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"If R is a commutative ring with unity and the graph Γ2(R)\J(R) is n-partite, then the number of maximal ideals of R is at most n." The proof of this result is not correct. In this paper we present a correct proof for this result. Also we generalize some results given in the aforementioned paper for the non-commutative rings.

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.