Research Article Nonderogatory Unicyclic Digraphs

A digraph is nonderogatory if its characteristic polynomial and minimal polynomial are equal. Relations between nonderogatory digraphs and their automorphism groups have appeared in the literature (see [1] and [2, Theorem 5.10]). More recently, several articles (see, e.g., [1, 3–8]) address the problem of determining when certain operations between nonderogatory digraphs are again nonderogatory. The aim of this paper is to characterize unicyclic nonderogatory digraphs (i.e., nonderogatory digraphs which have exactly one cycle), in terms of Hamiltonicity conditions. The results obtained by Gan [4] concerning the complete product of difans and diwheels are an immediate consequence of this characterization. A digraph (directed graph) Γ consists of a finite set of vertices together with a set of arcs, which are ordered pairs of vertices. The vertices and arcs are denoted VΓ and EΓ, respectively. If (u,v)∈ EΓ, then u and v are adjacent and (u,v) is an arc starting at vertex u and terminating at vertex v. A walk in Γ from vertex u to vertex v is a finite vertex sequence