We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This allows us to obtain an exact estimate for the co-rank.

We introduce the concept of a relative Tutte polynomial of coloured graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (and hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly.

A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with m edges can be separated into two parts of roughly equal size by the removal of vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges, where ct is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any ε > 0, there is an integer g(ε) such that every string graph with n vertices and girth at least g(ε) has at most (1 + ε)n edges. Furthermore, the number of such labelled graphs is at most (1 + ε)nT(n), where T(n) = nn−2 is the number of labelled trees on n vertices.

We study the number of random records in a binary search tree with n vertices (or equivalently, the number of cuttings required to eliminate the tree). We show that a classical limit theorem for convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. The asymptotic distribution of the (normalized) number of records or cuts is found to be weakly 1-stable.

We introduce and prove a family of inequalities satisfied by the Whitney rank generating function of a matroid in the positive quadrant of ℝ2. These can be interpreted as correlation inequalities at those points where the polynomial is known to count the number of independent sets, bases or spanning sets of the matroid. Our proofs also introduce an idea of rank dominating bijections in matroids, which are then used to obtain some simple extensions of the submodular property of matroid ranks.

We derive area limit laws for the various symmetry classes of staircase polygons on the square lattice, in a uniform ensemble where, for fixed perimeter, each polygon occurs with the same probability. This complements a previous study by Leroux and Rassart, where explicit expressions for the area and perimeter generating functions of these classes have been derived.

The number of closed tree-like walks in a graph is closely related to the moments of the roots of the matching polynomial for the graph. Thus, by counting these walks up to a given length it is possible to find approximations for the matching polynomial. This approach has been used in two separate problems involving asymptotic enumerations of 1-factorizations of regular graphs. Nevertheless, a systematic way to count the required walks had not previously been found.

In this paper we give an algorithm to count closed tree-like walks in a regular graph up to a given length. For small m, this provides expressions for the number of m-matchings in the graph in terms of the numbers of copies of certain small subgraphs that appear in the graph. The simplest of these expressions were already known, having been rediscovered by numerous authors using ad hoc methods. We offer the first general method for producing the expressions. We also find generating functions that isolate the contribution from the simplest kind of subgraph – namely a single cycle of arbitrary length.