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Let g (G) denote the girth of a graph G. In this paper, we determine the unique graph with the maximum least Q-eigenvalue among all unicyclic graphs of order n = 6k (k ≥ 8) with perfect matchings. For the cases when n = 6k + 2 and n = 6k + 4, we prove that g (G) = 3 if G is a graph with the maximum least Q-eigenvalue, and provide a conjecture and a problem on the sharp upper bound of the least Q-eigenvalue. [ABSTRACT FROM AUTHOR]

In this paper, we study the least Q-eigenvalue of a non-bipartite hamiltonian graph, and determine all non-bipartite hamiltonian graphs whose least Q-eigenvalue attains the minimum among all non-bipartite hamiltonian graphs on n vertices. [ABSTRACT FROM AUTHOR]

In this paper, for a nonbipartite graph with both order n and domination number γ , we show that it contains a nonbipartite unicyclic spanning subgraph with the same domination number γ . We also present some results about the changing of the domination number when the structure of a graph is changed. By investigating the relation between the domination number and the least Q -eigenvalue of a graph, we minimize the least Q -eigenvalue among all the nonbipartite graphs with both order n ⩾ 4 and given domination number γ ⩽ n + 1 3 . [ABSTRACT FROM AUTHOR]

Abstract: Let G be a simple connected graph on n vertices and m edges. Lima et al. (2011) in [2] posed the following conjecture on the least eigenvalue of the signless Laplacian of G: . In this paper we prove a stronger result: For any graph with n vertices and m edges, we have . [Copyright &y& Elsevier]

Abstract Denote by L g, 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 L g, 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 v g is attached with pendant vertex. Denote by q min 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 q min 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 q min are represented; (iv) among the nonbipartite graphs of order n and with given domination number n 2. [ABSTRACT FROM AUTHOR]