On the General Sum Distance Spectra of Digraphs

Abstract

Let G be a strongly connected digraph, and dG(vi,vj) denote the distance from the vertex vi to vertex vj and be defined as the length of the shortest directed path from vi to vj in G. The sum distance between vertices vi and vj in G is defined as sdG(vi,vj)=dG(vi,vj)+dG(vj,vi). The sum distance matrix of G is the n×n matrix SD(G)=(sdG(vi,vj))vi,vj∈V(G). For vertex vi∈V(G), the sum transmission of vi in G, denoted by STG(vi) or STi, is the row sum of the sum distance matrix SD(G) corresponding to vertex vi. Let ST(G)=diag(ST1,ST2,…,STn) be the diagonal matrix with the vertex sum transmissions of G in the diagonal and zeroes elsewhere. For any real number 0≤α≤1, the general sum distance matrix of G is defined as SDα(G)=αST(G)+(1−α)SD(G). The eigenvalues of SDα(G) are called the general sum distance eigenvalues of G, the spectral radius of SDα(G), i.e., the largest eigenvalue of SDα(G), is called the general sum distance spectral radius of G, denoted by μα(G). In this paper, we first give some spectral properties of SDα(G). We also characterize the digraph minimizes the general sum distance spectral radius among all strongly connected r-partite digraphs. Moreover, for digraphs that are not sum transmission regular, we give a lower bound on the difference between the maximum vertex sum transmission and the general sum distance spectral radius.