Art M. Duval

A directed graph version of strongly regular graphs

We study a directed graph version of strongly regular graphs whose adjacency matrices satisfy A^2 + (μ − λ)A − (t − μ)I = μJ, and AJ = JA = kJ. We prove existence (by construction), nonexistence, and necessary conditions, and construct homomorphisms for several families of parameter sets.

Simplicial matrix-tree theorems

First published in Transactions of the American Mathematical Society in volume 361 (2009), no. 11, 6073--6114, published by the American Mathematical Society

Shifted simplicial complexes are Laplacian integral

We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs.

A Non-Partitionable Cohen-Macaulay Simplicial Complex

Abstract A long-standing conjecture of Stanley states that every Cohen–Macaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.

Perron-Frobenius type results and discrete versions of nodal domain theorems

Abstract We prove discrete versions of nodal domain theorems; in particular, an eigenvector corresponding to the s th smallest eigenvalue of a graph Laplacian has at most s nodal domains. We compare our results to those of Courant and Pleijel on nodal domains of continuous Laplacians, and to those of Fiedler on nonnegative regions of graph Laplacians.

Semidirect product constructions of directed strongly regular graphs

We construct a new infinite family of directed strongly regular graphs, as Cayley graphs of certain semidirect product groups. This generalizes an earlier construction of Klin, Munemasa, Muzychuk, and Zieschang on some dihedral groups.

A combinatorial decomposition of simplicial complexes

We find a decomposition of simplicial complexes that implies and sharpens the characterization (due to Björner and Kalai) of thef-vector and Betti numbers of a simplicial complex. It generalizes a result of Stanley, who proved the acyclic case, and settles a conjecture of Stanley and Kalai.

Simplicial and cellular trees

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai, and Adin, and more recently by numerous authors, the fundamental topological properties of a tree — namely acyclicity and connectedness — can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley’s formula and the matrix-tree theorem. A subtlety of the higher-dimensional case is that enumeration must account for the possibility of torsion homology in trees, which is always trivial for graphs. Cellular trees are the starting point for further high-dimensional extensions of concepts from algebraic graph theory including the critical group, cut and flow spaces, and discrete dynamical systems such as the abelian sandpile model.

Iterated homology of simplicial complexes

We develop an iterated homology theory for simplicial complexes. Thistheory is a variation on one due to Kalai. For Δ a simplicial complex of dimension d − 1, and each r = 0, ...,d, we define rth iterated homology groups of Δ. When r = 0, this corresponds to ordinary homology. If Δ is a cone over Δ′, then when r = 1, we get the homology of Δ′. If a simplicial complex is (nonpure) shellable, then its iterated Betti numbers give the restriction numbers, hk,j, of the shelling. Iterated Betti numbers are preserved by algebraic shifting, and may be interpreted combinatorially in terms of the algebraically shifted complex in several ways. In addition, the depth of a simplicial complex can be characterized in terms of its iterated Betti numbers.

COVERING A CIRCULAR STRING WITH SUBSTRINGS OF FIXED LENGTH

A nonempty circular string C(x) of length n is said to be covered by a set Uk of strings each of fixed length k≤n iff every position in C(x) lies within an occurrence of some string u∈Uk. In this paper we consider the problem of determining the minimum cardinality of a set Uk which guarantees that every circular string C(x) of length n≥k can be covered. In particular, we show how, for any positive integer m, to choose the elements of Uk so that, for sufficiently large k, uk≈σk–m, where uk=|Uk| and σ is the size of the alphabet on which the strings are defined. The problem has application to DNA sequencing by hybridization using oligonucleotide probes.