Maxima of the $Q$-index of 2 leaves-free graphs with given size

TL;DR

This paper establishes sharp upper bounds on the $Q$-index for 2 leaves-free graphs with a specified size and characterizes the extremal graphs achieving these bounds.

Contribution

It provides new sharp bounds and characterizations for the $Q$-index of 2 leaves-free graphs, extending previous work on leaf-free graphs.

Findings

Sharp upper bounds on the $Q$-index for 2 leaves-free graphs

Characterization of extremal graphs achieving these bounds

Extension of bounds from leaf-free to 2 leaves-free graphs

Abstract

The $Q$-index of graph $G$ is the largest eigenvalue of the signless Laplacian matrix of $G$. Wang [Discrete Appl. Math. 356(2024)] proved the sharp upper bounds on the $Q$-index of leaf-free graphs with given size and characterized the corresponding extremal graphs. A graph is $2$ leaves-free if it has no two pendent vertices. In this paper, we give sharp upper bounds on the $Q$-index of 2 leaves-free graphs with given size and characterize the corresponding extremal graphs.