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.