Let G be a simple graph and let its vertex set be V (G) = {v1, v2, ..., vn}. The adjacency matrix A(G) of the graph G is a square matrix of order n whose (i, j)-entry is equal to unity if the vertices vi and vj are adjacent, and is equal to zero otherwise. The eigenvalues λ1, λ2, ..., λn of A(G), assumed in non increasing order, are the eigenvalues of the graph G. The energy of G was first defi...

We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighte...

Question 1 First, we have that λ − λ − 2 = (λ + 1)(λ − 2). Therefore, it is true that the eigenvalues of the corresponding adjacency matrix come in pairs of additive inverse. However, notice that ± √ −1 are eigenvalues, and therefore the associated matrix cannot be symmetric (there would only be real eigenvalues). We conclude that no undirected graph can have the above characteristic polynomial...

We will discuss a few basic facts about the distribution of eigenvalues of the adjacency matrix, and some applications. Then we discuss the question of computing the eigenvalues of a symmetric matrix. 1 Eigenvalue distribution Let us consider a d-regular graph G on n vertices. Its adjacency matrix AG is an n× n symmetric matrix, with all of its eigenvalues lying in [−d, d]. How are the eigenval...

In this paper we present an auto-detection corner based on eigenvalues product of covariance matrices (ADEPCM) of boundary points over multi-region of support. The algorithm starts with extracting the contour of an object, and then computes the eigenvalues product of covariance matrices of this contour at various regions of support. Finally determine automatically peaks of the graph of eigenval...