Related papers: On the Maximum Spread of Non-Negative Matrices

The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance between $v$ and any other vertex of $G$. The diameter of a graph $G$ is the maximum eccentricity of a vertex in $G$. The eccentric connectivity index of a connected…

We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erd{\H o}s-R{\'e}nyi graph $G(N,p)$. Recently, it was shown by Lee, up to an…

Let $G$ be a simple connected graph with vertex set $V(G)=\{v_{1}, v_{2}, \ldots, v_{n}\}$. The distance $d_G(v_i,v_j)$ between two vertices $v_i$ and $v_j$ of $G$ is the length of a shortest path between $v_i$ and $v_j$. The distance…

In [Curtin and Pourgholi, A group sum inequality and its application to power graphs, J. Algebraic Combinatorics, 2014], it is proved that among all directed power graphs of groups of a given order $ n $, the directed power graph of cyclic…

An {\em orientation} of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. Converting two-way traffic networks to one-way traffic networks and bidirectional communication networks to unidirectional…

Given a graph $G,$ a subset of vertices is called a maximum dissociation set of $G$ if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. The cardinality of a maximum dissociation set is called the…

For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $\lambda$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|$. Contributing to a…

The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix. Woo and Neumaier discovered that a connected graph $G$ with $\rho(G)\leq 3/2{\sqrt{2}}$ is either a dagger, an open quipu, or a closed quipu.…

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…

Let $G$ be a simple connected graph, and $D(G)$ be the distance matrix of $G$. Suppose that $D_{\max}(G)$ and $\lambda_1(G)$ are the maximum row sum and the spectral radius of $D(G)$, respectively. In this paper, we give a lower bound for…

In this paper, we introduce a new model to study network reliability with node failures. This model, strongly connected node reliability, is the directed variant of node reliability and measures the probability that the operational vertices…

For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_{\alpha}$-matrix of $G$ is defined as \begin{equation*} A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G), \text{ for any } \alpha \in [0,1].…

We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of…

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that \Delta-\lambda_1>\frac{1}{nD} where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of…

A nonsplit graph is a directed graph where each pair of nodes has a common incoming neighbor. We show that the radius of such graphs is in $O(\log \log n)$, where $n$ is the number of nodes. We then generalize the result to products of…

The spread of a connected graph G was introduced by Alon, Boppana and Spencer (1998) and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V(G) of the variance of f(X) when X is…

We show that for a graph $G$ with the vertex set $V$ and the largest eigenvalue $\lambda_{\max}(G)$, letting $$ M(G) := \max_{X,Y \subset V} \frac{e(X,Y)}{\sqrt{|X||Y|}} $$ (where $e(X,Y)$ denotes the number of edges between $X$ and $Y$),…

Let D be the distance matrix of a connected graph G and let nn(G), np(G) be the number of strictly negative and positive eigenvalues of D respectively. It was remarked in [1] that it is not known whether there is a graph for which np(G) >…

Let A be a square matrix with real entries. The spread of A is defined as the maximum of the distances among the eigenvalues of A. Let $S_m[a,b]$ denote the set of all $m\times m$ symmetric matrices with entries in the real interval $[a,b]$…

For a fixed positive integer $k$ and a graph $G$, let $\lambda_k(G)$ denote the $k$-th largest eigenvalue of the adjacency matrix of $G$. In 2017, Tait and Tobin proved that the maximum $\lambda_1(G)$ among all outerplanar graphs on $n$…