Characteristic polynomial and eigenvalues of the anti-adjacency matrix of cyclic directed prism graph

Abstract

A prism graph is a graph which corresponds to the skeleton of an n-prism and therefore it is a cyclic simple graph. It is denoted Yn (n ≥ 3) where n is half the number of vertices. An n-prism graph has 2n vertices and 3n edges. In this paper, only regularly-directed cyclic prism graphs are investigated. The anti-adjacency matrix is applied as the graph representation. An anti-adjacency matrix of graph representation is a 0–1 matrix of size m × m where m is the number of vertices. The entry bij of an anti-adjacency matrix B(G) of directed graph G is 0 if there exists a directed edge from vertex vi to vertex vj and is 1 otherwise. The characteristic polynomial of the anti-adjacency matrix B(Yn) of directed cyclic prism graph Yn are obtained. The characteristic polynomial will be proved by observing the both cyclic and acyclic induced subgraphs of the directed cyclic prims graph. Furthermore, the anti-adjacency matrix of directed cyclic prism graph is found to have both real eigenvalues and complex eigenvalues which appear in conjugate pairs.