Leap Zagreb Indices of Trees and Unicyclic Graphs

Abstract

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. ‎{\bf 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.

Keywords

• Leap Zagreb index‎
• ‎Zagreb index‎
• ‎degree (of vertex)