Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285120200601A study on some properties of leap graphs9171387610.22049/cco.2019.26430.1108ENAhmed MNajiDepartment of Mathematics, University of Mysore, Mysusu, India0000-0003-0007-8927B.DavvazDepartment of Mathematics, Yazd University, Yazd, IranSultan S.MahdeDepartment of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, IndiaN.D.SonerDepartment of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, IndiaJournal Article20190216In a graph $G$, the first and second degrees of a vertex $v$ are equal to the number of their first and second neighbors and are denoted by $d(v/G)$ and $d_2(v/G)$, respectively. The first, second and third leap Zagreb indices are the sum of squares of second degrees of vertices of $G$, the sum of products of second degrees of pairs of adjacent vertices in $G$ and the sum of products of first and second degrees of vertices of $G$, respectively. In this paper, we initiate in studying a new class of graphs depending on the relationship between first and second degrees of vertices and is so-called a leap graph. Some properties of the leap graphs are presented. All leap trees and ${C_3, C_4}$-free leap graphs are characterized.http://comb-opt.azaruniv.ac.ir/article_13876_3e34a313e1c9a12cdfc1edc950e25098.pdf