Characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph

Abstract

This research discussed the characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph. The entries of the antiadjacency matrix of a directed graph represent the presence or the absence of a directed arc from one vertex to the others. If A is the adjacency matrix of a graph , then the antiadjacency matrix B of graph is B = J − A, where J is the square matrix with all entries equal to one. The general form of characteristic polynomial coefficients of the antiadjacency matrix of directed unicyclic flower vase graph can be obtained by calculating the sum of determinants of the antiadjacency matrices of all induced cyclic and acyclic subgraphs, while the eigenvalues were obtained by using polynomial factorization and Horner’s method. This paper gives the characteristic polynomial coefficients and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph. The characteristic polynomial can be considered as a function that depends on the number of vertices.