Abstract This chapter introduces three hypergraph computation paradigms, including intra-hypergraph computation, inter-hypergraph and structure computation. Intra-hypergraph representation aims to conduct learning of a hypergraph, where each subject is represented by its components. Inter-hypergraph vertices in the vertex hypergraph. Hypergraph prediction, which find connections among vertices....

More details can be found in Some Notes on Parametrizing Representations on the atlas web site. In particular this has formulas for computing Cayley transforms and cross actions in various coordinates. The main point is the formula for y. See Marc's email in the Appendix. We have fixed G, G ∨ , and T, T ∨ as usual. Let X * = X * (T), which is canonically identified with X * (T ∨).

It is well known that, whenever k divides n, the complete k-uniform hypergraph on n vertices can be partitioned into disjoint perfect matchings. Equivalently, set of k-subsets an n-set parallel classes so that each class a partition n-set. This result as Baranyai's theorem, which guarantees existence Baranyai partitions. Unfortunately, proof theorem uses network flow arguments, making this none...

We consider strongly regular graphs r = (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman-Singleton graph, and the triangular graphs T(q) with q = 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We...

We determine under which conditions certain natural models of random constraint satisfaction problems have sharp thresholds of satisfiability. These models include graph and hypergraph homomorphism, the (d, k, t)-model, and binary constraint satisfaction problems with domain size 3.

Let H be a 3-uniform hypergraph of order n and size m, and let T be a subset of vertices of H. The set T is a strong transversal in H if T contains at least two vertices from every edge of H. The strong transversal number τs(H) of H is the minimum size of a strong transversal in H. We show that 7τs(H) ≤ 4n+ 2m, and we characterize the hypergraphs that achieve equality in this bound. In particul...

Let H be a hypergraph with n vertices. Suppose that d1, d2, . . . , dn are degrees of the vertices of H. The t-th graph entropy based on degrees of H is defined as

An upper bound for sorting permutations with an operation estimates the diameter of the corresponding Cayley graph and an exact upper bound equals the diameter. Computing tight upper bounds for various operations is of theoretical and practical (e.g. interconnection networks, genetics) interest. Akers and Krishnamurthy gave a Ω(n! n) time method that examines n! permutations to compute an upper...

Given a 3-uniform hypergraph H consisting of set V vertices, and T?V3 triples, null labelling is an assignment ±1 to the triples such that each vertex contained in equal number labelled +1 ?1. Thus, signed degree zero. A necessary condition for every even. The Null Labelling Problem determine whether has labelling. It proved this problem NP-complete. Computer enumerations suggest most hypergrap...

In this paper, the concept of hypergraph is extended to the fuzzy hypergraph. In the fuzzy hypergraph, the concepts of a-cut hypergraph, strength of edge and dual fuzzy hypergraph are developed. It is shown that the fuzzy hypergraph and a-cut hypergraph are useful to represent a fuzzy partition. An application example also shows that the strength of edge can be used to decompose the data set in...