一类直径为2的极大平面图的Mostar指数

图G的Mostar指数定义为Mo(G)=∑uv∈Ε(G)|nu-nv|,其中nu表示在G中到顶点u的距离比到顶点v的距离近的顶点个数,nv表示到顶点v的距离比到顶点u的距离近的顶点个数.若一个图G的任两点之间的距离至多为2,且不是完全图,则称G是一个直径为2的图.已知直径为2点数至少为4的极大平面图的最小度为3或4.本文研究了直径为2且最小度为4的极大平面图的Mostar指数.具体说,若G是一个点数为n,直径为2,最小度为4的极大平面图,则(1)当n≤12时,Mostar指数被完全确定;(2)当n≥13时,4/3n2-44/3n+94/3≤Mo(G)≤2n2-16n+24,且达到上,下界的极图同时被找到.

Mostar Index of Some Maximal Planar Graphs of Diameter Two

The Mostar index of a graph G is defined as Mo(G)=∑uv∈Ε(G)|n_u-n_v|,where n_u is the number of vertices of G closer to vertex u than to vertex v,and n_v is the number of vertices closer to vertex v than to vertex u.A non-complete graph G has diameter two if the distance of every pair of vertices in G is at most two.It was known that the minimum degree of a maximal planar graph of diameter two and having at least four vertices equals to 3 or 4.Let G be a maximal planar graph having n vertices,minimum degree 4,and diameter two.In this paper,we determine the value of Mo(G) if n ≤12,and show that,if n≥13,then 4/3 n~2-44/3 n+94/3≤Mo(G)≤2 n~2-16 n+24 and the corresponding extremal graphs attaining upper and lower bounds are given.