Dan Hefetz, Michael Krivelevich
https://arxiv.org/abs/2506.19731 https://arxiv.org/pdf/2506…

The Hamilton cycle space of random graphs
The cycle space of a graph $G$, denoted $C(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $C_n(G)$ denotes the subspace of $C(G)$, spanned by the incidence vectors of Hamilton cycles of $G$. A classical result in the theory of random graphs asserts that for $G \sim \mathbb{G}(n,p)$, asymptotically almost surely the necessary condition $δ(G) \geq 2$ is also sufficient to ensure Hamiltonicity. Resolving a…

The spectral Tur\'{a}n problem: Characterizing spectral-consistent graphs
Longfei Fang, Huiqiu Lin, Mingqing Zhai
https://arxiv.org/abs/2508.12070 https://

The spectral Turán problem: Characterizing spectral-consistent graphs
Let ${\rm EX}(n,H)$ and ${\rm SPEX}(n,H)$ denote the families of $n$-vertex $H$-free graphs with the maximum size and the maximum spectral radius, respectively. A graph $H$ is said to be spectral-consistent if ${\rm SPEX}(n,H)\subseteq {\rm EX}(n,H)$ for sufficiently large $n$. A fundamental problem in spectral extremal graph theory is to determine which graphs are spectral-consistent. Cioabă, Desai and Tait [European J. Combin. 99 (2022) 103420] proposed the following conjecture: Let $H$ be…

Spectral Tur\'{a}n problem of non-bipartite graphs: Forbidden books
Ruifang Liu, Lu Miao
https://arxiv.org/abs/2506.04884 https://

Spectral Turán problem of non-bipartite graphs: Forbidden books
A book graph $B_{r+1}$ is a set of $r+1$ triangles with a common edge, where $r\geq0$ is an integer. Zhai and Lin [J. Graph Theory 102 (2023) 502-520] proved that for $n\geq\frac{13}{2}r$, if $G$ is a $B_{r+1}$-free graph of order $n$, then $ρ(G)\leqρ(T_{n,2})$, with equality if and only if $G\cong T_{n,2}$. Note that the extremal graph $T_{n,2}$ is bipartite. Motivated by the above elegant result, we investigate the spectral Turán problem of non-bipartite $B_{r+1}$-free graphs of order $n$.…

On kernel isomorphisms of $m$-Cayley digraphs and finite $2$PCI-groups
Xing Zhang, Yan-Quan Feng, Jin-Xin Zhou, Fu-Gang Yin
https://arxiv.org/abs/2506.12306

On kernel isomorphisms of $m$-Cayley digraphs and finite $2$PCI-groups
The isomorphism problem for digraphs is a fundamental problem in graph theory. In this paper, we consider this problem for $m$-Cayley digraphs which are generalization of Cayley digraphs. Let $m$ be a positive integer. A digraph admitting a group $G$ of automorphisms acting semiregularly on the vertices with exactly $m$ orbits is called an $m$-Cayley digraph of $G$. In our previous paper, we developed a theory for $m$-Cayley isomorphisms of $m$-Cayley digraphs, and classified finite $m$CI-group…

Framing Triangulations and Framing Posets of Planar DAGs with Nontrivial Netflow Vectors
Jonah Berggren
https://arxiv.org/abs/2507.12684 https://

Framing Triangulations and Framing Posets of Planar DAGs with Nontrivial Netflow Vectors
The space of unit flows on a directed acyclic graph (DAG) with one source and one sink is known to admit regular unimodular triangulations induced by framings of the DAG. The dual graph of any of these triangulations may be given the structure of the Hasse diagram of a lattice, generalizing many variations of the Tamari lattice and the weak order. We extend this theory to flow polytopes of DAGs which may have multiple sources, multiple sinks, and nontrivial netflow vectors under certain planari…

Extending the Affirmative Action Problem: mixing numbers and integrated colorings of graphs
Charles Burnette, Broden Caton, Olivia Coward, Julian Davis, Austin Teter
https://arxiv.org/abs/2506.07192

Extending the Affirmative Action Problem: mixing numbers and integrated colorings of graphs
Consider a graph whose vertices are colored in one of two colors, say black or white. A white vertex is called integrated if it has at least as many black neighbors as white neighbors, and similarly for a black vertex. The coloring as a whole is integrated if every vertex is integrated. A classic exercise in graph theory, known as the Affirmative Action Problem, is to prove that every finite simple graph admits an integrated coloring. The solution can be neatly summarized with the one-liner: "m…

Integral Cayley graphs over a nonabelian group of order $8n$
Bei Ye, Xiaogang Liu
https://arxiv.org/abs/2508.10653 https://arxiv.org/pdf/2508.10653

Integral Cayley graphs over a nonabelian group of order $8n$
A graph is called an integral graph when all eigenvalues of its adjacency matrix are integers. We study which Cayley graphs over a nonabelian group $$ T_{8n}=\left\langle a,b\mid a^{2n}=b^8=e,a^n=b^4,b^{-1}ab=a^{-1} \right \rangle $$ are integral graphs. Based on the group representation theory, we first give the irreducible matrix representations and characters of $T_{8n}$. Then we give necessary and sufficient conditions for which Cayley graphs over $T_{8n}$ are integral graphs. As applicatio…

Walk Matrix-Based Upper Bounds on Generalized Cospectral Mates
Muhammad Raza, Mudassir Shabbir, Waseem Abbas
https://arxiv.org/abs/2507.06927 https://

Walk Matrix-Based Upper Bounds on Generalized Cospectral Mates
The problem of characterizing graphs determined by their spectrum (DS) or generalized spectrum (DGS) has been a longstanding topic of interest in spectral graph theory, originating from questions in chemistry and mathematical physics. While previous studies primarily focus on identifying whether a graph is DGS, we address a related yet distinct question: how many non-isomorphic generalized cospectral mates a graph can have? Building upon recent advances that connect this question to the propert…

Remarks on the Brouwer Conjecture
Oliver Knill
https://arxiv.org/abs/2508.07550 https://arxiv.org/pdf/2508.07550 …

Remarks on the Brouwer Conjecture
The Brouwer conjecture (BC) in spectral graph theory claims that the sum of the largest k Kirchhoff eigenvalues of a graph are bounded above by the number m of edges plus k(k+1)/2. We show that (BC) holds for all graphs with n vertices if n is larger or equal than 4 times the square of the maximal vertex degree. We also note that the weaker upper bound m+k(k+1) holds unconditionally. We also note that (BC) for graphs implies (BC) for quivers.

The Exponential Hyper-Zagreb Indices and Structural Properties of Trees and Bipartite Graphs
Jasem Hamoud
https://arxiv.org/abs/2508.14238 https://arxiv.or…

The Exponential Hyper-Zagreb Indices and Structural Properties of Trees and Bipartite Graphs
In this paper, we investigate the structural properties of trees and bipartite graphs through the lens of topological indices and combinatorial graph theory. We focus on the First and Second Hyper-Zagreb indices, $HM_1(G)$ and $HM_2(G)$, for trees $T \in T(n, Δ)$ with $n$ vertices and maximum degree $Δ$. Key propositions demonstrate that the presence of end-support or support vertices of degree at least three, distinct from a vertex of maximum degree, implies the existence of another tree $T'…

Flows on Gentle Algebras
Jonah Berggren
https://arxiv.org/abs/2507.12688 https://arxiv.org/pdf/2507.12688

Flows on Gentle Algebras
The space of unit flows on a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. Amply framed DAGs and their triangulated flow polytopes have recently been connected with the representation theory of certain gentle algebras. We expand on this connection by defining a flow on the fringed quiver of an arbitrary gentle algebra. We call the space of unit flows its turbulence polyhedron. We show that support tau-tilting modules of a gentle…

Exact Tur\'{a}n number of the Fano plane in the $\ell_2$-norm
Jianfeng Hou, Xizhi Liu, Yixiao Zhang
https://arxiv.org/abs/2507.12354 https://

Exact Turán number of the Fano plane in the $\ell_2$-norm
A classical object in hypergraph Turán theory is the Fano plane $\mathbb{F}$, the unique linear $3$-graph on seven vertices with seven edges. The Turán density and exact Turán number of $\mathbb{F}$, first proposed as a problem by Sós~\cite{Sos76} in the 1970s, were determined through a sequence of works by De Caen--Füredi \cite{DCF00}, Füredi-Simonovits \cite{FS05}, Keevash-Sudakov \cite{KS05}, and Bellmann-Reiher \cite{BR19}. Addressing a conjecture of Balogh--Clemen--Lidický \cite[C…