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 Gk n be the ...

The spectral theory of higher-order symmetric tensors is an important tool for revealing some important properties of a hypergraph via its adjacency tensor, Laplacian tensor, and signless Laplacian tensor. Owing to the sparsity of these tensors, we propose an efficient approach to calculate products of these tensors and any vectors. By using the state-of-the-art L-BFGS approach, we develop a fi...

We survey properties of spectra of signless Laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix. For regular graphs the whole existing theory of spectra of the adjacency matrix and of the Laplacian matrix transfers directly to the signless Laplacian, and so we consider arbitrary graphs with special emphasis on the non-regular case. The ...

Let qmin(G) stand for the smallest eigenvalue of the signless Laplacian of a graph G of order n: This paper gives some results on the following extremal problem: How large can qmin (G) be if G is a graph of order n; with no complete subgraph of order r + 1? It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds ...

Let G=(V,E), $V={v_1,v_2,ldots,v_n}$, be a simple connected graph with $%n$ vertices, $m$ edges and a sequence of vertex degrees $d_1geqd_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ and ${%D}=mathrm{diag }(d_1,d_2,ldots , d_n)$ be the adjacency and the diagonaldegree matrix of $G$, respectively. Denote by ${mathcal{L}^+}(G)={D}^{-1/2}(D+A) {D}^{-1/2}$ the normalized signles...

In this paper, various modifications of a connected graph G are regarded as perturbations of its signless Laplacian matrix. Several results concerning the resulting changes to the signless Laplacian spectral radius of G are obtained by solving intermediate eigenvalue problems of the second type. AMS subject classifications: 05C50

Abstract In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix, distance (signless) Laplacian matrix, to obtain some known and new results. Moreover, we propose some problems for further research. AMS Classification: ...