A study on some properties of leap graphs

In a graph G, the first and second degrees of a vertex v is equal to thenumber of their first and second neighbors and are denoted by d(v/G) andd 2 (v/G), respectively. The first, second and third leap Zagreb indices are thesum 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 firstand second degrees of vertices of G, respectively. In this paper, we initiate in studying a new class of graphs depending on the relationship between firstand second degrees of vertices and is so-called a leap graph. Some propertiesof the leap graphs are presented. All leap trees and {C 3, C 4 }-free leap graphsare characterized.