Geodesic
Chromatic number and some multiplicative vertex-degree-based indices of graphs
Kragujevac Journal of Mathematics, Tome 36 (2012) no. 2, p. 323 .

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For a (molecular) graph, the first and second Zagreb indices ($M_1$ and $M_2$) are two well-known topological indices in chemical graph theory introduced in 1972 by Gutman and Trinajstić. Multiplicative versions of Zagreb indices, such as Narumi-Katayama index, multiplicative Zagreb index and multiplicative sum Zagreb index, have been much studied in the past. Let ${\bf{G}}(n,k)$ be the set of connected graphs of order $n$ and with chromatic number $k$. In this paper we show that, in ${\bf{G}}(n,k)$, Turán graph $T_n(k)$ has the maximal Narumi-Katayama index, the maximal multiplicative Zagreb index and the maximal multiplicative sum Zagreb index. And the extremal graphs from ${\bf{G}}(n,k)$ with $k=2$ or $3$ are determined with minimal values of these above indices.
Classification : 05C07 05C15
Keywords: Vertex degree, Zagreb index, Multiplicative Zagreb index, Chromatic number 
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     title = {Chromatic number and some multiplicative vertex-degree-based indices of graphs},
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Kexiang Xu; Kechao Tang; Kinkar Ch. Das; Huansong Yue. Chromatic number and some multiplicative vertex-degree-based indices of graphs. Kragujevac Journal of Mathematics, Tome 36 (2012) no. 2, p. 323 . http://geodesic.mathdoc.fr/item/KJM_2012_36_2_a15/