单圈图的细分顶点Wiener指数的研究  

Wiener index of subdivision of unicyclic graph

作　　者：常旻宇 耿显亚 CHANG Minyu;GENG Xianya(School of Mathematics and Big Data,Anhui University of Science and Technology,Huainan 232001,China)

机构地区：[1]安徽理工大学数学与大数据学院,安徽淮南232001

出　　处：《哈尔滨商业大学学报（自然科学版）》2024年第4期438-440,共3页Journal of Harbin University of Commerce：Natural Sciences Edition

基　　金：安徽省自然科学基金(2008085MA01)。

摘　　要：细分顶点是一种用于修改图结构的方法,细分操作涉及将图中的边替换为由新顶点连接的路径,从而增加顶点数目并改变图的各种性质,例如直径、连通性、图的谱性质以及其他拓扑特性.细分顶点在化学图论、网络设计和电路理论中有着重要的应用.如果在一个图中用k个新的细分顶点替换一条边,则该边会被一条长度为(k+1)的路径取代.Wiener指数W(T)定义为树T所有顶点之间的距离之和,通过添加一条边构建一个单圈图U.用(k+2)阶的细分边更换单圈图U的一条边e构建出新图U_(e),则可构建一个W(U)和W(U_(1))+W(U_(2))+…W(U_(n))的关系.探讨了细分顶点的定义及其基本性质,分析细分操作对图的几何和谱性质的影响,并讨论细分顶点在实际应用中的一些典型案例.Subdivision vertices are a method used to modify graph structures.The new vertices,thereby increasing the number of vertices and altering various graph properties such as diameter,connectivity,spectral properties,and other topological characteristics.Subdivision vertices have significant applications in chemical graph theory,network design,and circuit theory.If a graph's edge is replaced with k new subdivision vertices,the edge is replaced by a path of length(k+1).The Wiener index is defined as the sum of distances between all vertices in a tree T.A unicyclic graph can be constructed by adding an edge to a tree.By replacing an edge of a unicyclic subdivision operation involves replacing edges in the graph with paths connected by graph U with a path on(k+2)vertices,a new graph U_(e) can be obtained.This allows the establishment of a relationship between quantities W(T)and W(U_(1))+W(U_(2))+…+W(U_(n)).This paper explored the definition and basic properties of subdivision vertices,analyzed the impact of subdivision operations on the geometric and spectral properties of graphs,and discusses some typical cases of subdivision vertices in practical applications.

关 键 词：单圈图 WIENER指数 细分边 距离之和 树 更换 

分 类 号：O157.6[理学—数学]