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

In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result for balanced bipartite graphs. Additionally, we construct infinitely many graphs to show that results proved in this paper give new strength for one to deter...