For a (molecular) graph G, the first and the second entire Zagreb indices are defined by the formulas M1εG=∑x∈VG∪EGdx2 and M2εG=∑xiseitheradjacentorincidenttoydxdy in which dx represents the degree of a vertex or an edge x. In the current manuscript, we establish some lower bounds on the first and the second entire Zagreb indices and determine the extremal trees which achieve these bounds.

Let k be a positive integer, and set $\left[k\right]:=\{1,2,\dots ,k\}$ . For a graph G, a k-rainbow dominating function (or kRDF) of G is a mapping $f:$ $V\left(G\right)\to 2^{\left[k\right]}$ in such a way that, for any vertex $v\in V\left(G\right)$ with the empty set under f, the condition ${\bigcup }_{u\in N_G\left(v\right)}f\left(u\right)=\left[k\right]$ always holds, where $N_G\left(v\right)$ is the open neighborhood of v. The weight of kRDF f of G is the summation of values of all vertices under f. The k-rainbow domination number of G, denoted by ${\gamma }_{rk}\left(G\right)$ , is the minimum weight of a kRDF of G. In this paper, we obtain the k-rainbow domination number of grid $P_3\square P_n$ for $k\in \{2,3,4\}$ .

We investigate some relationships between two vastly studied parameters of a simple graph G. These parameters include mixed domination number (denoted by gamma(m)(G)) and 2-independence number (beta(2) (G)). For a tree T, we obtain 3/4 beta(2) (T) >= gamma(m)(T) and characterized all those trees which attain the equality.