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

Suppose G = (V , E) is a simple graph and k is a fixed positive integer. A subset D ⊆ V is a distance k-dominating set of G if for every u ∈ V , there exists a vertex v ∈ D such that dG(u, v) ≤ k, where dG(u, v) is the distance between u and v in G. A set D ⊆ V is a distance k-paired-dominating set of G if D is a distance k-dominating set and the induced subgraph G[D] contains a perfect matchin...