相关论文: Dirac's theorem on simplicial matroids
Networks are often studied as graphs, where the vertices stand for entities in the world and the edges stand for connections between them. While relatively easy to study, graphs are often inadequate for modeling real-world situations,…
In this paper, we study the notion of chordality and cycles in hypergraphs from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. However, there is no…
We analyze Dirac spectra of two-dimensional QCD like theories both in the continuum and on the lattice and classify them according to random matrix theories sharing the same global symmetries. The classification is different from QCD in…
A $2$-matching complex is a simplicial complex which captures the relationship between $2$-matchings of a graph. In this paper, we will use discrete Morse Theory and the Matching Tree Algorithm to prove homotopical results. We will consider…
This paper computes the Dirac cohomology $H_D(\pi)$ of irreducible unitary Harish-Chandra modules $\pi$ of complex classical groups viewed as real reductive groups. More precisely, unitary representations with nonzero Dirac cohomology are…
In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {\it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and…
The deck of a graph $X$, $D(X)$, is defined as the multiset of all vertex-deleted subgraphs of $X$. Two graphs are said to be hypomorphic, if they have the same deck. Kelly-Ulam conjecture states that any two hypomorphic graphs on at least…
Inspired by work of Fr\"oberg (1990), and Eagon and Reiner (1998), we define the \emph{total $k$-cut complex} of a graph $G$ to be the simplicial complex whose facets are the complements of independent sets of size $k$ in $G$. We study the…
As a discretization of the Hodge Laplacian, the combinatorial Laplacian of simplicial complexes has garnered significant attention. In this paper, we study combinatorial Laplacians for complex pairs $(X, A)$, where $A$ is a subcomplex of a…
Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic…
Given a hypergraph $\mathcal{H}$, the dual hypergraph of $\mathcal{H}$ is the hypergraph of all minimal transversals of $\mathcal{H}$. The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner…
Let $V$ be a finite set. Let $\mathcal{K}$ be a simplicial complex with its vertices in $V$. In this paper, we discuss some differential calculus on $V$. We construct some constrained homology groups of $\mathcal{K}$ by using the…
The \emph{strong collapse} of a simplicial complex, proposed by Barmak and Minian (\emph{Disc. Comp. Geom. 2012}), is a combinatorial collapse of a complex onto its sub-complex. Recently, it has received attention from computational…
In 1963, Corr\'adi and Hajnal proved that for all $k \ge 1$ and $n \ge 3k$, every (simple) graph on n vertices with minimum degree at least 2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not…
We conjecture that the balanced complete bipartite graph $K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil}$ contains more cycles than any other $n$-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent…
We construct families of cell complexes that generalize expander graphs. These families are called non-$k$-hyperfinite, generalizing the idea of a non-hyperfinite (NH) family of graphs. Roughly speaking, such a complex has the property that…
Consider a graph $G$ with chromatic number $k$ and a collection of complete bipartite graphs, or bicliques, that cover the edges of $G$. We prove the following two results: \medskip \noindent $\bullet$ If the bicliques partition the edges…
A new approach to the problem of doubling is presented with the Dirac-Kahler (DK) theory as a starting point and using Geometric Discretisation providing us with a new way of extracting the Dirac field in the discrete setting of a…
An identification of two vertices $u$ and $v$ in a graph replaces them with a new vertex whose neighborhood is the union of the neighborhoods of $u$ and $v$. We study the {\sc ${\cal H}$-Identification} problem, which is to decide whether a…
A graph is $H$-free if it has no induced subgraph isomorphic to $H$. Brandst\"adt, Engelfriet, Le and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandst\"adt, Le and Mosca…