A lower bound for the sum of the two largest signless Laplacian eigenvalues

TL;DR

This paper establishes a new lower bound for the sum of the two largest signless Laplacian eigenvalues of a graph, characterizing the cases of equality and providing counterexamples for larger k.

Contribution

It proves the signless Laplacian version of Grone's inequality for k=2 and identifies the extremal graphs where equality holds.

Findings

Equality holds for star graphs and complete graphs K3 when k=2.

Counterexamples exist for k ≥ 3.

The inequality extends known results for k=1 to k=2.

Abstract

Let $G$ be a graph of order $n \geq 3$ with sequence degree given as $d_{1}(G) \geq ... \geq d_{n}(G)$ and let $\mu_1(G),..., \mu_n(G)$ and $q_1(G), ..., q_{n}(G)$ be the Laplacian and signless Laplacian eigenvalues of $G$ arranged in non increasing order, respectively. Here, we consider the Grone's inequality [R. Grone, Eigenvalues and degree sequences of graphs, Lin. Multilin. Alg. 39 (1995) 133--136] $$ \sum_{i=1}^{k} \mu_{i}(G) \geq \sum_{i=1}^{k} d_{i}(G)+1$$ and prove that for $k=2$, the equality holds if and only if $G$ is the star graph $S_{n}.$ The signless Laplacian version of Grone's inequality is known to be true when $k=1.$ In this paper, we prove that it is also true for $k=2,$ that is, $$q_{1}(G)+q_{2}(G) \geq d_1(G)+d_2(G)+1$$ with equality if and only if $G$ is the star $S_{n}$ or the complete graph $K_{3}.$ When $k \geq 3$, we show a counterexample.