Suppose that  is a finite group and  is a non-empty subset of  such that  and . Suppose that  is the Cayley graph whose vertices are all elements of  and two vertices  and  are adjacent if and only if . In this paper, we introduce the generalized Cayley graph denoted by  that  is a graph with vertex set consists of all column matrices  which all components are in  and two vertices  and  are adjacent if and only if , where  is a column matrix that each entry is the inverse of similar entry of  and  is  matrix with all entries in  ,  is the transpose of  and . In this paper, we clarify some basic properties of the new graph and assign the structure of  when  is complete graph , complete bipartite graph  and complete 3-partite graph  for every .

We introduce a new generalization of Cayley graphs. Moreover, we establish some basic properties of this new type of graph and we determine its structure under some assumptions.

In this paper, we study the notion of fuzzy sub-implicative ideal of a BH-algebra and we state and prove some theorems and propositions which determine the relationships among the sub-implicative ideal with the other subset in fuzzy senses of BH-algebra and also we give some properties of this ideal and relate it with other types of concepts of a BH-algebra.

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