Hypergraphs and hypermatrices with symmetric spectrum

It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to hypermatrices. To this end the concept of bipartiteness is generalized by a new monotone property of cubical hypermatrices, called odd-colorable matrices. It is shown that a nonnegative symmetric $r$-matrix $A$ has a symmetric spectrum if and only if $r$ is even and $A$ is odd-colorable. This result also solves a problem of Pearson and Zhang about hypergraphs with symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu. Separately, similar results are obtained for the $H$-spectram of hypermatrices.Comment: 17 pages. Corrected proof on p. 1

d-반순서의 경쟁그래프의 연구

학위논문 (박사)-- 서울대학교 대학원 : 사범대학 수학교육과, 2018. 2. 김서령.The \emph{competition graph} $C(D)$ of a digraph $D$ is defined to be a graph whose vertex set is the same as $D$ and which has an edge joining two distinct vertices $x$ and $y$ if and only if there are arcs $(x,z)$ and $(y,z)$ for some vertex $z$ in $D$. Competition graphs have been extensively studied for more than four decades. Cohen~\cite{cohen1968interval, cohen1977food, cohen1978food} empirically observed that most competition graphs of acyclic digraphs representing food webs are interval graphs. Roberts~\cite{roberts1978food} asked whether or not Cohen's observation was just an artifact of the construction, and then concluded that it was not by showing that if $G$ is an arbitrary graph, then $G$ together with additional isolated vertices as many as the number of edges of $G$ is the competition graph of some acyclic digraph. Then he asked for a characterization of acyclic digraphs whose competition graphs are interval graphs. Since then, the problem has remained elusive and it has been one of the basic open problems in the study of competition graphs. There have been a lot of efforts to settle the problem and some progress has been made. While Cho and Kim~\cite{cho2005class} tried to answer his question, they could show that the competition graphs of doubly partial orders are interval graphs. They also showed that an interval graph together with sufficiently many isolated vertices is the competition graph of a doubly partial order. In this thesis, we study the competition graphs of $d$-partial orders some of which generalize the results on the competition graphs of doubly partial orders. For a positive integer $d$, a digraph $D$ is called a \emph{$d$-partial order} if V(D) \subset \RR^d and there is an arc from a vertex $\mathbf{x}$ to a vertex $\mathbf{y}$ if and only if $\mathbf{x}$ is componentwise greater than $\mathbf{y}$. A doubly partial order is a $2$-partial order. We show that every graph $G$ is the competition graph of a $d$-partial order for some nonnegative integer $d$, call the smallest such $d$ the \emph{partial order competition dimension} of $G$, and denote it by $\dim_\text{poc}(G)$. This notion extends the statement that the competition graph of a doubly partial order is interval and the statement that any interval graph can be the competition graph of a doubly partial order as long as sufficiently many isolated vertices are added, which were proven by Cho and Kim~\cite{cho2005class}. Then we study the partial order competition dimensions of some interesting families of graphs. We also study the $m$-step competition graphs and the competition hypergraph of $d$-partial orders.1 Introduction 1 1.1 Basic notions in graph theory 1 1.2 Competition graphs 6 1.2.1 A brief history of competition graphs 6 1.2.2 Competition numbers 7 1.2.3 Interval competition graphs 10 1.3 Variants of competition graphs 14 1.3.1 m-step competition graphs 15 1.3.2 Competition hypergraphs 16 1.4 A preview of the thesis 18 2 On the competition graphs of d-partial orders 1 20 2.1 The notion of d-partial order 20 2.2 The competition graphs of d-partial orders 21 2.2.1 The regular (d − 1)-dimensional simplex △ d−1 (p) 22 2.2.2 A bijection from H d + to a set of regular (d − 1)-simplices 23 2.2.3 A characterization of the competition graphs of d-partial orders 25 2.2.4 Intersection graphs and competition graphs of d-partial orders 27 2.3 The partial order competition dimension of a graph 29 3 On the partial order competition dimensions of chordal graphs 2 38 3.1 Basic properties on the competition graphs of 3-partial orders 39 3.2 The partial order competition dimensions of diamond-free chordal graphs 42 3.3 Chordal graphs having partial order competition dimension greater than three 46 4 The partial order competition dimensions of bipartite graphs 3 53 4.1 Order types of two points in R 3 53 4.2 An upper bound for the the partial order competition dimension of a graph 57 4.3 Partial order competition dimensions of bipartite graphs 64 5 On the m-step competition graphs of d-partial orders 4 69 5.1 A characterization of the m-step competition graphs of dpartial orders 69 5.2 Partial order m-step competition dimensions of graphs 71 5.3 dim poc (Gm) in the aspect of dim poc (G) 76 5.4 Partial order competition exponents of graphs 79 6 On the competition hypergraphs of d-partial orders 5 81 6.1 A characterization of the competition hypergraphs of d-partial orders 81 6.2 The partial order competition hyper-dimension of a hypergraph 82 6.3 Interval competition hypergraphs 88 Abstract (in Korean) 99Docto

Realizability and uniqueness in graphs

AbstractConsider a finite graph G(V,E). Let us associate to G a finite list P(G) of invariants. To any P the following two natural problems arise: (R) Realizability. Given P, when is P=P(G) for some graph G?, (U) Uniqueness. Suppose P(G)=P(H) for graphs G and H. When does this imply G ≅ H? The best studied questions in this context are the degree realization problem for (R) and the reconstruction conjecture for (U). We discuss the problems (R) and (U) for the degree sequence and the size sequence of induced subgraphs for undirected and directed graphs, concentrating on the complexity of the corresponding decision problems and their connection to a natural search problem on graphs

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

Rank-based linkage I: triplet comparisons and oriented simplicial complexes

Rank-based linkage is a new tool for summarizing a collection $S$ of objects according to their relationships. These objects are not mapped to vectors, and ``similarity'' between objects need be neither numerical nor symmetrical. All an object needs to do is rank nearby objects by similarity to itself, using a Comparator which is transitive, but need not be consistent with any metric on the whole set. Call this a ranking system on $S$. Rank-based linkage is applied to the $K$-nearest neighbor digraph derived from a ranking system. Computations occur on a 2-dimensional abstract oriented simplicial complex whose faces are among the points, edges, and triangles of the line graph of the undirected $K$-nearest neighbor graph on $S$. In $|S| K^2$ steps it builds an edge-weighted linkage graph $(S, \mathcal{L}, \sigma)$ where $\sigma(\{x, y\})$ is called the in-sway between objects $x$ and $y$. Take $\mathcal{L}_t$ to be the links whose in-sway is at least $t$, and partition $S$ into components of the graph $(S, \mathcal{L}_t)$, for varying $t$. Rank-based linkage is a functor from a category of out-ordered digraphs to a category of partitioned sets, with the practical consequence that augmenting the set of objects in a rank-respectful way gives a fresh clustering which does not ``rip apart`` the previous one. The same holds for single linkage clustering in the metric space context, but not for typical optimization-based methods. Open combinatorial problems are presented in the last section.Comment: 37 pages, 12 figure

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete