A Brualdi-Hoffman-Tur\'{a}n problem for friendship graph

TL;DR

This paper determines the maximum spectral radius of graphs that do not contain a friendship graph as a subgraph, resolving a conjecture using the $k$-core technique.

Contribution

It solves a specific spectral extremal problem for friendship graphs, advancing understanding of $H$-free graphs in spectral graph theory.

Findings

Resolved a conjecture on spectral radius for $F_k$-free graphs.

Applied the $k$-core technique to a new class of extremal problems.

Established maximum spectral radius bounds for friendship graphs.

Abstract

A graph is said to be $H$-free if it does not contain $H$ as a subgraph. Brualdi-Hoffman-Tur\'{a}n type problem is to determine the maximum spectral radius of an $H$-free graph $G$ with give size $m$. The $F_k$ is the graph consisting of $k$ triangles that intersect in exactly one common vertex, which is known as the friendship graph. In this paper, we resolve a conjecture (the Brualdi-Hoffman-Tur\'{a}n-type problem for $F_k$) of Li, Lu and Peng [Discrete Math. 346 (2023) 113680] by using the $k$-core technique presented in Li, Zhai and Shu [European J. Combin, 120 (2024) 103966].