In this paper, we determine all harmonious graphs of order 6. All graphs in this paper are finite, simple and undirected. We shall use the basic notation and terminology of graph theory as in [1]. A graph G of vertex set V(G) and edge set E(G) and of size q = |E(G)| is said to be harmonious [2] if there exists an injective function f, called a harmonious labeling, f:V(G)→Z_q (=the group of integers modulo q) such that the induced function F~*:E(G)→Z_q defined by f~*(xy) = f(x) + f(y), for all xy ∈ E(G) is again injective. The function f is the vertex labeling function and the function f~* is the corresponding edge labeling function. The image of the function f (= I_m(f)) is called the corresponding set of vertex labels. This definition extends to the case where G is a tree by allowing exactly two vertices to have the same value under f. Graham and Sloana [3] determined the harmonious graphs of order ≤5. In this paper we extend this result to graphs of order 6. This paper is divided into two sections. In Section 1 we accumulate the results needed to establish our main theorems. In Section 2 we obtain our main theorems, Theorem 2.1 (resp. Theorem 2.2) which determine all connected (resp. disconnected) harmonious graphs of order 6.
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