The ratio and product of the multiplicative Zagreb indices

IF 1 Q4 CHEMISTRY, MULTIDISCIPLINARY Iranian journal of mathematical chemistry Pub Date : 2017-12-01 DOI:10.22052/IJMC.2017.53731.1198

R. Kazemi

引用次数: 0

Abstract

‎The first multiplicative Zagreb index $Pi_1(G)$ is equal to the‎ ‎product of squares of the degree of the vertices and the second‎ ‎multiplicative Zagreb index $Pi_2(G)$ is equal to the product of‎ ‎the products of the degree of pairs of adjacent vertices of the‎ ‎underlying molecular graphs $G$‎. ‎Also‎, ‎the multiplicative sum Zagreb index $Pi_3(G)$ is equal to the product of‎ ‎the sums of the degree of pairs of adjacent vertices of $G$‎. ‎In‎ ‎this paper‎, ‎we introduce a new version of the multiplicative sum‎ ‎Zagreb index and study the moments of the ratio and product of ‎all above‎ indices in a randomly chosen molecular graph with tree structure of order $n$. ‏Also, a ‎supermartingale is introduced by ‎‎Doob's supermartingale‎ ‎inequality.

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相乘的萨格勒布指数的比值和乘积

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *。同样，萨格勒布指数的乘法和$Pi_3(G)$等于$相邻顶点对的度数和的乘积$G$ $。在本文中，我们引入了一个新版本的乘法和萨格勒布指数，并研究了随机选择的n阶树结构分子图中上述所有指数之比和积的矩。此外，一个“超鞅”是由Doob的“超鞅”不等式引入的。

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