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 = [Aij] = {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 Aij. 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 GS 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 Ak = [aij〈k〉]. If S ⊆ V(G) then S(t) = ∑k = 0∞ cktk is called the formal generating function associated with GS, where ck = ∑i∈S ∑j∈S aij(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 GS1 and GS2 of G of order (n + 1) we give necessary and sufficient conditions under which GS1 and GS2 are cospectral.
