After introducing and developing fuzzy set theory, a lot of studies have been done in this field and then a result appeared as a Fuzzy Graph (Combination of graph theory and fuzzy set theory). This is now known as Fuzzy graph theory. In this article we review essential works on different types of fuzzy graph and fuzzy hyper graph. Mathematics Subject Classification (2010): 03E72, 05C72

The star valency of a graph G is the minimum, over all star decompositions , of the maximum number of elements in incident with a vertex. The maximum average degree of G, denoted by dmax-ave(G), is the maximum average degree of all subgraphs of G. In this paper, we prove that the star valency of G is either dmax-ave(G)=2 or dmax-ave(G)=2 +1, and provide a polynomial time algorithm for determini...

In this note we observe that in the hyper-graph removal lemma the edge removal can be done in a way that the symmetries of the original hyper-graph remain preserved. As an application we prove the following generalization of Szemerédi’s Theorem on arithmetic progressions. If in an Abelian group A there are sets S1, S2 . . . , St such that the number of arithmetic progressions x1, x2, . . . , xt...

We study the degenerate, the star and the degenerate star chromatic numbers and their relation to the genus of graphs. As a tool we prove the following strengthening of a result of Fertin et al. [8]: If G is a graph of maximum degree ∆, then G admits a degenerate star coloring using O(∆) colors. We use this result to prove that every graph of genus g admits a degenerate star coloring with O(g) ...

Broadcasting is an important collective communication operation in many applications which use parallel computing. In this paper, we focus on designing broadcasting algorithms for general incomplete star graphs. We propose two optimal one-to-all broadcasting algorithms for incomplete star graphs with a single-port communication model. An incomplete star graph with N nodes, where (n 1)! < N < n!...

This paper considers a kind of generalized measure of fault tolerance in the -star graph for and , and determines , which implies that at least edges of have to remove to get a disconnected graph that contains no vertices of degree less than . This result shows that the -star graph is robust when it is used to model the topological structure of a large-scale parallel processing system.

A p-star is a complete bipartite graph K1,p with one center node and p leaf nodes. In this paper we propose the first distributed self-stabilizing algorithm for graph decomposition into p-stars. For a graph G and an integer p ≥ 1, this decomposition provides disjoint components of G where each component forms a p-star. We prove convergence and correctness of the algorithm under an unfair distri...

The following statement was conjectured by Faudree, Rousseau, Schelp and Schuster: if a graph G is a non-star graph without cycles of length m 6 4 then G is a subgraph of its complement. So far the best result concerning this conjecture is that every non-star graph G without cycles of length m 6 6 is a subgraph of its complement. In this note we show that m 6 6 can be replaced by m 6 5.