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#### Article Title

The sum of the first two largest signless Laplacian eigenvalues of trees and unicyclic graphs

#### Keywords

The sum of eigenvalues, Signless Laplacian eigenvalues, Laplacian eigenvalues, Trees, Unicyclic graphs

#### Abstract

Let $G$ be a graph on $n$ vertices with $e(G)$ edges. The sum of eigenvalues of graphs has been receiving a lot of attention these years. Let $S_2 (G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and define $f(G) = e (G) +3 - S_2 (G)$. Oliveira et al. (2015) conjectured that $f(G) \geqslant f(U_{n})$ with equality if and only if $G \cong U_n$, where $U_n$ is the $n$-vertex unicyclic graph obtained by attaching $n-3$ pendent vertices to a vertex of a triangle. In this paper, it is proved that $S_2(G) < e(G) + 3 -\frac{2}{n}$ when $G$ is a tree, or a unicyclic graph whose unique cycle is not a triangle. As a consequence, it is deduced that the conjecture proposed by Oliveira et al. is true for trees and unicyclic graphs whose unique cycle is not a triangle.

#### Recommended Citation

Du, Zhibin.
(2019),
"The sum of the first two largest signless Laplacian eigenvalues of trees and unicyclic graphs",
*Electronic Journal of Linear Algebra*,
Volume 35, pp. 449-467.

DOI: https://doi.org/10.13001/1081-3810.3405

*Abstract*