A k -ended tree is a with at most leaves. In this note, we give simple proof for the following theorem. Let G be connected graph and an integer ( ≥ 2 ). S vertex subset of such that α ) ≤ + κ − 1 . Then, has which covers Moreover, condition sharp.

An independent broadcast on a connected graph $G$is a function $f:V(G)to mathbb{N}_0$such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$,and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$.The broadcast independence number $alpha_b(G)$ of $G$is the largest weight $sumlimits_{xin V(G)}f(x)$of an ind...

The domination subdivision number sdγ(G) of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upp...

The k-domination number γk(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is either in D or adjacent to at least k vertices in D. In 2010, the conjecture-generating computer program, Graffiti.pc, was queried for upperbounds on the 2-domination number. In this paper we prove new upper bounds on the 2-domination number of a g...

1.1. The independence ratio and the minimum eigenvalue. An independent set is a set of vertices in a graph, no two of which are adjacent. The independence ratio of a graph G is the size of its largest independent set divided by the total number of vertices. If G is regular, then the independence ratio is at most 1/2, and it is equal to 1/2 if and only if G is bipartite. The adjacency matrix of ...