Spectral analogues of Erdős’ and Moon–Moser’s theorems on Hamilton cycles

Abstract

In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number, and minimum degree of graphs which generalized Ore's theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper we present the spectral analogues of Erdős' theorem and Moon-Moser's theorem, respectively. Let G k n be the class of non-Hamiltonian graphs of order n and minimum degree at least k. We determine the maximum (signless Laplacian) spectral radius of graphs in G k n (for large enough n), and the minimum (signless Laplacian) spectral radius of the complements of graphs in G k n . All extremal graphs with the maximum (signless Laplacian) spectral radius and with the minimum (signless Laplacian) spectral radius of the complements are determined, respectively. We also solve similar problems for balanced bipartite graphs and the quasi-complements.

Stability Results on the Circumference of a Graph

“…Recently, Li and Ning [20], and independently, Füredi, Kostochka and Luo [12] proved a stability version of this theorem. Theorem 1.4 ( [20,12]). Let G be a non-Hamiltonian graph on n vertices with δ(G) ≥ k, where 1 ≤ k ≤ (n − 1)/2.…”

Wiener Index, Hyper-Wiener Index, Harary Index and Hamiltonicity Properties of graphs

“…Lemma 4.1 [13] Let k be an integer and G be a balanced bipartite graph on 2n vertices. If δ(G) k 1, n 2k + 3 and e(G) > n(n − k − 1) + (k + 1) 2 , then G is hamiltonian unless G ⊆ B k n .…”

“…6 Traceable of Graphs Lemma 6.1 [13] Let k be an integer and G be a graph of order n 6k + 10. If δ(G) k and e(G) > n−k−2 2 + (k + 1)(k + 2), then G is traceable, unless G ⊆ L k n or N k n .…”

“…Lemma 7.1 [13] Let k be an integer and G be a graph of order n 6k + 5. If δ(G) k and e(G) > n − k − 1 2 + (k + 1) 2 .…”

The formula for Turán number of spanning linear forests

“…Our proof of Theorem 1.5 is mainly based on the closure technique, which is initiated by Bondy and Chvátal [3] in 1976. But, the key ingredient is motivated by the counting technique from [15]. For more references on closure technique, we refer to [14,15,18,19].…”