University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 11 4 2020 12 01 On the Modified First Zagreb Connection Index of Trees of a Fixed Order and Number of Branching Vertices 213 226 110827 10.22052/ijmc.2020.240260.1514 EN Sadia Noureen National University of Computer and Emerging Sciences, Lahore, Pakistan Akhlaq Ahmad Bhatti National University of Computer and Emerging Sciences, Lahore, Pakistan Akbar Ali University of Hail, Hail, Saudi Arabia Journal Article 2020 08 30 The modified first Zagreb connection index $ZC_{1}^{*}$ for a graph $G$ is defined as $ZC_{1}^{*}(G)= \sum_{v\in V(G)}d_{v}\tau_{v}\,$, where $d_{v}$ is degree of the vertex $v$ and $\tau _{v}$ is the connection number of $v$ (that is, the number of vertices having distance 2 from $v$). By an $n$-vertex graph, we mean a graph of order $n$. A branching vertex of a graph is a vertex with degree greater than $2$. In this paper, the graphs with maximum and minimum $ZC_{1}^{*}$ values are characterized from the class of all $n$-vertex trees with a fixed number of branching vertices. https://ijmc.kashanu.ac.ir/article_110827_9963a63de0e8366746c57fc8870654ad.pdf