Closeness of Some Graph Operations

TL;DR

This paper investigates how graph operations affect the closeness centrality measure, providing formulas and calculations for various graph types and their transformations.

Contribution

It introduces formulas for closeness of shadow graphs and computes closeness for line graphs of well-known and composite graphs.

Findings

Derived a formula for the closeness of shadow graphs.

Calculated closeness of line graphs for key graph types.

Analyzed closeness in composite graphs like lollipop and bistar.

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).