The least Q-eigenvalue with fixed domination number

Abstract Denote by Lg, l the lollipop graph obtained by attaching a pendant path P = v g v g + 1 ⋯ v g + l (l ≥ 1) to a cycle C = v 1 v 2 ⋯ v g v 1 (g ≥ 3). A F g , l − g r a p h of order n ≥ g + 1 is defined to be the graph obtained by attaching n − g − l pendent vertices to some of the nonpendant vertices of Lg, l in which each vertex other than v g + l − 1 is attached at most one pendant vertex. A F g , l ∘ -graph is a F g , l − g r a p h in which vg is attached with pendant vertex. Denote by qmin the least Q − e i g e n v a l u e of a graph. In this paper, we proceed on considering the domination number, the least Q-eigenvalue of a graph as well as their relation. Further results obtained are as follows: (i) some results about the changing of the domination number under the structural perturbation of a graph are represented; (ii) among all nonbipartite unicyclic graphs of order n, with both domination number γ and girth g ( g ≤ n − 1 ), the minimum qmin attains at a F g , l -graph for some l; (iii) among the nonbipartite graphs of order n and with given domination number which contain a F g , l ∘ -graph as a subgraph, some lower bounds for qmin are represented; (iv) among the nonbipartite graphs of order n and with given domination number n 2 .