For a positive integer k, a set of vertices S in a graph G is said to be a k-dominating set if each vertex x in V (G) − S has at least k neighbors in S. The cardinality of a smallest k-dominating set of G is called the k-domination number of G and is denoted by γk(G). The independence number of a graph G is denoted by α(G). In [Australas. J. Combin. 40 (2008), 265–268], Fujisawa, Hansberg, Kubo...

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Caro, Davila, and Pepper recently proved δ(G)α(G)≤Δ(G)μ(G) for every graph G with minimum degree δ(G), maximum Δ(G), independence number α(G), matching μ(G). Answering some problems they posed, we characterize the extremal graphs δ(G)<Δ(G) as well δ(G)=Δ(G)=3.

let $g=(v,e)$ be a simple graph. a set $ssubseteq v$ isindependent set of $g$,  if no two vertices of $s$ are adjacent.the  independence number $alpha(g)$ is the size of a maximumindependent set in the graph. in this paper we study and characterize the independent sets ofthe zero-divisor graph $gamma(r)$ and ideal-based zero-divisor graph $gamma_i(r)$of a commutative ring $r$.