Spectra of digraph transformations

Abstract

Let D be a directed graph without loops or multiple arcs, with vertex set V(D)=V and arc set E(D)=E. Let D0 be the digraph with vertex set V and with no arcs, D1 the complete digraph with vertex set V, D+=D, and D- the complement Dc of D. For e=(x,y)∈E let x=t(e) and y=h(e). Let T(D)(Tcb(D)) be the digraph with vertex set V∪E and such that (v,e) is an arc in T(D) (resp., in Tcb(D)) if and only if v∈V,e∈E, and vertex v=t(e) (resp., v≠t(e)) in D. Similarly, let H(D) (Hcb(D)) be the digraph with vertex set V∪E and such that (e,v) is an arc in H(D) (resp., in Hcb(D)) if and only if v∈V,e∈E, and vertex v=h(e) (resp., v≠h(e)) in D. Given a digraph D and three variables x,y,z∈{0,1,+,-}, the xyz-transformation of D is the digraph Dxyz such that Dxy0=Dx∪(Dl)y and Dxyz=Dxy0∪W, where W=T(D)∪H(D) if z=+, W=Tcb(D)∪Hcb(D) if z=-, and W is the complete bipartite digraph with parts V and E if z=1. In this paper we obtain the adjacency characteristic polynomials of some xyz-transformations of an r-regular digraph D in terms of r, the number of vertices, and the adjacency spectrum of D. Similar results are obtained for some non-regular digraphs, named digraph-functions. We also give various constructions of non-isomorphic cospectral digraphs using xyz-transformations. Our notion of xyz-transformation is also valid for digraphs with multiple arcs and loops if x,y,z∈{0,+} and z∈{0,1,+,-}. We also extend the notion of xyz-transformation to digraphs (V,E) with loops and no multiple arcs.