The adjacency matrix of directed cyclic wheel graph

Abstract

An adjacency matrix is one of the matrix representations of a directed graph. In this paper, the adjacency matrix of a directed cyclic wheel graph is denoted by . From the matrix the general form of the characteristic polynomial and the eigenvalues of a directed cyclic wheel graph can be obtained. The norm of each coefficient of its characteristic polynomial is obtained by calculating the sum of the principal minors of subgraphs of the graph . It turns out that each coefficient ai, i = 1, 2, …, (n − 2), n equals zero except an−1 is equal to -1. In addition the matrix has real eigenvalues and also some complex eigenvalues that conjugate each other. The real eigenvalues are obtained by searching the real roots through the characteristic equation, then by factorization we get the polynomial factor that contains the complex roots. The complex eigenvalues of the adjacency matrix of a directed cyclic wheel graph or has a relation with the complex eigenvalues of the antiadjacency matrix of a directed cyclic wheel graph or , that is the complex eigenvalues of the matrix are equal to the negative of the complex eigenvalues of the matrix .