On a conjecture for the signless Laplacian eigenvalues

TL;DR

This paper investigates a conjecture relating the sum of the largest signless Laplacian eigenvalues of a graph to its edges, providing new bounds and confirming the conjecture for specific classes of graphs.

Contribution

The paper introduces new upper bounds for the sum of signless Laplacian eigenvalues and proves the conjecture for certain classes of graphs including connected, unicyclic, bicyclic, and some tricyclic graphs.

Findings

Conjecture holds for connected graphs with large k

Conjecture verified for all unicyclic and bicyclic graphs

Partial proof for tricyclic graphs when k ≠ 3

Abstract

Let $G$ be a simple graph with $n$ vertices and $e(G)$ edges, and $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)\geq0$ be the signless Laplacian eigenvalues of $G.$ Let $S_k^+(G)=\sum_{i=1}^{k}q_i(G),$ where $k=1, 2, \ldots, n.$ F. Ashraf et al. conjectured that $S_k^+(G)\leq e(G)+\binom{k+1}{2}$ for $k=1, 2, \ldots, n.$ In this paper, we give various upper bounds for $S_k^+(G),$ and prove that this conjecture is true for the following cases: connected graph with sufficiently large $k,$ unicyclic graphs and bicyclic graphs for all $k,$ and tricyclic graphs when $k\neq 3.$