Closeness of Some Graph Operations
Chavdar Dangalchev
TL;DR
This work develops a unified framework for closeness centrality under graph operations, deriving a general formula for the closeness of shadow graphs via $C(S(G))=4C(G)+\frac{n}{2}$ and listing how joins via an edge or vertex merge affect $C(A)$ and $C(B)$. Building on these, it computes exact closeness values for line graphs of fundamental graphs (cycles, paths, stars, complete graphs) and extends to more complex constructions formed by attaching graphs with bridges (e.g., lollipop, tadpole, broom, bistar). The authors provide explicit closed-form expressions for the closeness of both the line graphs and the base+bridge graphs, enabling efficient evaluation of centrality in networks assembled from standard components. The results have potential applications in network design and vulnerability analysis where graph operations model connectivity changes or modular compositions.
Abstract
Closeness is an important measure of network centrality. In this article we will calculate the closeness of graphs, created by using operations on graphs. We will prove a formula for the closeness of shadow graphs. We will calculate the closeness of line graphs of some wellknown graphs (like path, star, cycle, and complete graphs) and the closeness of line graphs of two of these graphs, connected by a bridge (like lollipop, tadpole, broom, and bistar graphs).