Let q (G) denote the spectral radius of the signless Laplacian matrix of a graph G, also known as the Q-index of G. The aim of this note is to study a general extremal problem: How large can q (G) be when G belongs to an abstract graph property? Even knowing very little about the graph property, this paper shows that useful conclusions about the asymptotics of q (G) can be made, which turn out ...

A set D of vertices in a graph G is a distance-k dominating set if every vertex of G either is in D or is within distance k of at least one vertex in D. A distance-k dominating set of G of minimum cardinality is called a minimum distance-k dominating set of G. For any graph G and for a subset F of the edge set of G the set F is an edge dominating set of G if every edge of G either is in D or is...

A graph G is said to be determined by its Q-spectrum if with respect to the signless Laplacian matrix Q , any graph having the same spectrum as G is isomorphic to G. The lollipop graph, denoted by Hn,p, is obtained by appending a cycle Cp to a pendant vertex of a path Pn−p. In this paper, it is proved that all lollipop graphs are determined by their Q -spectra. © 2008 Elsevier B.V. All rights r...