Let G = (V, E) be a (molecular) graph. For a family of graphs G, the first Zagreb index M ₁ and the second Zagreb index M ₂ have already studied. In particular, it has been presented, the first Zagreb index M ₁ and the second Zagreb index M ₂ of trees T in terms of domination parameter. In this paper, we present upper bounds on Zagreb indices of unicyclic and bicyclic graphs with a given domination number and also find upper bounds on the Zagreb indices of trees, unicyclic, and bicyclic graphs with a given total domination number.

We first determine the domination number for the zero-divisor graph of the product of two commutative rings with 1. We then calculate the domination number for the zero-divisor graph of any commutative artinian ring. Finally, we extend some of the results to non-commutative rings in which an element is a left zero-divisor if and only if it is a right zero-divisor.

For a graphG =(V(G),E(G)), an Italian dominating function (ID function)f:V(G) → {0,1,2} has the property that for every vertexv∈V(G) withf(v) = 0, eithervis adjacent to a vertex assigned 2 underforvis adjacent to least two vertices assigned 1 underf. The weight of an ID function is ∑_v∈V(G)f(v). The Italian domination number is the minimum weight taken over all ID functions ofG. In this paper, we initiate the study of a variant of ID functions. A restrained Italian dominating function (RID function)fofGis an ID function ofGfor which the subgraph induced by {v ∈V(G) |f(v) = 0} has no isolated vertices, and the restrained Italian domination numberγ_rI(G) is the minimum weight taken over all RID functions ofG. We first prove that the problem of computing this parameter is NP-hard, even when restricted to bipartite graphs and chordal graphs as well as planar graphs with maximum degree five. We prove that γrI(T) for a treeTof ordern≥ 3 different from the double starS_2,2can be bounded from below by (n+ 3)/2. Moreover, all extremal trees for this lower bound are characterized in this paper. We also give some sharp bounds on this parameter for general graphs and give the characterizations of graphsGwith small or largeγ_rI(G).

Let G = (V, E) be a graph. A set S ⊆ V is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this article is to study and characterize the dominating sets of the zero-divisor graph Γ(R) and ideal-based zero-divisor graph Γ _I (R) of a commutative ring R.