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Journal ArticleDOI

Spectral radius and Hamiltonian properties of graphs

29 Jan 2015-Linear & Multilinear Algebra (Taylor & Francis)-Vol. 63, Iss: 8, pp 1520-1530
TL;DR: In this paper, the spectral radius of a graph with minimum degree is shown to be the largest eigenvalue of the adjacency matrix of the graph, and two previous theorems due to Fiedler and Nikiforov and Lu et al. are obtained.
Abstract: Let be a graph with minimum degree . The spectral radius of , denoted by , is the largest eigenvalue of the adjacency matrix of . In this note, we mainly prove the following two results.(1) Let be a graph on vertices with . If , then contains a Hamilton path unless .(2) Let be a graph on vertices with . If , then contains a Hamilton cycle unless . As corollaries of our first result, two previous theorems due to Fiedler and Nikiforov and Lu et al. are obtained, respectively. Our second result refines another previous theorem of Fiedler and Nikiforov.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the spectral analogies of Erdős' and Moon and Moser's results for Hamilton cycles in balanced bipartite graphs are presented. But the spectral analogue is not a sufficient condition for graphs of order n and minimum degree k.
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 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 (for large enough n), and the minimum (signless Laplacian) spectral radius of the complements of graphs in . 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.

71 citations

Journal ArticleDOI
TL;DR: In this paper, Wu et al. gave sufficient conditions on the spectral radius for a bipartite graph being Hamiltonian and traceable, which improved the results of Yu and Fan.

45 citations

Journal ArticleDOI
TL;DR: In this paper, sufficient conditions for Hamiltonian paths and cycles in a graph of order n and the spectral radius of its adjacency matrix were studied, where n is the number of vertices in the graph.
Abstract: Let G be a graph of order n and λ(G) the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in G. One of the main results of the paper is the following theorem

43 citations

Journal ArticleDOI
TL;DR: In this paper, the spectral analogies of Erdős' and Moon-Moser's Hamilton cycles in balanced bipartite graphs are presented, and the maximum (signless Laplacian) spectral radius of graphs in the class of non-Hamiltonian graphs of order $n$ and minimum degree at least $k is determined.
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 $\mathcal{G}_n^k$ 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 $\mathcal{G}_n^k$ (for large enough $n$), and the minimum (signless Laplacian) spectral radius of the complements of graphs in $\mathcal{G}_n^k$. 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.

36 citations

Posted Content
TL;DR: In this article, the authors gave sufficient conditions on the spectral radius for a bipartite graph to be traceable and traceable, based on the results of Lu, Liu and Tian.
Abstract: In this paper, we give sufficient conditions on the spectral radius for a bipartite graph to Hamiltonian and traceable, which expand the results of Lu, Liu and Tian (2012) [10]. Furthermore, we also present tight sufficient conditions on the signless Laplacian spectral radius for a graph to Hamiltonian and traceable, which improve the results of Yu and Fan (2012) [12].

34 citations

References
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Journal ArticleDOI
TL;DR: In this paper, the best possible generalization of Dirac, Posa, and Bondy's necessary and sufficient conditions for a graph to be Hamiltonian was proved. But this generalization was only applicable to bipartite graphs.

298 citations

Journal ArticleDOI
TL;DR: It is shown that if G is Kp+1-free then if δ be the minimal degree of G then This inequality supersedes inequalities of Stanley and Hong and is sharp for regular graphs and for a class of graphs which are in some sense maximally irregular.
Abstract: Let λ(G) be the largest eigenvalue of the adjacency matrix of a graph G: We show that if G is Kp+1-free then ***** insert CODING here *****This inequality was first conjectured by Edwards and Elphick in 1983 and supersedes a series of previous results on upper bounds of λ(G).Let Ti denote the number of all i-cliques of G, λ = λ(G) and p = cl(G): We show ***** insert equation here *****Let δ be the minimal degree of G. We show ***** insert equation here *****This inequality supersedes inequalities of Stanley and Hong. It is sharp for regular graphs and for a class of graphs which are in some sense maximally irregular.

202 citations


"Spectral radius and Hamiltonian pro..." refers background in this paper

  • ...Lemma 6 (Hong et al. [18], Nikiforov [19]) For nonnegative integers p and q with 2q ≤ p(p − 1) and 0 ≤ x ≤ p − 1, the function f (x) = (x − 1)/2+√2q − px + (1+ x)2/4 is decreasing with respect to x....

    [...]

  • ...Theorem 3 (Fiedler and Nikiforov [2]) Let G be a graph on n vertices....

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  • ...Nikiforov [19] proved it for general graphs independently, and the case of equality was characterized in [20]....

    [...]

  • ...[18], Nikiforov [19]) For nonnegative integers p and q with 2q ≤ p(p − 1) and 0 ≤ x ≤ p − 1, the function f (x) = (x − 1)/2+√2q − px + (1+ x)2/4 is decreasing with respect to x....

    [...]

  • ...Corollary 1 (Fiedler and Nikiforov [2]) Let G be a graph on n vertices....

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Journal ArticleDOI
TL;DR: The following sharp upper bound is obtained of the minimum degree of vertices of G, which is either a regular graph or a bidegreed graph in which each vertex is of degree either ? or n?1.

157 citations


"Spectral radius and Hamiltonian pro..." refers background in this paper

  • ...[18] proved the following spectral inequality for connected graphs....

    [...]

  • ...[18], Nikiforov [19]) For nonnegative integers p and q with 2q ≤ p(p − 1) and 0 ≤ x ≤ p − 1, the function f (x) = (x − 1)/2+√2q − px + (1+ x)2/4 is decreasing with respect to x....

    [...]