The Conjugate Formal Product of a Graph

Let G be a simple graph of order n and let V (G) be its vertex set. 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 = [cij ] is the conjugate adjacency matrix of the graph G if cij = c for any two adjacent vertices i and j, cij = c for any two nonadjacent vertices i and j, and cij = 0 if i = j. Let P c G(λ) = |λI − A| denote the conjugate characteristic polynomial of G and let [Acij] = {λI−A}, where {M} denotes the adjoint matrix of a square matrix M . For any two subsets X,Y ⊆ V (G) define 〈X,Y 〉 = ∑