Bounds on the first leap Zagreb index of trees

The first leap Zagreb index $LM1(G)$ of a graph $G$ is the sum of the squares of its second vertex degrees, that is, $LM_1(G)=\sum_{v\in V(G)}d_2(v/G)^2$, where $d_2(v/G)$ is the number of second neighbors of $v$ in $G$. In this paper, we obtain bounds for the first leap Zagreb index of trees and determine the extremal trees achieving these bounds.