Related papers: On the matrix sequence $\{\Gamma(A^m)\}_{m=1}^\inf…

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 the fundamental group of a closed orientable surface of genus at least two. Consider the composition of a uniformly random element of $\mathrm{Hom}(\Gamma,S_n)$ with the $(n-1)$-dimensional irreducible representation of…

Let $G$ be a strongly connected digraph with $n$ vertices and $m$ arcs. For any real $\alpha\in[0,1]$, the $A_\alpha$ matrix of a digraph $G$ is defined as $$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$$ where $A(G)$ is the adjacency matrix of…

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D\geq 1$. For a vertex $x$ of $\Gamma$ the corresponding subconstituent algebra $T=T(x)$ is generated by the adjacency matrix $A$ of $\Gamma$ and the dual adjacency…

Let $G$ be a graph on $n$ vertices and $m$ edges. For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of $G$ is defined as $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal…

Let $G$ be a simple undirected graph, $\theta(G)$ be the circuit rank of $G$, $\eta_M(G)$ and $m_M(G,\lambda)$ be the nullity and the multiplicity of eigenvalue $\lambda$ of a graph matrix $M(G)$, respectively. In the case $M(G)$ is the…

The eccentricity matrix $\epsilon(G)$, of a connected graph $G$ is obtained by retaining the maximum distance from each row and column of the distance matrix of $G$ and the other entries are assigned with 0. In this paper, we discuss the…

In this paper, we completely characterize the digraphs of order $n$ whose $m$-step competition graphs are star graphs for positive integers $2\leq m < n$. This result in matrix version identifies the solution set to the matrix equation…

For a finite group $G,$ we investigate the direct graph $\Gamma(G),$ whose vertices are the non-hypercentral elements of $G$ and where there is an edge $x\mapsto y$ if and only if $[x,_ny]=1$ for some $n \in \mathbb N.$ We prove that…

A $\Gamma$-labeled graph is an oriented graph with edges invertibly labeled by a group $\Gamma$. We prove a structure theorem for $\Gamma$-labeled graphs which forbid a fixed $\Gamma$-labeled graph as an immersion, for any finite $\Gamma$.…

We prove an homogenization result, in terms of $\Gamma$-convergence, for energies concentrated on rectifiable lines in $\R^3$ without boundary. The main application of our result is in the context of dislocation lines in dimension $3$. The…

In matrix theory, a well established relation $(AB)^{T}=B^{T}A^{T}$ holds for any two matrices $A$ and $B$ for which the product $AB$ is defined. Here $T$ denote the usual transposition. In this work, we explore the possibility of deriving…

Let X be a non-empty finite set and alpha a symmetric bilinear form on a real finite dimensional vector space E. We say that a set GG={U_i | i in X} of linear lines in E is an isometric sheaf, if there exist generators u_i of the lines U_i,…

This paper is part of a program to understand topologies on spaces of valuations. We fix an ordered abelian group $\Gamma$ and an integral domain $R$. We study the relation between a topology on $\Gamma_\infty$ and the induced topology on…

Let $\Gamma$ be a simple graph of size $m$ and degree sequence $\delta_1\ge \delta_2\ge ... \ge \delta_n$. Let ${\cal L}(\Gamma)$ denotes the line graph of $\Gamma$. The aim of this paper is to study mathematical properties of the alliance…

Let $\Gamma$ be a connected regular graph with an eigenvalue $\lambda$ and corresponding idempotent $E_{\lambda}$. Let ${\cal E}_{\lambda}=\langle J,E_{\lambda}\rangle^\circ$ be the algebra generated by $J$ and $E_\lambda$ with respect to…

We study the $\Gamma$-convergence of a family of non-local, non-convex functionals in $L^p(I)$ for $p \ge 1$, where $I$ is an open interval. We show that the limit is a multiple of the $W^{1, p}(I)$ semi-norm to the power $p$ when $p>1$…

For a graph $G$ and for two distinct vertices $u$ and $v$, let $\kappa(u,v)$ be the maximum number of vertex-disjoint paths joining $u$ and $v$ in $G$. The average connectivity matrix of an $n$-vertex connected graph $G$, written…

Call the sum of the singular values of a matrix A the energy of A. We investigate graphs and matrices of energy close to the maximal one. We prove a conjecture of Koolen and Moulten and give a stability theorem characterizing all square…

We investigate the homogenization through Gamma-convergence for the L^2(\Omega)-weak topology of the conductivity functional with a zero-order term where the matrix-valued conductivity is assumed to be non strongly elliptic. Under proper…