Characteristic Polynomial of Antiadjacency Matrix of Directed Cyclic Wheel Graph

Abstract

A directed cyclic wheel graph with order n, where n ≥ 4 can be represented by an anti-adjacency matrix. The anti-adjacency matrix is a square matrix that has entries only 0 and 1. The number 0 denotes an edge that connects two vertices, whereas the number 1 denotes otherwise. The norm of every coefficient in characteristic polynomial of the anti-adjacency matrix of a directed cyclic wheel graph represents the number of Hamiltonian paths contained in the induced sub-graphs minus the number of the cyclic induced sub-graphs. In addition, the eigenvalues can be found through the anti-adjacency matrix of directed cyclic wheel graph. The result is, the anti-adjacency matrix of directed cyclic wheel graph has two real eigenvalues and some complex eigenvalues that conjugate to each other. The real eigenvalues are obtained by Horner method, while the complex eigenvalues are obtained by finding the complex roots from the factorization of the characteristic polynomial.