Spectral radius and Hamiltonian properties of graphs, II

TL;DR

This work investigates spectral conditions that force long cycles and Hamiltonicity in graphs and balanced bipartite graphs. It proves that for large $n$, a graph with $\lambda(G) > n-3$ (and, in the 2-connected case, $\lambda(G) > n-4$) contains a $C_{n-1}$ unless it lies in explicit extremal families, with analogous results for the signless Laplacian radius $q(G)$. In balanced bipartite graphs, a lower bound relative to $\sqrt{n(n-k)}$ ensures Hamiltonicity unless the graph equals the extremal $B^k_n$, extending Li–Ning’s results. The authors develop structural lemmas and Kelmans-operation based spectral inequalities to connect eigenvalue data with classical extremal graph bounds, and conclude with open problems on tight spectral criteria for even longer and consecutive cycles.

Abstract

In this paper, we first present spectral conditions for the existence of $C_{n-1}$ in graphs (2-connected graphs) of order $n$, which are motivated by a conjecture of Erdős. Then we prove spectral conditions for the existence of Hamilton cycles in balanced bipartite graphs. This result presents a spectral analog of Moon-Moser's theorem on Hamilton cycles in balanced bipartite graphs, and extends a previous theorem due to Li and the second author for $n$ sufficiently large. We conclude this paper with two problems on tight spectral conditions for the existence of long cycles of given lengths.

Figures (4)

• Figure 1: $L^3_n$ (left) and $N^3_n$ (right).
• Figure 2: The graph $\Lambda$.
• Figure 3: $\Gamma_1$, $\Gamma_2$, $\Psi_1$, $\Psi_2$ and $\Psi_3$.
• Figure 4: $X$, $Y$, $Z$, and $W$ in $B_n^k$.

Theorems & Definitions (36)

• Theorem 1.1: Fiedler and Nikiforov FN10
• Theorem 1.2
• Corollary 1
• Theorem 1.3
• Corollary 2
• Theorem 1.4
• Theorem 1.5: Li and Ning LN16
• Theorem 2.1: Bondy
• Theorem 2.2: Erdős E62
• Theorem 2.3: Ore O63