If G is a graph without isolated vertices, and if rand s are positive integers, then the (r, s)-domination number 'Yr,s(G) of G is the cardinality of a smallest vertex set D such that every vertex not in D is within distance r from some vertex in D, while every vertex in D is within distance s from another vertex in D. This generalizes the total domination number 'Yt(G) = 'Yl,l(G). Let T( G) de...

For a simple graph $G$ with vertex set $V(G)=\{v_1,...,v_n\}$, we define the closed neighborhood of $u$ as \\$N[u]=\{v \in V(G) \; | v \text{is adjacent to} u \text{or} v=u \}$ and matrix $N(G)$ whose $i$th column is characteristic vector $N[v_i]$. We say $S$ odd dominating if $N[u]\cap S$ for all $u\in V(G)$. prove that parity cardinality an equal to rank $G$, where defined dimension space $N(...

Let [Formula: see text] be a graph. A set [Formula: see text] is a distance k-dominating set of G if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text], where k is a positive integer. The distance k-domination number [Formula: see text] of G is the minimum cardinality among all distance k-dominating sets of G. The first Zagreb index of G is defined as ...