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.

A recursion due to Kook expresses the Laplacian eigenvalues of a matroid M in terms of the eigenvalues of its deletion M−e and contraction M/e by a fixed element e, and an error term. We show that this error term is given simply by the Laplacian eigenvalues of the pair (M−e, M/e). We further show that by suitably generalizing deletion and contraction to arbitrary simplicial complexes, the Lapla...

A signed graph is a graph where the edges are assigned either positive ornegative signs. Net degree of a signed graph is the dierence between the number ofpositive and negative edges incident with a vertex. It is said to be net-regular if all itsvertices have the same net-degree. Laplacian energy of a signed graph is defined asε(L(Σ)) =|γ_1-(2m)/n|+...+|γ_n-(2m)/n| where γ_1,...,γ_n are the ei...

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 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...