Ordering $Q$-indices of graphs: given size and girth

TL;DR

This paper investigates the ordering of the largest eigenvalues of the signless Laplacian matrix, called Q-indices, for graphs with specified size and girth, providing explicit orderings and extremal values.

Contribution

It establishes the ordering of the top Q-indices for graphs with given girth and size, extending spectral graph theory classifications.

Findings

Ordered the first (⌊g/2⌋+2) largest Q-indices for graphs with girth g.

Ordered the first (⌊g/2⌋+3) largest Q-indices for graphs with girth at least g.

Identified the first five largest Q-indices for graphs with girth 3.

Abstract

The signless Laplacian matrix in graph spectra theory is a remarkable matrix of graphs, and it is extensively studied by researchers. In 1981, Cvetkovi\'{c} pointed $12$ directions in further investigations of graph spectra, one of which is "classifying and ordering graphs". Along with this classic direction, we pay our attention on the order of the largest eigenvalue of the signless Laplacian matrix of graphs, which is usually called the $Q$-index of a graph. Let $\mathbb{G}(m, g)$ (resp. $\mathbb{G}(m, \geq g)$) be the family of connected graphs on $m$ edges with girth $g$ (resp. no less than $g$), where $g\ge3$. In this paper, we firstly order the first $(\lfloor\frac{g}{2}\rfloor+2)$ largest $Q$-indices of graphs in $\mathbb{G}(m, g)$, where $m\ge 3g\ge 12$. Secondly, we order the first $(\lfloor\frac{g}{2}\rfloor+3)$ largest $Q$-indices of graphs in $\mathbb{G}(m, \geq g)$, where $m\ge 3g\ge 12$. As a complement, we give the first five largest $Q$-indices of graphs in $\mathbb{G}(m, 3)$ with $m\ge 9$. Finally, we give the order of the first eleven largest $Q$-indices of all connected graphs with size $m$.