On general reduced second Zagreb index of graphs

Authors : Batmend HOROLDAGVA, Lkhagva BUYANTOGTOKH, Kinkar Ch. DAS

Pages : 1046-1056

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Publication Date : 2019-08-08

Article Type : Research Paper

Abstract :Recently, Furtula et al. [B. Furtula, I. Gutman, S. Ediz, On difference of Zagreb indices, Discrete Appl. Math., 2014] introduced a new vertex-degree-based graph invariant `reduced second Zagreb index` in chemical graph theory. Here we generalize the reduced second Zagreb index insert ignore into journalissuearticles values(call `general reduced second Zagreb index`);, denoted by $GRM_{\alpha}insert ignore into journalissuearticles values(G);$ and is defined as: $GRM_\alphainsert ignore into journalissuearticles values(G);=\sum_{uv\in Einsert ignore into journalissuearticles values(G);}insert ignore into journalissuearticles values(d_Ginsert ignore into journalissuearticles values(u);+\alpha);insert ignore into journalissuearticles values(d_Ginsert ignore into journalissuearticles values(v);+\alpha);,$ where $\alpha$ is any real number and $d_Ginsert ignore into journalissuearticles values(v);$ is the degree of the vertex $v$ of $G$. Let $\mathcal{G}_n^k$ be the set of connected graphs of order $n$ with $k$ cut edges. In this paper, we study some properties of $GRM_\alphainsert ignore into journalissuearticles values(G);$ for connected graphs $G$. Moreover, we obtain the sharp upper bounds on $GRM_\alphainsert ignore into journalissuearticles values(G);$ in $\mathcal{G}_n^{k}$ for $\alpha\geq-1/2$ and characterize the extremal graphs.
Keywords : Connected graph, Zagreb indices, general reduced second Zagreb index, cut edge, maximum degree