The conjugate formal product of a graph

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Let G be a simple graph of order n and let V (G) be its vertex set. Let
and
, 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 Ac = [cij] is the conjugate adjacency matrix of the graph G if cij = c for any two adjacent vertices i and j, cij =
for any two nonadjacent vertices i and j and cij = 0 if i = j. Let
denote the conjugate characteristic polynomial of G and let
, where {M} denotes the adjoint matrix of a square matrix M. For any two subsets X,Y
V (G) define
. The expression
is called the conjugate formal product of the sets X and Y, associated with the graph G. Using the conjugate formal product we continue our previous investigations of some properties of the conjugate characteristic polynomial of G.