On arithmetic graphs

Abstract

A (p, q)-graph G = (V, E) is said to be (k, d)-arithmetic, where k and d are positive integers if its p vertices admits a labeling of distinct non-negative integers such that the values of the edges obtained as the sums of the labels of their end vertices form the set {k, k + d, k + 2d, ..., k + (q - 1)d}. In this paper we prove that for all odd n, the generalized web graph W (t, n) and some cycle related graphs are (k, d)-arithmetic. Also we prove that a class of trees called Tp-trees and subdivision of Tp-trees are (k + q - 1) (d, d)-arithmetic for all positive integers k and d.