Some improved bounds on two energy-like invariants of some derived graphs

TL;DR

This paper derives improved mathematical bounds for energy-like graph invariants, specifically the Laplacian-energy-like invariant and incidence energy, for certain derived graphs of regular and semiregular graphs.

Contribution

It introduces new bounds on $LEL$ and $IE$ for $ ext{R}$-graphs, $ ext{Q}$-graphs, and line graphs, refining previous results using inequalities.

Findings

Improved bounds on $LEL$ and $IE$ for $ ext{R}$-graphs and $ ext{Q}$-graphs.

New lower bounds for $LEL$ and $IE$ of line graphs of semiregular graphs.

Theoretical analysis confirms these bounds are tighter than existing ones.

Abstract

Given a simple graph $G$, its Laplacian-energy-like invariant $LEL(G)$ and incidence energy $IE(G)$ are the sum of square root of its all Laplacian eigenvalues and signless Laplacian eigenvalues, respectively. Applying the Cauchy-Schwarz inequality and the Ozeki inequality, along with its refined version, we obtain some improved bounds on $LEL$ and $IE$ of the $\mathcal {R}$-graph and $\mathcal{Q}$-graph for a regular graph. Theoretical analysis indicates that these results improve some known results. In addition, some new lower bounds on $LEL$ and $IE$ of the line graph of a semiregular graph are also given.