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 Ue can be obtained.This allows the establishment of a relationship between quantities W(T)and W(U1)+W(U2)+…+W(Un).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.
unicyclic graphWiener indexsubdivisionsum of distancestreereplacing
万方数据
维普
