Maxima of the Q-index: graphs without long paths

TL;DR

This paper establishes tight upper bounds on the maximum eigenvalue of the signless Laplacian for graphs lacking long paths, using a stability result related to graphs with large minimum degree.

Contribution

It introduces a new stability result and extends previous bounds on the signless Laplacian eigenvalues for graphs without long paths.

Findings

Derived tight upper bounds for q(G) in graphs without long paths

Proved a stability result for graphs with large minimum degree and no long paths

Extended previous work by Ali and Staton on spectral graph theory

Abstract

This paper gives tight upper bound on the largest eigenvalue q(G) of the signless Laplacian of graphs with no paths of given order. The main ingredient of our proof is a stability result of its own interest, about graphs with large minimum degree and with no long paths. This result extends previous work of Ali and Staton.