We study the signless Laplacian spectral radius of graphs and prove three conjectures of Cvetkovic, Rowlinson, and Simic [Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math., Nouv. S?r. 81(95) (2007), 11-27].

Let G be a simple connected graph with the vertex set V (G). The eccentric distanceis the sum of all distances from the vertex v. In this paper we characterize the extremal unicyclic graphs among n-vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum. In addition, we characterize the extremal trees with given diameter and minimal eccentric distance sum.

In this paper, we show that of all graphs of order n with matching number β, the graphs with maximal spectral radius are K n if n = 2β or 2β + 1; K 2β+1 ∪ K n−2β−1 if 2β + 2 n < 3β + 2; K β K n−β or K 2β+1 ∪ K n−2β−1 if n = 3β + 2; K β K n−β if n > 3β + 2, where K t is the empty graph on t vertices.

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