Related papers: Generalizations of interval and proper interval gr…
We introduce the notion of sortability and $t$-sortability for a simplicial complex and study the graphs for which their independence complexes are either sortable or $t$-sortable. We show that the proper interval graphs are precisely the…
We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Posa's sufficient…
For any natural number $d$, a graph $G$ is a (disjoint) $d$-interval graph if it is the intersection graph of (disjoint) $d$-intervals, the union of $d$ (disjoint) intervals on the real line. Two important subclasses of $d$-interval graphs…
We unify several seemingly different graph and digraph classes under one umbrella. These classes are all broadly speaking different generalizations of interval graphs, and include, in addition to interval graphs, also adjusted interval…
This paper makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group which are preserved by automorphisms of the…
For each fixed $d\ge 1$, we obtain asymptotic estimates for the number of $d$-representable simplicial complexes on $n$ vertices as a function of $n$. The case $d=1$ corresponds to counting interval graphs, and we obtain new results in this…
We first develop a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. This includes, among others, the graph Laplacian, the combinatorial Laplacian on simplicial complexes, the…
In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where…
Theoretical development and applications of graph signal processing (GSP) have attracted much attention. In classical GSP, the underlying structures are restricted in terms of dimensionality. A graph is a combinatorial object that models…
The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatic problems. Some of those applications imply restrictions on the 2-interval graphs, and justify…
Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d=2), and strictly contains the class of…
This research focuses on analyzing the depth of generalized binomial edge ideals. We extend the notion of $d$-compatible map for the pairs of a complete graph and an arbitrary graph, and using it, we give a combinatorial lower bound for the…
In [Discrete differential calculus on simplicial complexes and constrained homology, Chin. Ann. Math. Ser. B 44(4), 615-640, 2023], the constrained (co)homology for simplicial complexes and independence hypergraphs is constructed via…
Hypergraphs have seen widespread application in network and data science communities in recent years. We present a survey of recent work to construct auxiliary structures from hypergraphs -- specifically simplicial, relative, and chain…
The generalized Dehn-Sommerville relations determine the odd subalgebra of the combinatorial Hopf algebra. We introduce a class of eulerian hypergraphs that satisfy the generalized Dehn-Sommerville relations for the combinatorial Hopf…
A cocomparability graph is a graph whose complement admits a transitive orientation. An interval graph is the intersection graph of a family of intervals on the real line. In this paper we investigate the relationships between interval and…
We work out the graph limit theory for dense interval graphs. The theory developed departs from the usual description of a graph limit as a symmetric function $W(x,y)$ on the unit square, with $x$ and $y$ uniform on the interval $(0,1)$.…
We consider a variety of connections between threshold graphs, shifted complexes, and simplicial complexes naturally formed from a graph. These graphical complexes include the independent set, neighborhood, and dominance complexes. We…
Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their…
In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their…