Adjacency polynomials of digraph transformations.

Let A ( λ , D ) be the adjacency characteristic polynomial of a digraph D . In the paper Deng and Kelmans (2013) the so-called ( x y z ) -transformation D x y z of a simple digraph D was considered, where x , y , z ∈ { 0 , 1 , + , − } , and the formulas of A ( λ , D x y z ) were obtained for every r -regular digraph D in terms of r , the number of vertices of D , and A ( λ , D ) . In this paper we define the so-called ( x y a b ) -transformation D x y a b of a simple digraph D , where x , y , a , b ∈ { 0 , 1 , + , − } . This notion generalizes the previous notion of the ( x y z ) -transformation D x y z , namely, D x y a b = D x y z if and only if a = b = z . We extend our previous results on A ( λ , D x y z ) to the ( x y a b ) -transformation D x y a b by obtaining the formulas of A ( λ , D x y a b ) , where x , y , a , b ∈ { 0 , 1 , + , − } and a ≠ b , for every simple r -regular digraph D in terms of r , the number of vertices of D , and A ( λ , D ) . We also use ( x y a b ) -transformations to describe various constructions providing infinitely many examples of adjacency cospectral non-isomorphic digraphs. [ABSTRACT FROM AUTHOR]