Tootfinder

Classification of polyhedral graphs by numbers of common neighbours
Riccardo W. Maffucci
https://arxiv.org/abs/2508.01349 https://arxiv.org/pdf/2508.01349

Classification of polyhedral graphs by numbers of common neighbours
We propose a classification of polyhedra (planar, $3$-connected graphs) according to their type i.e., their set of quantities of common neighbours for each pair of distinct vertices. For every (finite) set of non-negative integers, we either classify all the polyhedra of that type, or construct infinitely many polyhedra of that type, or prove that none exist. This problem is related to the theory of strongly regular and Deza graphs, distances in graphs, and degree sequences. There is potent…

Applications of the factorization theorem of conformal blocks to vertex operator algebras
Xu Gao, Jianqi Liu
https://arxiv.org/abs/2508.01294 https://arxiv…

Applications of the factorization theorem of conformal blocks to vertex operator algebras
We apply the factorization and vector bundle propositionerty of the sheaves of conformal blocks on $\overline{\mathscr{M}}_{g,n}$. defined by vertex operator algebras (VOAs) and give geometric proofs of essential results in the representation theory of strongly rational VOAs. We first prove the associativity of the fusion tensor product by investigating four-point conformal blocks on genus zero curves. Then we give a formula to calculate the ranks of VOA-conformal block bundles. Finally, by inv…

Flux tube formation and the Weingarten representation of center vortices and chains
David R. Junior, Luis E. Oxman
https://arxiv.org/abs/2508.02341 https://

Flux tube formation and the Weingarten representation of center vortices and chains
We review some recent results regarding the formulation of mixed ensembles of oriented and nonoriented center vortices based on the Weingarten representation for the sum over surfaces. This framework enabled a partition function for the Abelian-projected ensemble of vortex worldsurfaces, previously formulated as a wavefunctional peaked at center-vortex loops. In particular, we showed how an Abelian ensemble can account for $N$-ality while supporting a ``dual superconductor'' model for confineme…

GPT-NL, een mooi initiatief dat voortgang maakt, let ook op het doel "Tot slot is het goed om in het achterhoofd te houden dat GPT-NL wordt ontwikkeld voor specifieke taken: samenvatten, versimpelen, en het extraheren van informatie. Het doel van GPT-NL is niet om een generiek kennismodel te ontwikkelen."
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Monitoring tijdens de training: de verwachte oogst van GPT-NL
GPT-NL is een ambitieus project: wij geloven dat technologie betrouwbaar en transparant moet zijn, een wederkerige bijdrage moet leveren, en soevereiniteit van Nederland moet versterken. Vanuit deze kernwaarden willen wij het AI innovatielandschap versterken. Met deze updates houden we je op de hoogte van de voortgang van de training.

Recognizing flag varieties and reductive groups
I. Grojnowski, N. I. Shepherd-Barron
https://arxiv.org/abs/2509.03504 https://arxiv.org/pdf/2509.03504

Recognizing flag varieties and reductive groups
Fix a flat and projective morphism $X\rightarrowΣ$ of schemes. We show, first, that any set of $\mathbb{P}^1$-fibrations on $X$ defines a set of simple roots, a set of simple coroots and a Cartan matrix $C$. Second, $X$ is an étale $F$-bundle over some projective $Σ$-scheme, where $F$ is the flag variety of the adjoint Chevalley group over the integers defined by $C$. In particular, if the simple roots generate the Néron--Severi group of $X$ relative to $Σ$ and $X$ is cohomologically flat …

The generalized Tur\'an number for K_3 in graphs without suspensions of a path on five vertices
Doudou Hei, Xinmin Hou, Yue Ma
https://arxiv.org/abs/2509.03851 https://

The generalized Turán number for K_3 in graphs without suspensions of a path on five vertices
Given graphs $H$ and $F$, the generalized Turán number $\ex(n, H, F)$ is defined as the maximum number of copies of $H$ in an $n$-vertex graph that contains no copy of $F$. The suspension $\widehat{F}$ of a graph $F$ is obtained by adding a new vertex that is adjacent to every vertex of $F$. Mubayi and Mukherjee (2023, DM) conjectured that $\ex(n, K_3, \widehat{P_k})=\left\lfloor \frac{k-2}{2}\right\rfloor \cdot \frac{n^2}{8}+o(n^2)$, where $P_k$ is a path on $k\ge 4$ vertices. Using…

Reconfiguration graph of vertex colorings for some generalized split graphs
Manoj Belavadi, T. Karthick
https://arxiv.org/abs/2509.03190 https://arxiv.org/…

Reconfiguration graph of vertex colorings for some generalized split graphs
Given a graph $G$, the \emph{reconfiguration graph of the $\ell$-colorings} of $G$, denoted by ${\cal R}_\ell(G)$, is the graph whose vertices are the $\ell$-colorings of $G$ and two $\ell$-colorings are joined by an edge if they differ on exactly one vertex of $G$. A graph $G$ is \emph{$\ell$-mixing} if ${\cal R}_\ell(G)$ is connected and the \emph{$\ell$-recoloring diameter} of $G$ is the diameter of ${\cal R}_\ell(G)$. An interesting classification problem that recently gathered a wide atten…

On the difference of the intersection power graph and the power graph of a finite group
Sudip Bera, Peter J. Cameron
https://arxiv.org/abs/2509.03919 https://

On the difference of the intersection power graph and the power graph of a finite group
The two graphs of the title both have vertex set G. In the intersection power graph, x and y are joined if some non-identity element is a power of both; in the power graph, x and y joined if one is a power of the other. Thus the power graph is a spanning subgraph of the intersection power graph, and we define the edges of the difference graph to be the difference of these edge sets. In this paper, we give a number of results about the difference graph. We examine groups whose power graph and in…

Coalition graphs of connected domination partitions in subcubic graphs
Andrey A. Dobrynin, Aleksey N. Glebov
https://arxiv.org/abs/2509.04204 https://arxiv…

Coalition graphs of connected domination partitions in subcubic graphs
A graph is subcubic if it is connected and its maximum vertex degree does not exceed 3. Two disjoint vertex subsets of a graph $G$ form a connected coalition in $G$ if neither of them is a connected dominating set but their union is a connected dominating set. A connected coalition partition of $G$ is a partition of its vertices $π(G) = \{V_1, V_2,..., V_k \}$, such that each $V_i$ is either a connected dominating set consisting of a single vertex or forms a coalition with some set of $π(G)$.…

On the $Z_q$-forcing number: computational approach and exact values
Aida Abiad, Maryam Moghaddas
https://arxiv.org/abs/2509.03967 https://arxiv.org/pdf/25…

On the $Z_q$-forcing number: computational approach and exact values
Zero forcing is a graph coloring process that is used to model spreading phenomena in real-world scenarios. It can also be viewed as a single-player combinatorial game on a graph, where the player's goal is to select a subset of vertices of minimum cardinality that eventually leads to all vertices of the graph being colored. A variant of this game, called the $q$-analogue of zero forcing, was later introduced. In this version, the player again seeks to choose the smallest number of vertices tha…