Eccentricity energy change of coalescence of graphs due to edge deletion

TL;DR

This paper studies how removing edges affects the eccentricity energy of a specific class of coalesced graphs, proving that eccentricity energy always increases with edge deletion.

Contribution

It demonstrates that for a class of coalesced graphs, eccentricity energy monotonically increases when edges are deleted, a novel insight into graph energy behavior.

Findings

Eccentricity energy increases with edge deletion in the studied graph class.

Identifies a class of graphs with monotonic eccentricity energy growth under edge removal.

Provides theoretical proof of energy increase upon edge deletion.

Abstract

The eccentricity matrix of a graph is obtained from the distance matrix by keeping the largest entries in their row or column, and the remaining entries are replaced by zeros. The eccentricity energy of a graph is the sum of the absolute values of the eigenvalues of its eccentricity matrix. In this paper, we investigate the effect of edge deletion on the eccentricity energy of graphs of the form $$G=K_{2n}\circ_{n} K_{2n}\circ_{n}\cdots \circ_{n} K_{2n}, (\text{\textit{l} copies of } K_{2n}),$$ where $n\geq 3,$ $l\geq 2,$ and $\circ_{n}$ denotes the $n-$coalescence of graphs, and prove that the eccentricity energy increases whenever an edge is removed. This result identifies a class of graphs whose eccentricity energy exhibits monotonic growth under edge deletion.