a-角链的Szeged指数和边Szeged指数

Szeged Index and Edge-Szeged Index of a-polygonal Chains

定义连通图的Szeged指数为Sz(G)=∑vv∈E(G)nu(uv|G)nv(uv|G),edge-Szeged指数为Sze(G)=∑uv∈E(G)mu (uv|G)mv(uv|G),其中nu(uv|G)(nv(uv|G)和mu(uv|G)(mv(uv|G)分别是到点u(v)的距离比到点v(u)的距离小的顶点的数量和边的数量.a-角链是由边长为1的正a边形构造的,其中a为偶数.本文给出了任意a-角链的Szeged指数和边Szeged指数的精确值.作为推论,多种多角链的Szeged指数和边Szeged指数都可以直接得到,例如已知的六角链(见[MATCH Commun.Math.Comput.Chem.,2001,43:7-15]).

The Szeged index and edge-Szeged index of a connected graph are defined as Sz(G)=∑uv∈E(G)nu(uv|G)nv(uv|G)and Sze(G)=∑uv∈E(G)mu(uv|G)mv(uv|G)respectively,where nu(uv|G)(resp.,nv(uv|G)) and mu(uv|G)(resp.,mv(uv|G)) are the number of vertices and edges whose distances to vertex u(resp.,v) are smaller than the distances to vertex v(resp.,u),respectively.The a-polygonal chains are recursively constructed by a-regular polygon of length one for every even number a.In this paper,the exact values of the Szeged index and edge-Szeged index of a-polygonal chains for any even number a are determined.As corollaries,many kinds of polygonal chains including the known result about hexagonal chains(see [MATCH Commun.Math.Comput.Chem.,2001,43:7-151) can be gotten directly.