The maximum spectral radius of wheel-free graphs

TL;DR

This paper determines the maximum spectral radius of wheel-free graphs of a given order and characterizes the extremal graphs, addressing a specific problem in spectral graph theory related to forbidden subgraphs.

Contribution

It solves a Brualdi-Solheid-Turán type problem by finding the maximum spectral radius for wheel-free graphs and characterizing the extremal graphs.

Findings

Maximum spectral radius of wheel-free graphs is established.

Extremal graphs achieving the maximum spectral radius are characterized.

Results contribute to spectral graph theory and forbidden subgraph problems.

Abstract

A wheel graph is a graph formed by connecting a single vertex to all vertices of a cycle. A graph is called wheel-free if it does not contain any wheel graph as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Tur\'{a}n type problem: what is the maximum spectral radius of a graph of order $n$ that does not contain subgraphs of particular kind. In this paper, we study the Brualdi-Solheid-Tur\'{a}n type problem for wheel-free graphs, and we determine the maximum (signless Laplacian) spectral radius of a wheel-free graph of order $n$. Furthermore, we characterize the extremal graphs.