With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properies (the eigenvalues and eigenvectors) of these matrices pro...

The anti-adjacency matrix of a graph is constructed from the distance by keeping each row and column only largest distances. This can be interpreted as opposite adjacency matrix, which instead in distances equal to 1. (anti-)adjacency eigenvalues are those its matrix. Employing novel technique introduced Haemers (2019) [9], we characterize all connected graphs with exactly one positive eigenval...

A graph G is said to be nonsingular (resp., singular) if its adjacency matrix A(G) is nonsingular (resp., singular). The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of ±1s. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contract...

Detection of fake accounts on social networks is a challenging process. The previous methods in identification of fake accounts have not considered the strength of the users’ communications, hence reducing their efficiency. In this work, we are going to present a detection method based on the users’ similarities considering the network communications of the users. In the first step, similarity ...

The characteristic polynomial of the adjacency matrix of a graph is noted in connection with a quantity characterizing the topological nature of structural isomers saturated hydrocarbons [S], a set of numbers that are the same for all graphs isomorphic to the graph, and others [l]. Many properties of the characteristic polynomials of the adjacency matrices of a graph and its line graph [3] have...

Our goal is to use the properties of the adjacency/Laplacian matrix of graphs to first understand the structure of the graph and, based on these insights, to design efficient algorithms. The study of algebraic properties of graphs is called algebraic graph theory. One of the most useful algebraic properties of graphs are the eigenvalues (and eigenvectors) of the adjacency/Laplacian matrix.

We consider simple labeled graphs, with non-zero labels in a ring. If the adjacency matrix of a labeled graph is invertible, the inverse matrix is a (labeled) adjacency matrix of another graph, called the inverse of the original graph. If the labeling takes place in an ordered ring, then balanced inverses—those with positive products of labels along every cycle—are of interest. We introduce the...

Received: date / Accepted: date Abstract We prove a central limit theorem for the components of the largest eigenvectors of the adjacency matrix of a finite-dimensional random dot product graph whose true latent positions are unknown. We use the spectral embedding of the adjacency matrix to construct consistent estimates for the latent positions, and we show that the appropriately scaled differ...

Graph algorithms that test adjacencies are usually implemented with an adjacency-matrix representation because the adjacency takes constant time matrices, but it linear in degree of vertices lists. In this article, we review adjacency-map representation, which supports tests expected time, and show graph run faster maps than lists by a small factor if they do not one or two orders magnitude per...

‎let $g$ be a finite group‎. ‎an element $gin g$ is called non-vanishing‎, ‎if for‎ ‎every irreducible complex character $chi$ of $g$‎, ‎$chi(g)neq 0$‎. ‎the bi-cayley graph $bcay(g,t)$ of $g$ with respect to a subset $tsubseteq g$‎, ‎is an undirected graph with‎ ‎vertex set $gtimes{1,2}$ and edge set ${{(x,1),(tx,2)}mid xin g‎, ‎ tin t}$‎. ‎let $nv(g)$ be the set‎ ‎of all non-vanishing element...