Extensions of the Graham-Hoffman-Hosoya Type Theorems for the Exponential Distance Matrices and q-Distance Matrices  

作　　者：Jie WANG 

机构地区：[1]School of Mathematical Sciences,Capital Normal University,Beijing 100048,P.R.China

出　　处：《Journal of Mathematical Research with Applications》2024年第6期723-734,共12页数学研究及应用（英文版）

摘　　要：Let G be a strongly connected directed weighted graph with vertex set{v_(1),v_(2),...,v_(n)},in which each edge e is assigned with an arbitrary nonzero weight w(e).For any two vertices v_i,v_(j )of G,the distance d_(ij )from v_(i)to v_(j)is defined as dij=P∈P(v_(i),v_(j))min∑w(e) where P(v_(i),v_(j)) denotes the set consisting of all the directed paths from v_(i)to v_(j)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 F^(q)_(G)=(q^(dij))_(n×n),and define the q-distance matrix of G as D_(G)^(q)=(d_(ij)^(q))_(n×n)with d_(ij)^(q)={1-q^(dij)/1-q,if q≠1,dij,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-HoffmanHosoya theorem (J.Graph Theory,1977).In this paper,we present some Graham-HoffmanHosoya type theorems for the exponential distance matrix F_(G)^(q)and q-distance matrix D_(G)^(q),extending all the known Graham-Hoffman-Hosoya type theorems.

关 键 词：distance matrices exponential distance matrices q-distance matrices determinants COFACTORS cofactor sums 

分 类 号：O157.5[理学—数学] O151.21[理学—基础数学]