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

Entire Zagreb indices of graphs

The Zagreb indices have been introduced in 1972 to explain some properties of chemical compounds at molecular level mathematically. Since then, the Zagreb indices have been studied extensively due to their ease of calculation and their numerous applications in place of the existing chemical methods which needed more time and increased the costs. Many new kinds of Zagreb indices are recently introduced for several similar reasons. In this paper, we introduce the entire Zagreb indices by adding incidency of edges and vertices to the adjacency of the vertices. Our motivation in doing so was the following fact about molecular graphs: The intermolecular forces do not only exist between the atoms, but also between the atoms and bonds, so one should also take into account the relations (forces) between edges and vertices in addition to the relations between vertices to obtain better approximations to intermolecular forces. Exact values of these indices for some families of graphs are obtained and some important properties of the entire Zagreb indices are established