On the Modified First Zagreb Connection Index of Trees of a Fixed Order and Number of Branching Vertices

The modified first Zagreb connection index $ZC_{1}^{*}$ for a graph $G$ is defined as $ZC_{1}^{*}(G)= sum_{vin V(G)}d_{v}tau_{v},$, 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.