On the Sumset Labeling of Graphs N. K. Sudev Department of Mathematics Vidya Academy of Science & Technology Thalakkottukara, Thrissur - 680501, Kerala, India. E-mail: sudevnk@gmail.com Abstract For a positive integer n, let Z n be the set of all non-negative integers modulo n and P(Z n ) be its power set. A sumset valuation or a sumset labeling of a given graph G is an injective function f : V (G) → P(Z n ) such that the induced function f + : E(G) → P(Z n ) defined by f + (uv)= f (u)+ f (v). A sumset indexer of a graph G is an injective sumset valued function f : V (G) → P(Z n ) such that the induced function f + : E(G) → P(Z n ) is also injective. In this paper, some properties and characteristics of this type of sumset labeling of graphs are being studied. Key Words: Sumset graphs; weak sumset graphs; strong sumset graphs, maximal sumset graphs exquisite sumset graphs; sumset number of a graph. Mathematics Subject Classification: 05C78. 1 Introduction For all terms and definitions, not defined specifically in this paper, we refer to [3], [9] and [15]. For graph classes, we further refer to [4], [6] and [16] and for notions and results in number theory, we refer to [2] and [10]. Unless mentioned otherwise, all graphs considered here are simple, finite and have no isolated vertices. Let A and B be two sets. The sumset of A and B is denoted by A + B and is defined as A + B = {a + b : a ∈ A, b ∈ B}. If either A or B is countably infinite, then A + B is also countably infinite. We denote the cardinality of a set A by |A|. Then, we have the following theorem on the cardinality of the sumset of two sets. Theorem 1.1. [10] For two non-empty sets A and B, |A| + |B|− 1 ≤|A + B|≤ |A||B|. Another theorem on sumsets of two sets of integers proved in [10] is given below. Theorem 1.2. Let A and B be two non-empty sets of integers. Then, |A + B| = |A| + |B|− 1 if and only if A and B are arithmetic progressions with the same common difference. 1