Related papers: Repr\'esentations lin\'eaires des graphes finis

For a metric space $X$ with a compatible measure $\mu$, Genevois and Tessera defined the Scaling Group of $(X,\mu)$ as the subgroup $\Gamma$ of $\mathbb{R}_{>0}$ of positive real numbers $\gamma$ for which there are quasi-isometries of $X$…

The $G$-graph $\Gamma(G,S)$ is a graph from the group $G$ generated by $S\subseteq G$, where the vertices are the right cosets of the cyclic subgroups $\langle s \rangle, s\in S$ with $k$-edges between two distinct cosets if there is an…

Given a group $\Gamma$, we establish a connection between the unitarisability of its uniformly bounded representations and the asymptotic behaviour of the isoperimetric constants of Cayley graphs of $\Gamma$ for increasingly large…

The solubility graph $\Gamma_S(G)$ associated with a finite group $G$ is a simple graph whose vertices are the elements of $G$, and there is an edge between two distinct vertices if and only if they generate a soluble subgroup. In this…

Let $\Gamma=(X,\mathcal{R})$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set $X$ and edge set $\mathcal{R}$. Fix a vertex $x \in X$, and define $\mathcal{R}_f = \mathcal{R} \setminus \{yz \mid…

Let $\Gamma$ be a quiver on n vertices $v_1, v_2, ..., v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\alpha \in \N^n$. Hua gave a formula for $A_{\Gamma}(\alpha, q)$, the number of isomorphism classes of absolutely…

If $\Gamma$ is a graph for which every edge is in exactly one clique of order $\omega$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $\Gamma$. We discover many general…

The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G) + mu(G) equals its order, then G is a Koenig-Egervary graph. We call G…

We investigate the relationship between finite groups and incidence geometries through their automorphism structures. Building upon classical results on the realizability of groups as automorphism groups of graphs, we develop a general…

We prove that for every countable string graph $S$, there is a planar graph $G$ with $V(G)=V(S)$ such that \[ \frac{1}{23660800}d_S(u,v) \le d_G(u,v) \le 162 d_S(u,v) \] for all $u,v\in V(S)$, where $d_S(u,v)$, $d_G(u,v)$ denotes the…

We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs $(\Gamma,\Sigma)$ is stable if $Aut(\Gamma\times\Sigma) \cong…

Let $G$ be a finite insoluble group with soluble radical $ R(G)$. The solubility graph $\Gamma_{\rm S}(G)$ of $G$ is a simple graph whose vertices are the elements of $G\setminus R(G) $ and two distinct vertices $x$ and $y$ are adjacent if…

A regular equivalence between two graphs $\Gamma,\Gamma'$ is a pair of uniformly proper Lipschitz maps $V\Gamma\to V\Gamma'$ and $V\Gamma'\to V\Gamma$. Using separation profiles we prove that there are $2^{\aleph_0}$ regular equivalence…

Let $\Gamma$ be a finite graph and let $\Gamma^{\mathrm{e}}$ be its extension graph. We inductively define a sequence $\{\Gamma_i\}$ of finite induced subgraphs of $\Gamma^{\mathrm{e}}$ through successive applications of an operation called…

The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. A graph is well-covered if every maximal stable set has the same size. G is a Koenig-Egervary graph if its order equals…

We develop the geometric and homological framework for non-commutative $n$-ary $\Gamma$-semirings by constructing a sheaf and derived theory over their non-commutative $\Gamma$-spectrum. Starting with a non-commutative $n$-ary…

Let $G$ be a finite, non-abelian group of the form $G = A N$, where $A \leq G$ is abelian, and $N \trianglelefteq G$ is cyclic. We prove that the commuting graph $\Gamma(G)$ of $G$ is either a connected graph of diameter at most four, or…

Let $\Gamma=(X,\mathcal{R})$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set $X$ and edge set $\mathcal{R}$. Fix a vertex $x \in X$, and define $\mathcal{R}_f = \mathcal{R} \setminus \{yz \mid…

Let $G$ be a finite non-cyclic group. The non-cyclic graph $\Gamma_G$ of $G$ is the graph whose vertex set is $G\setminus Cyc(G)$, two distinct vertices being adjacent if they do not generate a cyclic subgroup, where $Cyc(G)=\{a\in G:…

Given a graph $\Gamma$, one may conside the set $X$ of its vertices as a metric space by assuming that all edges have length one. We consider two versions of homology theory of $\Gamma$ and their $K$-theory counterparts -- the $K$-theory of…