(1, N)-arithmetic graphs

Abstract

A (p, q)-graph G is said to be (1, N)-arithmetic if there is a function from the vertex set V(G) to so that the values obtained as the sums of the labeling assigned to their end vertices, can be arranged in the arithmetic progression . In this paper, we prove that Stars, Paths, complete bipartite graph , highly irregular graph and Cycle are (1, N)-arithmetic, is not (1, N)-arithmetic. We also prove that no graph G containing an odd cycle is (1, N)-arithmetic for every positive integer N.