The signless Laplacian spectral radius of graphs with no intersecting triangles

TL;DR

This paper characterizes the graph with the largest signless Laplacian spectral radius among graphs of order n that do not contain a specific intersecting triangles structure, identifying the complete split graph as unique extremal.

Contribution

It establishes the extremal graph for the maximum signless Laplacian spectral radius in graphs excluding a certain triangle configuration, extending spectral graph theory results.

Findings

Complete split graph $S_{n,k}$ uniquely maximizes the spectral radius.

Maximum spectral radius achieved by $S_{n,k}$ for graphs without $F_k$.

Results hold for $k ext{≥}2$ and sufficiently large $n$.

Abstract

Let $F_k$ denote the $k$-fan consisting of $k$ triangles which intersect in exactly one common vertex, and $S_{n,k}$ the complete split graph of order $n$ consisting of a clique on $k$ vertices and an independent set on the remaining vertices in which each vertex of the clique is adjacent to each vertex of the independent set. In this paper, it is shown that $S_{n,k}$ is the unique graph attaining the maximum signless Laplacian spectral radius among all graphs of order $n$ containing no $F_k$, provided that $k\geq 2$ and $n\geq 3k^2-k-2$.