A t-Cayley hypergraph X = t-Cay(G; S) is called normal for a finite group G, if the right regular representationR(G) of G is normal in the full automorphism group Aut(X) of X. In this paper, we investigate the normality of t-Cayley hypergraphs of abelian groups, where S < 4.

We present a simple mechanism, which can be randomised, for constructing sparse 3-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over Z2 and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expans...

We give lower bounds on the maximum possible girth of an r-uniform, d-regular hypergraph with at most n vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between 3/2 + o(1) and 2 + o(1)). We also define a random r-uniform ‘Cayley’ hypergraph on the symmetric group Sn which has girth Ω...

‎let $s$ be a subset of a finite group $g$‎. ‎the bi-cayley graph ${rm bcay}(g,s)$ of $g$ with respect to $s$ is an undirected graph with vertex set $gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid xin g‎, ‎ sin s}$‎. ‎a bi-cayley graph ${rm bcay}(g,s)$ is called a bci-graph if for any bi-cayley graph ${rm bcay}(g,t)$‎, ‎whenever ${rm bcay}(g,s)cong {rm bcay}(g,t)$ we have $t=gs^alpha$ for some $...

We prove hypergraph variants of the celebrated Alon–Roichman theorem on spectral expansion of sparse random Cayley graphs. One of these variants implies that for every odd prime p and any ε > 0, there exists a set of directions D ⊆ Fp of size Op,ε(p (1−1/p+o(1))n) such that for every set A ⊆ Fp of density α, the fraction of lines in A with direction in D is within εα of the fraction of all line...

We propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. To this end we construct groupoids whose Cayley graphs have large girth not just in the usual sense, but with respect to a discounted distance measure that contracts arbitrarily long sequences of edges within the same sub-groupoid (coset) and only co...

We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r ≥ 3. Similar to the bijection between spanning trees of the complete graph on (n + 1) vertices and Parking functions of length n, we derive a bijection from spanning trees of the complete (r + 1)-uniform hypergraph which arise from a fixed r-perfect matching (see Section 2) and r-P...

We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r ≥ 3. Similar to the bijection between spanning trees in complete graphs and Parking functions, we derive a bijection from spanning trees of the complete (r + 1)-uniform hypergraph which arise from a fixed r-perfect matching (see Section 2) and r-Parking functions. We observe a simp...