Spectral radius and Hamiltonicity of graphs with large minimum degree

TL;DR

This paper establishes spectral radius-based conditions for Hamiltonian paths and cycles in graphs with large minimum degree, providing tight bounds and characterizing exceptional cases.

Contribution

It introduces spectral radius criteria for Hamiltonicity in graphs with large minimum degree, extending classical results with precise bounds and specific structural exceptions.

Findings

Spectral radius ≥ n−k−1 implies Hamiltonian cycle unless specific graph structures occur.

Spectral radius ≥ n−k−2 implies Hamiltonian path unless specific graph structures occur.

Bounds on n are shown to be nearly tight within an additive constant.

Abstract

This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $\lambda\left( G\right) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $k\geq1,$ $n\geq k^{3}/2+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $\delta\left( G\right) \geq k.$ If \[ \lambda\left( G\right) \geq n-k-1, \] then $G$ has a Hamiltonian cycle, unless $G=K_{1}\vee(K_{n-k-1}+K_{k})$ or $G=K_{k}\vee(K_{n-2k}+\overline{K}_{k})$. (2) Let $k\geq1,$ $n\geq k^{3}/2+k^{2}/2+k+5,$ and let $G$ be a graph of order $n$, with minimum degree $\delta\left( G\right) \geq k.$ If \[ \lambda\left( G\right) \geq n-k-2, \] then $G$ has a Hamiltonian path, unless $G=K_{k}\vee(K_{n-2k-1}+\overline {K}_{k+1})$ or $G=K_{n-k-1}+K_{k+1}$ In addition, it is shown that in the above statements, the bounds on $n$ are tight within an additive term not exceeding $2$.