Abstract
Let G = (V, E) be a (molecular) graph. For a family of graphs G, the first Zagreb index M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and the second Zagreb index M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> have already studied. In particular, it has been presented, the first Zagreb index M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and the second Zagreb index M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 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.
Highlights
Throughout this paper, all graphs are assumed to be simple connected, undirected with n ≥ 1 vertices and m edges
We obtain the upper bound on Zagreb indices of trees, unicycles and bicycles with a given total domination number
We obtain upper bound for Zagreb indices of unicyclic and bicyclic graphs with given domination number and maximum degree of graph G
Summary
Throughout this paper, all graphs are assumed to be simple connected, undirected with n ≥ 1 vertices and m edges. INDEX TERMS Domination, total domination, Zagreb indices, tree, unicyclic, bicyclic graph. Mojdeh et al.: Zagreb Indices of Trees, Unicyclic, and Bicyclic Graphs With Given (Total) Domination respectively if it has only one cycle (exactly two cycles) and denoted by Un (Bn).
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