Related papers: A combinatorial statistic for labeled threshold gr…
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep…
A barcode is a finite multiset of intervals on the real line, $B = \{ (b_i, d_i)\}_{i=1}^n$. Barcodes are important objects in topological data analysis, where they serve as summaries of the persistent homology groups of a filtration. The…
In a paired threshold graph, each vertex has a weight, and two vertices are adjacent if their weight sum is large enough and their weight difference is small enough. It generalizes threshold graphs and unit interval graphs, both very well…
Given a combinatorial structure, a ``twin'' is a pair of disjoint substructures which are isomorphic (or look the same in some sense). In recent years, there have been many problems about finding large twins in various combinatorial…
The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk…
This work considers new entropy-based proofs of some known, or otherwise refined, combinatorial bounds for bipartite graphs. These include upper bounds on the number of the independent sets, lower bounds on the minimal number of colors in…
A graph $G=(V,E)$ is called a \emph{$k$-threshold graph} with \emph{thresholds} $\theta_1<\theta_2<...<\theta_k$ if we can assign a real number $r(v)$ to each vertex $v\in V$, such that for any $u,v\in V$, we have $uv\in E$ if and only if…
This paper studies \emph{Dirichlet arrangements}, a generalization of graphic hyperplane arrangements arising from electrical networks and order polytopes of finite posets. We generalize descriptions of combinatorial features of graphic…
Determining the number of embeddings of Laman graph frameworks is an open problem which corresponds to understanding the solutions of the resulting systems of equations. In this paper we investigate the bounds which can be obtained from the…
We will start from the beginning and define a matroid and its Orlik-Solomon algebra and holonomy Lie algebra, but first we give some background from topology and cohomology. A (central) hyperplane arrangement is a finite number of subspaces…
By using the Poisson formula for resultants and the variants of chip-firing game on graphs, we provide a combinatorial method for computing a class of of resultants, i.e. the characteristic polynomials of the adjacency tensors of starlike…
Using a notation of corner between edges when graph has a fixed rotation, i.e. cyclical order of edges around vertices, we define combinatorial objects - combinatorial maps as pairs of permutations, one for vertices and one for faces.…
Let K(X) be the collection of all non-zero finite dimensional subspaces of rational functions on an n-dimensional irreducible variety X. For any n-tuple L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the number of…
We analyze statistical properties of the complex system with conditions which manifests through specific constraints on the column/row sum of the matrix elements. The presence of additional constraints besides symmetry leads to new…
In this paper, we introduce and generalize some combinatorial invariants of graphs such as matching number and induced matching number to hypergraphs. Then we compare them together and present some upper bounds for the regularity of…
Closed form expressions are given for computing the parameters and vectors that identify and define the $n-1$ dimensional conic section that results from the intersection of a hyperplane with an $n$-dimensional conic section: cone,…
De Finetti's classical result of [18] identifying the law of an exchangeable family of random variables as a mixture of i.i.d. laws was extended to structure theorems for more complex notions of exchangeability by Aldous [1,2,3], Hoover…
This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was…
The Bell numbers count the number of different ways to partition a set of $n$ elements while the graphical Bell numbers count the number of non-equivalent partitions of the vertex set of a graph into stable sets. This relation between graph…
Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…