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Eric Babson

Publications and source records attributed to Eric Babson.

At least 19 recordsLinked to original sources
arXiv2024

Models of random spanning trees

There are numerous randomized algorithms to generate spanning trees in a given ambient graph; several target the uniform distribution on trees (UST), while in practice the fastest and most frequently used draw random weights on the edges and then employ a greedy algorithm to choose the minimum-weight spanning tree (MST). Though MST is a workhorse in applications, the mathematical properties of random MST are far less explored than those of UST. In this paper we develop tools for the quantitative study of random MST. We consider the standard case that the weights are drawn i.i.d. from a single distribution on the real numbers, as well as successive generalizations that lead to \emph{product measures}, where the weights are independently drawn from arbitrary distributions.

cs.DM↗
arXiv2023

From Erdos-Renyi graphs to Linial-Meshulam complexes via the multineighbor construction

The $m$-neighbor complex of a graph is the simplicial complex in which faces are sets of vertices with at least $m$ common neighbors. We consider these complexes for Erdos-Renyi random graphs and find that for certain explicit families of parameters the resulting complexes are with high probability $(t-1)$-dimensional with all $(t-2)$-faces and each $(t-1)$-face present with a fixed probability. Unlike the Linial-Meshulam measure on the same complexes there can be correlations between pairs of $(t-1)$-faces but we conjecture that the two measures converge in total variation for certain parameter sequences.

math.CO↗
arXiv2023

Homological algebra and poset versions of the Garland method

Garland introduced a vanishing criterion for a characteristic zero cohomology group of a locally finite and locally connected simplicial complex. The criterion is based on the spectral gaps of the graph Laplacians of the links of faces and has turned out to be effective in a wide range of examples. In this note we extend the approach to include a range of non-simplicial (co)chain complexes associated to combinatorial structures we call Garland posets and elaborate further on the case of cubical complexes.

math.CO↗
arXiv2019

Higher dimensional connectivity and minimal degree of random graphs with an eye towards minimal free resolutions

In this note we define and study graph invariants generalizing to higher dimension the maximum degree of a vertex and the vertex-connectivity (our $0$-dimensional cases). These are known to coincide almost surely in any regime for Erdoes-Renyi random graphs. We show the same in the one dimensional case for a middle density regime and show the easier inequality for all dimensions in the same regime. Our original motivation comes from the study of minimal free resolutions of Stanley-Reisner rings of clique complexes of graphs in commutative algebra, In that setting the higher dimensional vertex connectivities determine the lengths of the strands. Through our results we aim to replace (asymptotically) vertex connectivity by a simpler invariant.

math.CO↗
arXiv2013

Fundamental Groups of Random Clique Complexes

Clique complexes of Erdős-Rényi random graphs with edge probability between $n^{-{1\over 3}}$ and $n^{-{1\over 2}}$ are shown to be aas not simply connected. This entails showing that a connected two dimensional simplicial complex for which every subcomplex has fewer than three times as many edges as vertices must have the homotopy type of a wedge of circles, two spheres and real projective planes. Note that $n^{-{1\over 3}}$ is a threshold for simple connectivity and $n^{-{1\over 2}}$ is one for vanishing first $\F_2$ homology.

math.CO↗
arXiv2012

Meunier Conjecture

Frédéric Meunier's question about a multicolored Sperner lemma is addressed, leaving the question of connectivity for the color hypergraphs of such a multicolored simplex. Sperner's lemma asserts the existence of a simplex using all the colors for any vertex coloring of a subdivision of a large simplex with appropriate boundary conditions. Meunier's questions generalizes this to the situation of having several such colorings and asserts the existence of a simplex using enough different colors from each coloring.

math.CO↗
arXiv2012

Dull cut off for circulants

Families of symmetric simple random walks on Cayley graphs of Abelian groups with a bound on the number of generators are shown to never have sharp cut off in the sense of [1], [3], or [5]. Here convergence to the stationary distribution is measured in the total variation norm. This is a situation of bounded degree and no expansion. Sharp cut off or the cut off phenomenon has been shown to occur in families such as random walks on a hypercube [1] in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion [4]. Our examples agree with Peres' conjecture in [3] relating sharp cut off, spectral gap, and mixing time.

math.PR↗
arXiv2012

Random Walks and Mixed Volumes of Hypersimplices

Below is a method for relating a mixed volume computation for polytopes sharing many facet directions to a symmetric random walk. The example of permutahedra and particularly hypersimplices is expanded upon.

math.CO↗
arXiv2012

A Central Limit Theorem for Repeating Patterns

We prove a central limit theorem for the length of the longest subsequence of a random permutation which follows one of a class of repeating patterns. This class includes every fixed pattern of ups and downs having at least one of each, such as the alternating case considered by Stanley in arXiv:math/0511419 and Widom in arXiv:math/0511533. In every case considered the convergence in the limit of long permutations is to normal with mean and variance linear in the length of the permutation.

math.CO↗
arXiv2012

Counting Group Valued Graph Colorings

There are many variations on partition functions for graph homomorphisms or colorings. The case considered here is a counting or hard constraint problem in which the range or color graph carries a free and vertex transitive Abelian group action so that the colors are identified with the elements of this group. A Fourier transform is used to obtain an expansion for the numbers of colorings with terms indexed by isthmus free subgraphs of the domain. The terms are products of a polynomial in the edge density a of the color graph and the number of colorings of the indexing subgraph of the domain into the complementary color graph. The polynomial in a is independent of the color group and the term has order (1-a) to the r where r is the number of vertices minus the number of components in the indexing subgraph. Thus if (1-a) is small there is a main term indexed by the empty subgraph which is a polynomial in a and the first dependence on the coloring group occurs in the lowest order corrections which are indexed by the shortest cycles in the graph and are of order (1-a) to the g-1 where g is the length of these shortest cycles. The main theorem is stated as a reciprocity law. Examples are given in which the coloring groups are long cycles and products of short cycles and adjacent vertices are required to have distant rather than distinct colors. The chromatic polynomial of a graph corresponds to using any group and taking the allowed set to be the complement of the identity.

math.CO↗
arXiv2011

The fundamental group of random 2-complexes

We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erdős-Rényi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log(n)/n. We use a variant of Gromov's local-to-global theorem for linear isoperimetric inequalities to show that when p = O(n^{-1/2 -ε}) the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2-dimensional simplicial complexes and establish isoperimetric inequalities for such complexes. These intermediate results do not involve randomness and may be of independent interest.

math.CO↗
arXiv2011

The fundamental group of random 2-complexes

We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erdős-Rényi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log(n)/n. We use a variant of Gromov's local-to-global theorem for linear isoperimetric inequalities to show that when p = O(n^{-1/2 -ε}) the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2-dimensional simplicial complexes and establish isoperimetric inequalities for such complexes.

math.GR↗
arXiv2010

Distributions of order patterns of interval maps

A permutation $σ$ describing the relative orders of the first $n$ iterates of a point $x$ under a self-map $f$ of the interval $I=[0,1]$ is called an \emph{order pattern}. For fixed $f$ and $n$, measuring the points $x\in I$ (according to Lebesgue measure) that generate the order pattern $σ$ gives a probability distribution $μ_n(f)$ on the set of length $n$ permutations. We study the distributions that arise this way for various classes of functions $f$. Our main results treat the class of measure preserving functions. We obtain an exact description of the set of realizable distributions in this case: for each $n$ this set is a union of open faces of the polytope of flows on a certain digraph, and a simple combinatorial criterion determines which faces are included. We also show that for general $f$, apart from an obvious compatibility condition, there is no restriction on the sequence $\{μ_n(f)\}$ for $n=1,2,...$. In addition, we give a necessary condition for $f$ to have \emph{finite exclusion type}, i.e., for there to be finitely many order patterns that generate all order patterns not realized by $f$. Using entropy we show that if $f$ is piecewise continuous, piecewise monotone, and either ergodic or with points of arbitrarily high period, then $f$ cannot have finite exclusion type. This generalizes results of S. Elizalde.

math.CO↗
arXiv2008

Lefschetz Properties and Basic Constructions on Simplicial Spheres

The well known $g$-conjecture for homology spheres follows from the stronger conjecture that the face ring over the reals of a homology sphere, modulo a linear system of parameters, admits the strong-Lefschetz property. We prove that the strong-Lefschetz property is preserved under the following constructions on homology spheres: join, connected sum, and stellar subdivisions. The last construction is a step towards proving the $g$-conjecture for piecewise-linear spheres.

math.CO↗
arXiv2005

Reverse Lexicographic and Lexicographic Shifting

A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, $Δ_{\lex}$ -- an operation that transforms a monomial ideal of $S=\field[x_i: i\in\N]$ that is finitely generated in each degree into a squarefree strongly stable ideal -- is defined and studied. It is proved that (in contrast to the reverse lexicographic case) a squarefree strongly stable ideal $I\subset S$ is fixed by lexicographic shifting if and only if $I$ is a universal squarefree lexsegment ideal (abbreviated USLI) of $S$. Moreover, in the case when $I$ is finitely generated and is not a USLI, it is verified that all the ideals in the sequence $\{Δ_{\lex}^i(I)\}_{i=0}^{\infty}$ are distinct. The limit ideal $\barΔ(I)=\lim_{i\to\infty}Δ_{\lex}^i(I)$ is well defined and is a USLI that depends only on a certain analog of the Hilbert function of $I$.

math.CO↗
arXiv2005

Proof of the Lovasz Conjecture

To any two graphs G and H one can associate a cell complex Hom(G,H) by taking all graph multihomorphisms from G to H as cells. In this paper we prove the Lovasz Conjecture which states that if Hom(C_{2r+1},G) is k-connected, then χ(G)\geq k+4, where r,k\in Z, r\geq 1, k\geq -1, and C_{2r+1} denotes the cycle with 2r+1 vertices. The proof requires analysis of the complexes Hom(C_{2r+1},K_n). For even n, the obstructions to graph colorings are provided by the presence of torsion in H^*(Hom(C_{2r+1},K_n);Z). For odd n, the obstructions are expressed as vanishing of certain powers of Stiefel-Whitney characteristic classes of Hom(C_{2r+1},K_n), where the latter are viewed as $\zz$-spaces with the involution induced by the reflection of C_{2r+1}.

math.CO↗
arXiv2005

Complexes of graph homomorphisms

$Hom(G,H)$ is a polyhedral complex defined for any two undirected graphs $G$ and $H$. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that $Hom(K_m,K_n)$ is homotopy equivalent to a wedge of $(n-m)$-dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graph $G$, and integers $m\geq 2$ and $k\geq -1$, we have $\varpi_1^k(\thom(K_m,G))\neq 0$, then $χ(G)\geq k+m$; here $Z_2$-action is induced by the swapping of two vertices in $K_m$, and $\varpi_1$ is the first Stiefel-Whitney class corresponding to this action. Furthermore, we prove that a fold in the first argument of $Hom(G,H)$ induces a homotopy equivalence. It then follows that $Hom(F,K_n)$ is homotopy equivalent to a direct product of $(n-2)$-dimensional spheres, while $Hom(\bar{F},K_n)$ is homotopy equivalent to a wedge of spheres, where $F$ is an arbitrary forest and $\bar{F}$ is its complement.

math.CO↗