Leap Zagreb indices of trees and unicyclic graphs

Abstract

By d(v|G) and d2(v|G) are denoted the number of first and second neighbors of the vertex v of the graph G. The first, second, and third leap Zagreb indices of G are defined as LM1(G) = Pv∈V (G) d2(v|G)2, LM2(G) = Puv∈E(G) d2(u|G) d2(v|G), and LM3(G) = Pv∈V (G) d(v|G) d2(v|G), respectively. In this paper, we generalize the results of Naji et al. [Commun. Combin. Optim. 2 (2017), 99–117], pertaining to trees and unicyclic graphs. In addition, we determine upper and lower bounds on these leap Zagreb indices and characterize the extremal graphs.