Spectral radius and traceability of connected claw-free graphs

TL;DR

This paper investigates the spectral radius and traceability of connected claw-free graphs, establishing conditions under which such graphs are traceable unless they are a specific exceptional graph.

Contribution

It provides new spectral conditions for traceability in connected claw-free graphs, extending previous theorems to this class of graphs.

Findings

If spectral radius ≥ n-4, graph is traceable unless it is N_{n-3,3}.

If the spectral radius of the complement ≤ that of N_{n-3,3}, then the graph is traceable unless it is N_{n-3,3}.

Results extend prior theorems to claw-free graphs.

Abstract

Let $G$ be a connected claw-free graph on $n$ vertices and $\overline{G}$ be its complement graph. Let $\mu(G)$ be the spectral radius of $G$. Denote by $N_{n-3,3}$ the graph consisting of $K_{n-3}$ and three disjoint pendent edges. In this note we prove that: (1) If $\mu(G)\geq n-4$, then $G$ is traceable unless $G=N_{n-3,3}$. (2) If $\mu(\overline{G})\leq \mu(\overline{N_{n-3,3}})$ and $n\geq 24$, then $G$ is traceable unless $G=N_{n-3,3}$. Our works are counterparts on claw-free graphs of previous theorems due to Lu et al., and Fiedler and Nikiforov, respectively.