On formal products and spectra of graphs

Abstract

For any non-singular matrix M we denote by M the matrix formed by the algebraic cofactors of order ( n − 1) so that { M} T = | M| M −1. Let G be an arbitrary simple graph of order n and let A = [ A ij ] = { λI − A}, where A is the adjacency matrix of G. Besides, let X, Y be any two subsets of the vertex set V( G) and define 〈 X, Y〉 = ∑ i∈ X ∑ j∈ Y A ij . The expression 〈 X, Y〉 is called the formal product of the sets X and Y associated with the graph G. For any S ⊆ V( G), denote by G S the graph obtained from the graph G by adding a new vertex x which is adjacent exactly to the vertices from S, which is called the overgraph of G. Further, for any adjacency matrix A of G, let A k = [ a ij 〈 k〉 ]. If S ⊆ V( G) then ▪ S(t) = ∑ k = 0 ∞ c kt k is called the formal generating function associated with G S , where c k = ∑ i∈S ∑ j∈S a ij (k) (k = 0, 1, 2,…) . In this paper, using the formal product and the formal generating functions, some results about cospectral graphs are proved. In particular, for any two overgraphs G S 1 and G S 2 of G of order ( n + 1) we give necessary and sufficient conditions under which G S 1 and G S 2 are cospectral.