Abstract Hypergraph modeling is the fundamental task in hypergraph computation, which targets on establishing a high-quality structure to accurately formulate high-order correlation among data. In this section, we introduce different methods show how build hypergraphs using various pieces of information, such as features, attributes, and/or graphs. These are organized into two broad categories,...

For a collection of (not necessarily distinct) matchingsM = (M1,M2, . . . ,Mq) in a hypergraph, where each matching is of size t, a matching M of size t contained in the union ∪i=1Mi is called a rainbow matching if there is an injective mapping from M to M assigning to each edge e of M a matching Mi ∈M containing e. Let f(r, t) denote the maximum k for which there exists a collection of k match...

‎let $g$ be a finite group‎. ‎an element $gin g$ is called non-vanishing‎, ‎if for‎ ‎every irreducible complex character $chi$ of $g$‎, ‎$chi(g)neq 0$‎. ‎the bi-cayley graph $bcay(g,t)$ of $g$ with respect to a subset $tsubseteq g$‎, ‎is an undirected graph with‎ ‎vertex set $gtimes{1,2}$ and edge set ${{(x,1),(tx,2)}mid xin g‎, ‎ tin t}$‎. ‎let $nv(g)$ be the set‎ ‎of all non-vanishing element...

In this paper we find a field such that the algebras obtained by the Cayley-Dickson process are division algebras of dimension 2t,∀t ∈ N. Subject Classification: 17D05; 17D99. From Frobenius Theorem and from the remark given by Bott and Milnor in 1958, we know that for n ∈ {1, 2, 4} we find the real division algebras over the real field R. These are: R, C, H(the real quaternion algebra), O(the ...

A graph is called symmetric if its full automorphism group acts transitively on the set of arcs. The Cayley graph $Gamma=Cay(G,S)$ on group $G$ is said to be normal symmetric if $N_A(R(G))=R(G)rtimes Aut(G,S)$ acts transitively on the set of arcs of $Gamma$. In this paper, we classify all connected tetravalent normal symmetric Cayley graphs of order $p^2q$ where $p>q$ are prime numbers.

in this work, using eigenvalues and eigenvectors of unitary cayley graphs over finite local rings and elementary linear algebra, we characterize which local rings allowing pst occurring in its unitary cayley graph. moreover, we have some developments when $r$ is a product of local rings.

A mixed hypergraph is a triple H = (X, C,D), where X is the vertex set, and each of C, D is a list of subsets of X. A strict k-coloring of H is a surjection c : X → {1, . . . , k} such that each member of C has two vertices assigned a common value and each member of D has two vertices assigned distinct values. The feasible set of H is {k : H has a strict k-coloring}. Among other results, we pro...

A mixed hypergraph is a triple H = (X, C,D) where X is the vertex set and each of C, D is a family of subsets of X, the C-edges and D-edges, respectively. A k-coloring of H is a mapping c : X → [k] such that each C-edge has two vertices with the same color and each D-edge has two vertices with distinct colors. H = (X, C,D) is called a mixed hypertree if there exists a tree T = (X, E) such that ...

It is exponentially unlikely that a sparse random graph or hypergraph is connected, but such graphs occur commonly as the giant components of larger random graphs. This simple observation allows us to estimate the number of connected graphs, and more generally the number of connected d-uniform hypergraphs, on n vertices with m = O(n) edges. We also estimate the probability that a binomial rando...