Two Kinds Of Balaban Indices Of Graphs

Graph theory,originated from the famous problem on seven bridges of Konigs-berg,which has rich contents and extensive application background and becomes one of the active branches of mathematics.Among them,topological indices of graphs have many applications.Specially,the Balaban index of a graph,as a kind of impor-tant topological index based on distances,was proposed by the famous mathematical chemist A.T.Balaban in 1982.The Balaban index of a connected graph G is defined as follow.#12 where m and n are the number of edges and vertices of G,respectively.DG(u)is denote as the sum of distances between the vertex u and other one.In G order to research other different structure properties of graphs,the Sum-Balaban index of graphs was also proposed by A.T.Balaban at al.,i.e.,#12 These two kinds of topological indices have been found lots of applications such as cheminformatics,QSAR and QSPR studies and drug design and synthesis.In this work,some researches are shown as follows.(1)A lower bound on the Balaban index of a graph is given.Based on the small world network graph and the Cartesian product K2×Ct,a kind of 4-regular graphs is constructed,whose formulas on the Balaban index and Sum-Balaban index of them are also presented.(2)Compare their Balaban indices among straight chain polyomino systems and give their different upper bounds and lower ones on the Balaban indices.(3)Compute the Balaban indices of the trees with small diameter and perfect matchings.Then by comparing them,the trees with the largest and smallest Balaban indices are found,respectively.