树和单圈图的带和权Mostar指标

Sum-weighted Mostar Index of Trees and Unicyclic Graphs

不包含圈的简单连通图称为树,顶点数和边数相同的简单连通图称为单圈图。在Mostar指标的基础上,为求得树和单圈图的带和权Mostar指标的上下界,根据割边的特殊性,通过计算发现将图的非平凡割边变为悬挂边,其带和权Mostar指标变大,在此基础上,对两类图的结构进行分析,对树进行移边变换,对单圈图进行缩圈变换,最终确立了树和单圈图的带和权Mostar指标的上下界并刻画了达到相应界值的极值图,同时给出了圈长固定时,单圈图的带和权Mostar指标的上界和其达到上界的极值图。

A graph without cycles is called tree if is connected,while a graph is called unicyclic graphs if it is connected and has the same number of vertices and edges.Based on the Mostar index,to obtain the upper and lower bounds of the sum-weighted Mostar index of trees and unicyclic graphs,according to the particularity of the cut edge,it is found through calculation that the non-trivial cut edge of the graph is changed into pendant edge,and the sum-weighted Mostar index becomes larger.On this basis,by analyzing the structure of these two kinds of graphs,the edge-shifting transformation is performed on the tree,and the cycle-shrinking transformation is performed on the unicyclic graph,the upper and lower bounds of the sum-weighted Mostar index of trees and unicyclic graphs are established and the extremal graphs reaching the corresponding bounds are characterized.At the same time,the upper bound of the sum-weighted Mostar index of the unicyclic graphs and the extreme value graph reaching the upper bound are given when the circle length is fixed.