We review the concepts of hypertree decomposition and hypertree width from a graph theoretical perspective and report on a number of recent results related to these concepts. We also show – as a new result – that computing hypertree decompositions is fixed-parameter intractable. 1 Hypertree Decompositions: Definition and Basics This paper reports about the recently introduced concept of hypertr...

Linde, Moore, and Nordahl introduced a generalisation of the honeycomb dimer model to higher dimensions. The purpose this article is describe number structural properties generalised model. First, it shown that samples are in one-to-one correspondence with perfect matchings hypergraph. This leads Kasteleyn theory: partition function equals Cayley hyperdeterminant adjacency hypermatrix Second, w...

Graph signal processing (GSP) techniques are powerful tools that model complex relationships within large datasets, being now used in a myriad of applications different areas including data science, communication networks, epidemiology, and sociology. Simple graphs can only pairwise among which prevents their application modeling networks with higher-order relationships. For this reason, some e...

Each t-norm can be identified with its Cayley tomonoid, which consists of pairwise commuting order-preserving functions from the real unit interval to itself. Cayley tomonoids provide an easily manageable, yet versatile tool for the construction of t-norms. To give evidence to this claim, we review and reformulate several construction methods that are known in the literature. We adopt, on the o...

For a finite group G and a subset S ⊆ G (possibly, S contains the identity of G), the bi-Cayley graph BCay(G, S) of G with respect to S is the graph with vertex set G × {0, 1} and with edge set {(x, 0), (sx, 1)|x ∈ G, s ∈ S}. A bi-Cayley graph BCay(G, S) is called a BCI-graph if, for any bi-Cayley graph BCay(G, T ), whenever BCay(G, S) ∼= BCay(G, T ) we have T = gS , for some g ∈ G, α ∈ Aut(G)....

We develop the general Theory of Cayley Hamilton algebras and we compare this with theory pseudocharacters. finally characterize prime T-ideals for discuss some aspects their geometry.

Let H=(V,E) be a hypergraph. A panchromatic t-colouring of H is a t-colouring of its vertices such that each edge has at least one vertex of each colour; and H is panchromatically t-choosable if, whenever each vertex is given a list of t colours, the vertices can be coloured from their lists in such a way that each edge receives at least t different colours. The Hall ratio of H is h(H) =min {∣∣...

For a graph G = (V,E), a subset F ⊂ V (G) is called an Rk-vertex-cut of G if G − F is disconnected and each vertex u ∈ V (G) − F has at least k neighbors in G − F . The Rk-vertex-connectivity of G, denoted by κ (G), is the cardinality of the minimum Rk-vertex-cut of G, which is a refined measure for the fault tolerance of network G. In this paper, we study κ2 for Cayley graphs generated by k-tr...

Fix integers t ≥ r ≥ 2 and an r-uniform hypergraph F . We prove that the maximum number of edges in a t-partite r-uniform hypergraph on n vertices that contains no copy of F is ct,F ( n r ) + o(nr), where ct,F can be determined by a finite computation. We explicitly define a sequence F1, F2, . . . of r-uniform hypergraphs, and prove that the maximum number of edges in a t-chromatic r-uniform hy...

Let H = (V, E) be a hypergraph, where V is set of vertices and E non-empty subsets called edges. If all edges have the same cardinality r, then an r-uniform hypergraph; if consists r-subsets V, complete denoted by $$K_n^r$$ , n |V|. A hypergraph H′ (V′, E′) subhypergraph V′ ⊆ E′ E. The edge-connectivity minimum edge F such that − not connected, F). An k-edge-maximal every has at most k, but for...