Tootfinder

This https://arxiv.org/abs/2211.06352 has been replaced.
link: https://scholar.google.com/scholar?q=a

Spectral Triadic Decompositions of Real-World Networks
A fundamental problem in mathematics and network analysis is to find conditions under which a graph can be partitioned into smaller pieces. The most important tool for this partitioning is the Fiedler vector or discrete Cheeger inequality. These results relate the graph spectrum (eigenvalues of the normalized adjacency matrix) to the ability to break a graph into two pieces, with few edge deletions. An entire subfield of mathematics, called spectral graph theory, has emerged from these results.…

This https://arxiv.org/abs/2304.04856 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Graph Convex Hull Bounds as generalized Jensen Inequalities
Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function $f\colon K \to \mathbb{R}$ defined on a convex domain $K \subseteq \mathbb{R}^{d}$ and any random variable $X$ taking values in $K$, $\mathbb{E}[f(X)] \geq f(\mathbb{E}[X])$. In this paper, sharp upper and lower bounds on $\mathbb{E}[f(X)]$, termed ``graph convex hull bounds'', are deri…

Extremal problems about the order and size of nonhamiltonian locally linear graphs
Feng Liu, Leilei Zhang
https://arxiv.org/abs/2402.16066 https://<…

Extremal problems about the order and size of nonhamiltonian locally linear graphs
The study of the relationship between local and global properties of mathematical objects has always been a key subject of investigation in different areas of mathematics. A graph $G$ is called locally linear if the neighbourhood of every vertex of $G$ induces a path. And $G$ is called locally hamiltonian (traceable) if the neighbourhood of every vertex of $G$ induces a hamiltonian (traceable) graph. The local properties of graphs are being studied extensively. For example, the minimum order of…

Reflexive graph lenses in univalent foundations
Jonathan Sterling
https://arxiv.org/abs/2404.07854 https://arxiv.org/pdf/2404.07854…

Reflexive graph lenses in univalent foundations
Martin-Löf's identity types provide a generic (albeit opaque) notion of identification or "equality" between any two elements of the same type, embodied in a canonical reflexive graph structure $(=_A, \mathbf{refl})$ on any type $A$. The miracle of Voevodsky's univalence principle is that it ensures, for essentially any naturally occurring structure in mathematics, that this the resultant notion of identification is equivalent to the type of isomorphisms in the category of such structures. Cha…

This https://arxiv.org/abs/2401.05348 has been replaced.
initial toot: https://mastoxiv.page/@arX…

Approach to Toric Code Anyon Excitation, Indirect Effects of Kitaev Spin in Local Social Opinion Models
The study of Opinion Dynamics, which explores how individual opinions and beliefs evolve and how societal consensus is formed, has been examined across social science, physics, and mathematics. Historically based on statistical physics models like the Ising model, recent research integrates quantum information theory concepts, such as Graph States, Stabilizer States, and Toric Codes. These quantum approaches offer fresh perspectives for analyzing complex relationships and interactions in opinio…

On the matching complexes of categorical product of path graphs
Raju Kumar Gupta, Sourav Sarkar, Sagar S. Sawant, Samir Shukla
https://arxiv.org/abs/2403.15298

On the matching complexes of categorical product of path graphs
The matching complex $\mathsf{M}(G)$ of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. These complexes appears in various places and found applications in many areas of mathematics including; discrete geometry, representation theory, combinatorics, etc. In this article, we consider the matching complexes of categorical product $P_n \times P_m$ of path graphs $P_n$ and $P_m$. For $m = 1$, $P_n \times P_m$ is a discrete graph and therefore its matching complex is the vo…

A 2-distance $(2\Delta 7)$-coloring of planar graphs
Zakir Deniz
https://arxiv.org/abs/2403.12302 https://arxiv.org/pdf/2403.12302

A 2-distance $(2Δ+7)$-coloring of planar graphs
A vertex coloring of a graph $G$ is called a $2$-distance coloring if any two vertices at a distance at most $2$ from each other receive different colors. Recently, Bousquet et al. (Discrete Mathematics, 346(4), 113288, 2023) proved that $2Δ+7$ colors are sufficient for the $2$-distance coloring of planar graphs with maximum degree $Δ\geq 9$. In this paper, we strengthen their result by removing the maximum degree constraint and show that all planar graphs admit a 2-distance $(2Δ+7)$-colorin…