Signless Laplacian eigenvalue problems of Nordhaus-Gaddum type

TL;DR

This paper investigates bounds on the second-largest signless Laplacian eigenvalues of a graph and its complement, establishing new inequalities and characterizing extremal graphs for these bounds.

Contribution

It introduces new bounds for the sum of second-largest signless Laplacian eigenvalues and characterizes extremal graphs achieving these bounds.

Findings

Proves that for certain graphs, $q_2(G)+q_2(ar{G})\

establishes lower bounds for $q_2(G)+q_2(ar{G})$,

characterizes graphs attaining the bounds.

Abstract

Let $G$ be a graph of order $n$, and let $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)$ denote the signless Laplacian eigenvalues of $G$. Ashraf and Tayfeh-Rezaie [Electron. J. Combin. 21 (3) (2014) \#P3.6] showed that $q_1(G)+q_1(\overline{G})\leq 3n-4$, with equality holding if and only if $G$ or $\overline{G}$ is the star $K_{1,n-1}$. In this paper, we discuss the following problem: for $n\geq6$, does $q_2(G)+q_2(\overline{G})\leq 2n-5$ always hold? We provide positive answers to this problem for the graphs with disconnected complements and the bipartite graphs, and determine the graphs attaining the bound. Moreover, we show that $q_2(G)+q_2(\overline{G})\geq n-2$, and the extremal graphs are also characterized.