On formal products and angle matrices of a graph

Abstract

Let G be a simple graph of order n and let P G ∗(λ)=|λI−A ∗| denote the Seidel characteristic polynomial, where A ∗ is the Seidel adjacency matrix of G. Let P ∗(G) be the collection of Seidel characteristic polynomials P G i ∗(λ) of vertex deleted subgraphs G⧹i (i=1,2,…,n) . If G and H are two switching equivalent graphs, using the Seidel formal product and the Seidel angle matrices, we prove that P ∗(G)= P ∗(H) . Further, let P G ( λ)=| λI− A| be the characteristic polynomial of the graph G, where A is the adjacency matrix of G. Let S be any subset of the vertex set V( G) and let G S be the graph obtained from the graph G by adding a new vertex x which is adjacent exactly to the vertices from S. In particular, if G is a regular graph of degree r, we prove that P G S (λ)= (−1) n+1 λ+r+1 (λ− r ̄ )P G S ( λ ̄ )+ (λ+r+1−|S|) 2 λ+r+1 P G( λ ̄ ) , where G S denotes the complement of G S, r ̄ =(n−1)−r and λ ̄ =−λ−1 . Using the last relation we prove that the polynomial reconstruction conjecture is true for all graphs G S for which G is regular.