<abstract><p>Let $ A(G) and D(G) be the adjacency matrix degree diagonal of a graph G $, respectively. For any real number \alpha \in[0, 1] Nikiforov defined A_{\alpha} $-matrix as A_{\alpha}(G) = D(G)+(1-\alpha)A(G) $. Let S_k(A_{\alpha}(G)) sum k largest eigenvalues In this paper, some bounds on are obtained, which not only extends results signless Laplacian matrix, but it also gi...

We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computer-generated conjectures.

Received by the editors on July 22, 2010. Accepted for publication on February 17, 2011. Handling Editor: Bryan Shader. School of Mathematical Science, South China Normal University, Guangzhou, 510631, P.R. China ([email protected], Zhifu You; [email protected], Bolian Liu). This work was supported by the NNSF of China (No. 11071088). Electronic Journal of Linear Algebra ISSN 1081-3810 A publi...

In this paper, we investigate the relation between the Q -spectrum and the structure of G in terms of the circumference of G. Exploiting this relation, we give a novel necessary condition for a graph to be Hamiltonian by means of its Q -spectrum. We also determine the graphs with exactly one or two Q -eigenvalues greater than or equal to 2 and obtain all minimal forbidden subgraphs and maximal ...

AMS Mathematics Subject Classification (2000): 05C50