Leap Zagreb indices of trees and unicyclic graphs

By d(v|G) and d_2(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 LM_1(G) = sum_{v in V(G)} d_2(v|G)^2, LM_2(G) = sum_{uv in E(G)} d_2(u|G) d_2(v|G), and LM_3(G) = sum_{v in V(G)} d(v|G) d_2(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 for these leap Zagreb indices and characterize the extremal graphs.