Elsevier

Discrete Applied Mathematics

Volume 206, 19 June 2016, Pages 15-38
Discrete Applied Mathematics

Adjacency polynomials of digraph transformations

https://doi.org/10.1016/j.dam.2016.01.032Get rights and content
Under an Elsevier user license
open archive

Abstract

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

Keywords

Adjacency polynomial
Regular digraph
(xyab)-transformation
Cospectral digraphs

Cited by (0)

This work is supported by the Fund of Science and Technology Commission of Shanghai Municipality   13ZR1400100, the Fund of China Scholarship Council   201306635014, and the National Natural Science Foundation of China grant 11371086.