Comer introduced a class of hypergroups, using the name of polygroups. He emphasized the importance of polygroups, by analyzing them in connections to graphs, relations, Boolean and cylindric algebras. Indeed, polygroups are multi valued systems that satisfy group like axioms. Given a polygroup with a finite generating set, we can form a Cayley Hypergraph for that polygroup with respect to that generating set. This helps us to better understand and investigate polygroup structures. More precisely,in this paper, we introduce the construction of Cayley Hypergraphs over polygroups, say $CH(\mathbf{P},S)$ such that $\mathbf{P}$ is a polygroup and $\langle S\rangle =P$. We investigate some properties of them. It is well known to give a constructing for building a big polygroup from two small ones. This structure is called extensionof polygroups. In particular, we describe the connection between Cayley Hypergraphs over extension of two polygroups and Cartesian product of two Cayley Hypergraphs.

For an associated combinatorial structure with Lie algebra $\mathbf{g}_n$ of upper triangular matrices, an allowable, forbidden, and the graphs that are not associated with $\mathbf{g}_n$ of any three vertices are determined. This work also introduces a neoteric association of Hypergraph with Lie algebra of upper triangular matrix $\mathcal{G}_n$ for an element of Lie algebra $\mathbf{g}_n$. The properties of this structure are analyzed, characterized and have been presented as an algorithm for finite order.

In a visual cryptography scheme, a secret image is encoded inton shares, in the form of transparencies. The shares are then distributed to n participants. Qualified subsets of participants can recover the secret image by superimposing their transparencies, but non-qualified subsets of participants have no information about the secret image. Pixel expansion, which represents the number of subpixels in the encoding of the secret image, should be as small as possible. Optimal schemes are those that have the minimum pixel expansion. In this paper we study the pixel expansion of Hypergraph access structures and introduce a number of upper bounds on the pixel expansion of special kinds of access structures. Also we demonstrate the minimum pixel expansion of induced matching Hypergraph is sharp when every qualified subset is exactly one edge with odd size. Furthermore we explain that the minimum pixel expansion of every graph access structure Pn is exactly [n+1/2]. It indicates the lower bound mentioned in [4] is sharp.

We use the recursive method of construction large sets of t-designs given by Qiu-rong Wu (A note on extending t-designs, Australas. J. Combin., 4 (1991) 229 {235.), and present a similar method for constructingt -subset-regular self-complementary k -uniform Hypergraphs of order v. As an application we show the existence of a new family of 2-subset-regular self-complementary 4-uniform Hypergraphs with u=16 m+3.