2017/07/22 by Guanglong Yu, Yu, Guanglong · 1 citation
Mathematics · #A.0 #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.1707.07123
30 pages
arxiv created 2017/09/04 · arxiv updated 2017/09/05
Denote by Lg, l the lollipop graph obtained by attaching a pendant path ℙ=vgvg+1⋯ vg+l (l≥ 1) to a cycle ℂ=v1v2⋯ vgv1 (g≥ 3). A \mathcal Fg, l-graph 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 vg+l-1 is attached at most one pendant vertex. A \mathcal F∘g, l-graph is a \mathcal Fg, l-graph in which vg is attached with pendant vertex. Denote by qmin the least Q-eigenvalue 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 \mathcal Fg, l-graph for some l; (iii) among the nonbipartite graphs of order n and with given domination number which contain a \mathcal 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), (n-1)/(2), the minimum qmin is completely determined respectively; (v) among the nonbipartite graphs of order n≥ 4, and with both domination number (n+1)/(3)<γ≤ (n)/(2) and odd-girth (the length of the shortest odd cycle) at most 5, the minimum qmin is completely determined.