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Multiplicative Zagreb indices of $$k$$-trees. (English) Zbl 1303.05034
Summary: Let $$G$$ be a graph with vertex set $$V(G)$$ and edge set $$E(G)$$. The first generalized multiplicative Zagreb index of $$G$$ is $$\prod_{1, c}(G) = \prod_{v \in V(G)} d(v)^c$$, for a real number $$c > 0$$, and the second multiplicative Zagreb index is $$\prod_2(G) = \prod_{u v \in E(G)} d(u) d(v)$$, where $$d(u), d(v)$$ are the degrees of the vertices of $$u, v$$. The multiplicative Zagreb indices have been the focus of considerable research in computational chemistry dating back to Narumi and Katayama in 1980s. In this paper, we generalize Narumi-Katayama index and the first multiplicative index, where $$c = 1, 2$$, respectively, and extend the results of Gutman to the generalized tree, the $$k$$-tree, where the results of Gutman are for $$k = 1$$. Additionally, we characterize the extremal graphs and determine the exact bounds of these indices of $$k$$-trees, which attain the lower and upper bounds.

MSC:
 05C07 Vertex degrees 05C05 Trees 92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
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References:
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