Closeness Centralities of Lollipop Graphs
Chavdar Dangalchev
TL;DR
The paper analyzes closeness centralities for the lollipop family $L_{m,n}$, focusing on how network growth and resilience respond to vertex and edge deletions as well as link additions. It derives exact closed-form expressions for $C(L_{m,n})$, vertex residual closeness $VR(L_{m,n})$, link residual closeness $LR(L_{m,n})$, and additional closeness $A(L_{m,n})$, plus extensive case analyses of deletion and addition scenarios. The results identify which local modifications maximize or minimize closeness and provide practical guidance for maintaining or expanding such networks; the appendices support detailed comparisons among candidate graphs. The findings deepen understanding of how a bridged clique-plus-path structure behaves under perturbations, with implications for design of robust, scalable networks.
Abstract
Closeness is one of the most studied characteristics of networks. Residual closeness is a very sensitive measure of graphs robustness. Additional closeness is a measure of growth potentials of networks. In this article we calculate the closeness, vertex residual closeness, link residual closeness, and additional closeness of lollipop graphs.