Nanotube by Gap Program

Let G be a connected graph. The vertex-set and edge-set of G denoted by V(G) and E(G) respectively. The distance between the vertices u and v, d(u,v), in a graph is the number of edges in a shortest path connecting them. Two graph vertices are adjacent if they are joined by a graph edge. The degree of a vertex ( ) i V G ∈ is the number of vertices joining to i and denoted by ( ) v i . The ( , ) i j entry of the adjacency matrix of G is denoted by ( , ) A i j . Let e be an edge of a graph G connecting the vertices u and v. Define two sets 1( ) N e G and 2( ) N e G as follows: 1( ) { ( ) ( , ) ( , )} N e G x V G d x u d x v = ∈ < and 2( ) { ( ) ( , ) ( , )} N e G x V G d x v d x u = ∈ < . The number of elements of 1( ) N e G and 2 ( ) N e G are denoted by 1( ) n e G and 2( ) n e G respectively. A topological index was introduced by Gutman and called the Szeged index, abbreviated as Sz [1] and defined as