scholarly journals Minimal Non-Odd-Transversal Hypergraphs and Minimal Non-Odd-Bipartite Hypergraphs

10.37236/9519 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Yi-Zheng Fan ◽  
Yi Wang ◽  
Jiang-Chao Wan
Keyword(s):  
Incidence Matrix ◽  
Upper Bounds ◽  
Simple Graphs ◽  
Adjacency Tensor ◽  
Uniform Hypergraphs ◽  
Equivalent Characterization ◽  
Vertex Set

Among all uniform hypergraphs with even uniformity, the odd-transversal or odd-bipartite hypergraphs are closer to bipartite simple graphs than bipartite hypergraphs from the viewpoint of both structure and spectrum. A hypergraph is called odd-transversal if it contains a subset of the vertex set such that each edge intersects the subset in an odd number of vertices, and it is called minimal non-odd-transversal if it is not odd-transversal but deleting any edge results in an odd-transversal hypergraph. In this paper we give an equivalent characterization of the minimal non-odd-transversal hypergraphs by means of the degrees and the rank of its incidence matrix over $\mathbb{Z}_2$. If a minimal non-odd-transversal hypergraph is uniform, then it has even uniformity, and hence is minimal non-odd-bipartite. We characterize $2$-regular uniform  minimal non-odd-bipartite hypergraphs, and give some examples of $d$-regular uniform hypergraphs which are minimal non-odd-bipartite. Finally we give upper bounds for the least H-eigenvalue of the adjacency tensor of minimal non-odd-bipartite hypergraphs.

scholarly journals The Bounds of the Edge Number in Generalized Hypertrees

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 2 ◽  
Author(s):  
Ke Zhang ◽  
Haixing Zhao ◽  
Zhonglin Ye ◽  
Yu Zhu ◽  
Liang Wei
Keyword(s):  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Number Of Components ◽  
Uniform Hypergraphs ◽  
Vertex Set

A hypergraph H = ( V , ε ) is a pair consisting of a vertex set V , and a set ε of subsets (the hyperedges of H ) of V . A hypergraph H is r -uniform if all the hyperedges of H have the same cardinality r . Let H be an r -uniform hypergraph, we generalize the concept of trees for r -uniform hypergraphs. We say that an r -uniform hypergraph H is a generalized hypertree ( G H T ) if H is disconnected after removing any hyperedge E , and the number of components of G H T − E is a fixed value k   ( 2 ≤ k ≤ r ) . We focus on the case that G H T − E has exactly two components. An edge-minimal G H T is a G H T whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r -uniform G H T on n vertices has at least 2 n / ( r + 1 ) edges and it has at most n − r + 1 edges if r ≥ 3   and   n ≥ 3 , and the lower and upper bounds on the edge number are sharp. We then discuss the case that G H T − E has exactly k   ( 2 ≤ k ≤ r − 1 ) components.

scholarly journals Vertex decomposable graph

2016 ◽  
Vol 99 (113) ◽  
pp. 203-209 ◽  
Author(s):  
N. Hajisharifi ◽  
S. Yassemi
Keyword(s):  
Simple Graph ◽  
Simple Graphs ◽  
Vertex Set ◽  
Vertex Decomposable ◽  
Vertex Sets

Let G be a simple graph on the vertex set V (G) and S = {x11,...,xn1} a subset of V (G). Let m1,...,mn ? 2 be integers and G1,...,Gn connected simple graphs on the vertex sets V (Gi) = {xi1,..., ximi} for i = 1,..., n. The graph G(G1,...,Gn) is obtained from G by attaching Gi to G at the vertex xi1 for i = 1,...,n. We give a characterization of G(G1,...,Gn) for being vertex decomposable. This generalizes a result due to Mousivand, Seyed Fakhari, and Yassemi.

Matrix Rational Completions Satisfying Generalized Incidence Equations

1965 ◽  
Vol 17 ◽  
pp. 1-12 ◽  
Author(s):  
E. C. Johnsen
Keyword(s):  
Incidence Matrix ◽  
Equivalent Characterization

Let us consider the following problem. Let there be v elements x1 , . . . , xv and v sets S1, . . . , Sv such that every set contains exactly k distinct elements and every pair of sets has exactly λ distinct elements in common. To avoid trivial situations we shall in general assume that 0 < λ < k < v — 1. This is known as a v, k, λconfiguration or design. We can give an equivalent characterization of a configuration in terms of a matrix A = [aij], called its incidence matrix, by writing the elements x1 , . . . , xv row and the sets S1, . . . , Sv in a column and setting aij = 1 if xj is in Si and aij = 0 if xj is not.

scholarly journals On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 116
Author(s):  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
The Other ◽  
Inner Product ◽  
Equivalent Characterization ◽  
Geometric Constant ◽  
Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

Induced L-bornological vector spaces and L-Mackey convergence1

2021 ◽  
Vol 40 (1) ◽  
pp. 1277-1285
Author(s):  
Zhen-yu Jin ◽  
Cong-hua Yan
Keyword(s):  
Complete Lattice ◽  
Vector Spaces ◽  
Specific Description ◽  
Equivalent Characterization ◽  
Fuzzy Setting

Motivated by the concept of lattice-bornological vector spaces of J. Paseka, S. Solovyov and M. Stehlík, which extends bornological vector spaces to the fuzzy setting over a complete lattice, this paper continues to study the theory of L-bornological vector spaces. The specific description of L-bornological vector spaces is presented, some properties of Lowen functors between the category of bornological vector spaces and the category of L-bornological vector spaces are discussed. In addition, the notions and some properties of L-Mackey convergence and separation in L-bornological vector spaces are showed. The equivalent characterization of separation in L-bornological vector spaces in terms of L-Mackey convergence is obtained in particular.

scholarly journals Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 484
Author(s):  
Anita Keszler ◽  
Zsolt Tuza
Keyword(s):  
Element Subset ◽  
Uniform Hypergraphs ◽  
Vertex Set ◽  
Recursive Constructions

In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C(r,k,v) of order v is a collection of r-uniform k-cycles on a v-element vertex set, such that each r-element subset is an edge in precisely one of those k-cycles. We present cyclic hypercycle systems C(3,5,v) of orders v=25,26,31,35,37,41,46,47,55,56, a highly symmetric construction for v=40, and cyclic 2-split constructions of orders 32,40,50,52. As a consequence, all orders v≤60 permitted by the divisibility conditions admit a C(3,5,v) system. New recursive constructions are also introduced.

scholarly journals Tight Quotients and Double Quotients in the Bruhat Order

10.37236/1871 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
John R. Stembridge
Keyword(s):  
Coxeter Groups ◽  
Upper Bounds ◽  
Bruhat Order ◽  
Weyl Groups ◽  
Order Dimension ◽  
Parabolic Subgroups ◽  
Affine Weyl Groups ◽  
Positive Side ◽  
Maximal Parabolic Subgroups

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., "double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are "tight" in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups.

scholarly journals New Results on Degree Sequences of Uniform Hypergraphs

10.37236/3414 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Sarah Behrens ◽  
Catherine Erbes ◽  
Michael Ferrara ◽  
Stephen G. Hartke ◽  
Benjamin Reiniger ◽  
...  
Keyword(s):  
Efficient Algorithm ◽  
Necessary Conditions ◽  
Degree Sequence ◽  
Sufficient Conditions ◽  
Degree Sequences ◽  
Uniform Hypergraph ◽  
Graphic Sequence ◽  
Uniform Hypergraphs ◽  
Open Question

A sequence of nonnegative integers is $k$-graphic if it is the degree sequence of a $k$-uniform hypergraph. The only known characterization of $k$-graphic sequences is due to Dewdney in 1975. As this characterization does not yield an efficient algorithm, it is a fundamental open question to determine a more practical characterization. While several necessary conditions appear in the literature, there are few conditions that imply a sequence is $k$-graphic. In light of this, we present sharp sufficient conditions for $k$-graphicality based on a sequence's length and degree sum.Kocay and Li gave a family of edge exchanges (an extension of 2-switches) that could be used to transform one realization of a 3-graphic sequence into any other realization. We extend their result to $k$-graphic sequences for all $k \geq 3$. Finally we give several applications of edge exchanges in hypergraphs, including generalizing a result of Busch et al. on packing graphic sequences.

scholarly journals Monochromatic Path and Cycle Partitions in Hypergraphs

10.37236/2631 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
András Gyárfás ◽  
Gábor N. Sárközy
Keyword(s):  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Vertex Set

Here we address the problem to partition edge colored hypergraphs by monochromatic paths and cycles generalizing a well-known similar problem for graphs.We show that $r$-colored $r$-uniform complete hypergraphs can be partitioned into monochromatic Berge-paths of distinct colors. Also, apart from $2k-5$ vertices, $2$-colored $k$-uniform hypergraphs can be partitioned into two monochromatic loose paths.In general, we prove that in any $r$-coloring of a $k$-uniform hypergraph there is a partition of the vertex set intomonochromatic loose cycles such that their number depends only on $r$ and $k$.

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