The signless Laplacian spectral radius of graphs without intersecting odd cycles

TL;DR

This paper establishes a sharp upper bound for the signless Laplacian spectral radius of graphs that do not contain certain intersecting odd cycles, and characterizes the extremal graphs achieving this bound.

Contribution

It provides the first sharp upper bound for the spectral radius of graphs avoiding intersecting odd cycles and characterizes all extremal graphs.

Findings

Sharp upper bound for the signless Laplacian spectral radius of $F_{a_1,\dots,a_k}$-free graphs.

Characterization of all extremal graphs attaining the bound.

Application of stability methods and eigenvalue structure analysis.

Abstract

Let $F_{a_1,\dots,a_k}$ be a graph consisting of $k$ cycles of odd length $2a_1+1,\dots, 2a_k+1$, respectively which intersect in exactly a common vertex, where $k\geq1$ and $a_1\ge a_2\ge \cdots\ge a_k\ge 1$. In this paper, we present a sharp upper bound for the signless Laplacian spectral radius of all $F_{a_1,\dots,a_k}$-free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof.