On conjugate adjacency matrices of a graph

Abstract

Let G be a simple graph of order n. Let c = a + b m and c ¯ = a - b m , where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A c = [ c ij ] is the conjugate adjacency matrix of the graph G if c ij = c for any two adjacent vertices i and j, c ij = c ¯ for any two nonadjacent vertices i and j, and c ij = 0 if i = j . Let P G ( λ ) = | λ I - A | and P G c ( λ ) = | λ I - A c | denote the characteristic polynomial and the conjugate characteristic polynomial of G, respectively. In this work we show that if P G c ( λ ) = P H c ( λ ) then P G ¯ c ( λ ) = P H ¯ c ( λ ) , where G ¯ denotes the complement of G. In particular, we prove that P G c ( λ ) = P H c ( λ ) if and only if P G ( λ ) = P H ( λ ) and P G ¯ ( λ ) = P H ¯ ( λ ) . Further, let P c ( G ) be the collection of conjugate characteristic polynomials P G i c ( λ ) of vertex-deleted subgraphs G i = G ⧹ i ( i = 1 , 2 , … , n ) . If P c ( G ) = P c ( H ) we prove that P G c ( λ ) = P H c ( λ ) , provided that the order of G is greater than 2.