On two energy-like invariants of line graphs and related graph operations

Abstract

Abstract For a simple graph G of order n, let μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n = 0 $\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}=0$ be its Laplacian eigenvalues, and let q 1 ≥ q 2 ≥ ⋯ ≥ q n ≥ 0 $q_{1}\geq q_{2}\geq\cdots\geq q_{n}\geq0$ be its signless Laplacian eigenvalues. The Laplacian-energy-like invariant and incidence energy of G are defined as, respectively, LEL ( G ) = ∑ i = 1 n − 1 μ i and IE ( G ) = ∑ i = 1 n q i . $$\mathit{LEL}(G)=\sum_{i=1}^{n-1}\sqrt{ \mu_{i}} \quad\mbox{and}\quad \mathit {IE}(G)=\sum_{i=1}^{n} \sqrt{q_{i}}. $$ In this paper, we present some new upper and lower bounds on LEL and IE of line graph, subdivision graph, para-line graph and total graph of a regular graph, some of which improve previously known results. The main tools we use here are the Cauchy-Schwarz inequality and the Ozeki inequality.

Keywords

• Laplacian-energy-like invariant
• incidence energy
• line graph
• subdivision graph
• para-line graph
• total graph