Abstract
Let G be a strongly connected directed weighted graph with vertex set {vi,V2,…,vn},in which each edge e is assigned with an arbitrary nonzero weight w(e).For any two vertices vi,vj of G,the distance dij from vi to vj is defined as dij=min P∈P(vi,vj)∑e∈Pw(e),where P(vi,vj)denotes the set consisting of all the directed paths from vi to vj in G.Given a nonzero indeterminant q,following the definitions from Yan and Yeh(Adv.Appl.Math.,2007),and Bapat et al.(Linear Algebra Appl.,2006),one can define the exponential distance matrix of G as FqG=(qdij)n×n,and define the q-distance matrix of G as DqG=(dqij)n×n with dqij={1-qdij/1-q,if q≠1,dif,if q=1,extending the original definitions only for the undirected unweighted connected graphs.One of the remarkable results about the distance matrices of graphs is due to the Graham-Hoffman-Hosoya theorem(J.Graph Theory,1977).In this paper,we present some Graham-Hoffman-Hosoya type theorems for the exponential distance matrix FqG and q-distance matrix DqG,ex-tending all the known Graham-Hoffman-Hosoya type theorems.
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