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.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for forthcoming IMA volume "Recent Trends in Combinatorics

Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory. Published July 201

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

The lattice of integer flows of a regular matroid

For a finite multigraph G, let \Lambda(G) denote the lattice of integer flows of G -- this is a finitely generated free abelian group with an integer-valued positive definite bilinear form. Bacher, de la Harpe, and Nagnibeda show that if G and H are 2-isomorphic graphs then \Lambda(G) and \Lambda(H) are isometric, and remark that they were unable to find a pair of nonisomorphic 3-connected graphs for which the corresponding lattices are isometric. We explain this by examining the lattice \Lambda(M) of integer flows of any regular matroid M. Let M_\bullet be the minor of M obtained by contracting all co-loops. We show that \Lambda(M) and \Lambda(N) are isometric if and only if M_\bullet and N_\bullet are isomorphic.Comment: 18 pages, no figures. Revised version to appear in J. Combin. Theory Series

Excluding Kuratowski graphs and their duals from binary matroids

We consider some applications of our characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor. We characterise the internally 4-connected binary matroids with no minor in some subset of {M(K3,3),M*(K3,3),M(K5),M*(K5)} that contains either M(K3,3) or M*(K3,3). We also describe a practical algorithm for testing whether a binary matroid has a minor in the subset. In addition we characterise the growth-rate of binary matroids with no M(K3,3)-minor, and we show that a binary matroid with no M(K3,3)-minor has critical exponent over GF(2) at most equal to four.Comment: Some small change

Ordinary and Generalized Circulation Algebras for Regular Matroids

Let E be a finite set, and let R(E) denote the algebra of polynomials in indeterminates (x_e)_{e in E}, modulo the squares of these indeterminates. Subalgebras of R(E) generated by homogeneous elements of degree 1 have been studied by many authors and can be understood combinatorially in terms of the matroid represented by the linear equations satisfied by these generators. Such an algebra is related to algebras associated to deletions and contractions of the matroid by a short exact sequence, and can also be written as the quotient of a polynomial algebra by certain powers of linear forms. We study such algebras in the case that the matroid is regular, which we term circulation algebras following Wagner. In addition to surveying the existing results on these algebras, we give a new proof of Wagner's result that the structure of the algebra determines the matroid, and construct an explicit basis in terms of basis activities in the matroid. We then consider generalized circulation algebras in which we mod out by a fixed power of each variable, not necessarily equal to 2. We show that such an algebra is isomorphic to the circulation algebra of a "subdivided" matroid, a variation on a result of Nenashev, and derive from this generalized versions of many of the results on ordinary circulation algebras, including our basis result. We also construct a family of short exact sequences generalizing the deletion-contraction decomposition