In this paper, we present certain algorithms for clustering the vertices of fuzzy graphs(FGs) and intuitionistic fuzzy graphs(IFGs). These algorithms are based on the edge density of the given graph. We apply the algorithms to practical problems to derive the most prominent cluster among them. We also introduce parameters for intuitionistic fuzzy graphs.

Abstract. In this paper, we introduced a definition of edge domination set using strong edges and edge independent sets of intuitionistic fuzzy graphs. We determine the domination number ) (G γ and edge domination number ) ( ' G γ for several classes of intuitionistic fuzzy graphs and the relation between them are discussed. Also we introduce a regular dominating set and regular independent set in intuitionistic fuzzy graphs with suitable illustrations.

In this paper, we introduce the notion of balanced intuitionistic fuzzy graphs and present some of their properties. We also prove that G 1 G 2 is balanced if and only if D(G 1) = D(G 2) = D(G 1 G 2).

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A Hamiltonian decomposition of G is a partition of its edge set into disjoint Hamilton cycles. Manikandan and Paulraja conjectured that if G and H are Hamilton cycle decomposable circulant graphs with at least one of them is nonbipartite, then their tensor product is Hamilton cycle decomposable. In this paper, we have proved that, if G is a Hamilton cycle decomposable circulant graph with certain properties and H is a Hamilton cycle decomposable multigraph, then their tensor product is Hamilton cycle decomposable. In particular, tensor products of certain sparse Hamilton cycle decomposable circulant graphs are Hamilton cycle decomposable. * A circulant X = Circ(n; L) is a graph with vertex set V(X) = {u 0 , u 1 , . . . , u n−1 } and edge set E(X) = {u i u i+ℓ | i ∈ Z n , ℓ ∈ L}, where L ⊆ 1, 2, . . . , ⌊ n 2 ⌋ and Z n is the set of integers modulo n. The elements of L are called jumps. Clearly, every circulant graph of order n is a Cayley graph with the underlying group being Z n .The problem of finding Hamilton cycle decompositions of product graphs is not new. Hamilton cycle decompositions of various product graphs, including digraphs, have been studied by many authors; see, for example, [1,6,[9][10][11][13][14][15][16][17][18]. It has been conjectured [6] that if both G and H are Hamilton cycle decomposable graphs, then G H is Hamilton cycle decomposable, where denotes the cartesian product of graphs. This conjecture has been verified to be true for a large classes of graphs [18]. Baranyai and Szász [5] proved that if both G and H are even regular Hamilton cycle decomposable graphs, then G • H is Hamilton cycle decomposable. In [16], Ng has obtained a partial solution to the following conjecture of Alspach et al. [1]: If D 1 and D 2 are directed Hamilton cycle decomposable digraphs, then D 1 • D 2 is directed Hamilton cycle decomposable. Jha [10] conjectured the following Conjecture 1.1 [10] If both G and H are Hamilton cycle decomposable graphs and G × H is connected, then G × H is Hamilton cycle decomposable. Conjecture 1.1 was disproved, see [3]. Because of this, finding Hamilton cycle decompositions of the tensor products of Hamilton cycle decomposable graphs is considered to be difficult. Though Conjecture 1.1 has been disproved, we believe that if the graphs G and H are suitably chosen, that is, with some suitable conditions imposed on them, then G × H may have Hamilton cycle decomposition. In [2] and [13] it has been proved that K r × K s and K r,r × K m are Hamilton cycle decomposable; in [14] it is shown that the tensor product of two regular complete multipartite graphs is Hamilton cycle decomposable. Hamilton cycle decompositions of the tensor products of complete bipartite graphs and complete multipartite graphs are dealt with in [15]. Also in [17], Paulraja and Sivasankar proved that are directed Hamilton cycle decomposable. It can be observed that K r , K r, r , K r • K s are circulant graphs. Based on the results of [10, 11, 13-15], Manikandan and Paulraja conjectured the following: Conjecture 1...