Abstract :
Let G be a simple graph of order n and let PG∗(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 PGi∗(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 PG(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 GS 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 thatPGS(λ)=(−1)n+1λ+r+1(λ−r̄)PGS(λ̄)+(λ+r+1−|S|)2λ+r+1PG(λ̄),where GS denotes the complement of GS, r̄=(n−1)−r and λ̄=−λ−1. Using the last relation we prove that the polynomial reconstruction conjecture is true for all graphs GS for which G is regular.
