Abstract The F-index F(G) of a graph G is obtained by the sum of cubes of the degrees of all the vertices in G. It is defined in the same paper of 1972 where the first and second Zagreb indices are introduced to study the structure-dependency of total π-electron energy. Recently, Furtula and Gutman [J. Math. Chem. 53 (2015), no. 4, 1184–1190] reinvestigated F-index and proved its various properties. A connected graph with order n and size m, such that m = n + 2, is called a tricyclic graph. In this paper, we characterize the extremal graphs and prove the ordering among the different subfamilies of graphs with respect to F-index in Ωnα $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$, where Ωnα $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$ is a complete class of tricyclic graphs with three, four, six and seven cycles, such that each graph has α ≥ 1 pendant vertices and n ≥ 16 + α order. Mainly, we prove the bounds (lower and upper) of F(G), i.e 8n+12α+76≤F(G)≤8(n−1)−7α+(α+6)3 for each G∈Ωnα. $$\begin{array}{} \displaystyle 8n+12\alpha +76\leq F(G)\leq 8(n-1)-7\alpha + (\alpha+6)^3 ~\mbox{for each}~ G\in {\it\Omega}^{\alpha}_n. \end{array}$$

中文翻译：

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具有固定悬垂顶点的三环图的 F-index 的边界

摘要 图 G 的 F 指数 F(G) 由 G 中所有顶点的度数的立方和得到。它在 1972 年的同一篇论文中定义，其中引入了第一和第二萨格勒布指数来研究总 π 电子能量的结构依赖性。最近，富图拉和古特曼 [J. 数学。化学 53 (2015)，没有。4, 1184–1190] 重新研究了 F 指数并证明了它的各种性质。阶为 n 且大小为 m 的连通图，使得 m = n + 2，称为三环图。在本文中，我们描述了极值图的特征，并证明了图的不同子族之间关于 Ωnα 中 F-index 的排序 $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \ end{array}$，其中 Ωnα $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$ 是具有三、四、六个和七个循环，这样每个图都有 α ≥ 1 个悬垂顶点和 n ≥ 16 + α 阶。主要是证明F(G)的上界和下界，即8n+12α+76≤F(G)≤8(n−1)−7α+(α+6)3 对于每个G∈Ωnα。$$\begin{array}{} \displaystyle 8n+12\alpha +76\leq F(G)\leq 8(n-1)-7\alpha + (\alpha+6)^3 ~\mbox{for每个}~ G\in {\it\Omega}^{\alpha}_n。\end{数组}$$