The structured coalescent is a continuous-time Markov chain which describes the genealogy of a sample of homologous genes from a subdivided population. Assuming this model, some results are proved relating to the genealogy of a pair of genes and the extent of subpopulation differentiation, which are valid under certain graph-theoretic symmetry and regularity conditions on the structure of the population. We first review and extend earlier results stating conditions under which the mean time since the most recent common ancestor of a pair of genes from any single subpopulation is independent of the migration rate and equal to that of two genes from an unstructured population of the same total size. Assuming the infinite alleles model of neutral mutation with a small mutation rate, we then prove a simple relationship between the migration rate and the value of Wright's coefficient F ST for a pair of neighbouring subpopulations, which does not depend on the precise structure of the population provided that this is sufficiently symmetric.

It is well known that the maximal displacement of a random walk indexed by an m-ary tree with bounded independent and identically distributed edge weights can reliably yield much larger asymptotics than a classical random walk whose summands are drawn from the same distribution. We show that, if the edge weights are mean-zero, then nonclassical asymptotics arise even when the tree grows much more slowly than exponentially. Our conditions are stated in terms of a Minkowski-type logarithmic dimension of the boundary of the tree.

The binary interval tree is a random structure that underlies interval division and parking problems. Five incomplete one-sided variants of binary interval trees are considered, providing additional flavors and variations on the main applications. The size of each variant is studied, and a Gaussian tendency is proved in each case via an analytic approach. Differential equations on half scale and delayed differential equations arise and can be solved asymptotically by local expansions and Tauberian theorems. Unlike the binary case, in an incomplete interval tree the size determines most other parameters of interest, such as the height or the internal path length.

We distribute the points and lines of PG(2, 2n+1) according to a special structure that we call the daisy structure. This distribution is intimately related to a special block design which turns out to be isomorphic to PG(n, 2).

We show a blocking set of 3q points in PG(2, 2n+1)that intersects each line in at least two points and we apply this to find a lower bound for the heterochromatic number of the projective plane.

In this paper, we introduce a compound random mapping model which can be viewed as a generalization of the basic random mapping model considered by Ross and by Jaworski. We investigate a particular example, the Poisson compound random mapping, and compare results for this model with results known for the well-studied uniform random mapping model. We show that, although the structure of the components of the random digraph associated with a Poisson compound mapping differs from the structure of the components of the random digraph associated with the uniform model, the limiting distribution of the normalized order statistics for the sizes of the components is the same as in the uniform case, i.e. the limiting distribution is the Poisson-Dirichlet (½) distribution on the simplex {{x i } : ∑ x i ≤ 1, x i ≥ x i+1 ≥ 0 for every i ≥ 1}.

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of Sbecomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.

Finite graphs of valency 4 and girth 4 admitting ½-transitive group actions, that is, vertex- and edge- but not arc-transitive group actions, are investigated. A graph is said to be ½-transitive if its automorphism group acts ½-transitively. There is a natural orientation of the edge set of a ½-transitive graph induced and preserved by its automorphism group. It is proved that in a finite ½-transitive graph of valency 4 and girth 4 the set of 4-cycles decomposes the edge set in such a way that either every 4-cycle is alternating or every 4-cycle is directed relative to this orientation. In the latter case vertex stabilizers are isomorphic to Z2.

Uniform sequential tree-building aggregation of n particles is analyzed together with the effect of the avalanche that takes place when a subtree rooted at a uniformly chosen vertex is removed. For large n, the expected subtree size is found to be ≃ logn both for the tree of size n and the tree that remains after an avalanche. Repeated breakage-restoration cycles are seen to give independent avalanches which attain size k(1 ≤ k ≤ n-1) with probability (k(k+1))-1 and restored trees that are recursive.

Suppose that a graph process begins with n isolated vertices, to which edges are added randomly one by one so that the maximum degree of the induced graph is always at most d. In a previous article, the authors showed that as n → ∞, with probability tending to 1, the result of this process is a d-regular graph. This graph is shown to be connected with probability asymptotic to 1.

This paper inverstigates the automorphism groups of Cayley graphs of metracyclic p-gorups. A characterization is given of the automorphism groups of Cayley grahs of a metacyclic p-group for odd prime p. In particular, a complete determiniation of the automophism group of a connected Cayley graph with valency less than 2p of a nonabelian metacyclic p-group is obtained as a consequence. In subsequent work, the result of this paper has been applied to solve several problems in graph theory.

Following Füredi and Komlós, we develop a graph theory method to study the high moments of large random matrices with independent entries. We apply this method to sparse N × N random matrices A N,p that have, on average, p non-zero elements per row. One of our results is related to the asymptotic behaviour of the spectral norm ∥A N,p ∥ in the limit 1 ≪ p ≪ N. We show that the value pc = log N is the critical one for lim ∥A N,p /√p∥ to be bounded or not. We discuss relations of this result with the Erdős–Rényi limit theorem and properties of large random graphs. In the proof, the principal issue is that the averaged vertex degree of plane rooted trees of k edges remains bounded even when k → ∞. This observation implies fairly precise estimates for the moments of A N,p . They lead to certain generalizations of the results by Sinai and Soshnikov on the universality of local spectral statistics at the border of the limiting spectra of large random matrices.

Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path.

We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

The random triangle model was recently introduced as a random graph model that captures the property of transitivity that is often found in social networks, i.e. the property that given that two vertices are second neighbors, they are more likely to be neighbors. For parameters p ∊ [0,1] and q ≥ 1, and a finite graph G = (V, E), it assigns to elements η of {0,1}E probabilities which are proportional to where t(η) is the number of triangles in the open subgraph. In this paper the behavior of the random triangle model on the two-dimensional triangular lattice is studied. By mapping the system onto an Ising model with external field on the hexagonal lattice, it is shown that phase transition occurs if and only if p = (q−1)−2/3 and q > q c for a critical value q c which turns out to equal It is furthermore demonstrated that phase transition cannot occur unless p = p c (q), the critical value for percolation of open edges for given q. This implies that for q ≥ q c , p c (q) = (q−1)−2/3.

Let X i : i ≥ 1 be i.i.d. points in ℝd , d ≥ 2, and let T n be a minimal spanning tree on X 1,…,X n . Let L(X 1,…,X n ) be the length of T n and for each strictly positive integer α let N(X 1,…,X n ;α) be the number of vertices of degree α in T n . If the common distribution satisfies certain regularity conditions, then we prove central limit theorems for L(X 1,…,X n ) and N(X 1,…,X n ;α). We also study the rate of convergence for EL(X 1,…,X n ).

In a cubic multigraph certain restrictions on the paths are made to define what is called a railway. On the tracks in the railway (edges in the multigraph) an equivalence relation is defined. The number of equivalence classes induced by this relation is investigated for a random railway achieved from a random cubic multigraph, and the asymptotic distribution of this number is derived as the number of vertices tends to infinity.

We approach sphere of influence graphs (SIGs) from a probabilistic perspective. Ordinary SIGs were first introduced by Toussaint as a type of proximity graph for use in pattern recognition, computer vision and other low-level vision tasks. A random sphere of influence graph (RSIG) is constructed as follows. Consider n points uniformly and independently distributed within the unit square in d dimensions. Around each point,X i , draw an open ball (‘sphere of influence’) with radius equal to the distance to X i 's nearest neighbour. Finally, draw an edge between two points if their spheres of influence intersect. Asymptotically exact values for the expected number of edges in a RSIG are determined for all values of d; previously, just upper and lower bounds were known for this quantity. A modification of the Azuma-Hoeffding exponential inequality is employed to exhibit the sharp concentration of the number of edges around its expected value.

A random mapping (T n ;q) of a finite set V, V = {1,2,…,n}, into itself assigns independently to each i ∊ V its unique image j ∊ V with probability q if i = j and with probability P = (1-q)/(n−1) if i ≠ j. Three versions of epidemic processes on a random digraph G T representing (T n ;q) are studied. The exact probability distributions of the total number of infected elements as well as the threshold functions for these epidemic processes are determined.

Consider the basic location problem in which k locations from among n given points X 1,…,X n are to be chosen so as to minimize the sum M(k; X 1,…,X n ) of the distances of each point to the nearest location. It is assumed that no location can serve more than a fixed finite number D of points. When the X i , i ≥ 1, are i.i.d. random variables with values in [0,1]d and when k = ⌈n/(D+1)⌉ we show that

where α := α(D,d) is a positive constant, f is the density of the absolutely continuous part of the law of X 1, and c.c. denotes complete convergence.

The random triangle model is a Markov random graph model which, for parameters p ∊ (0,1) and q ≥ 1 and a graph G = (V,E), assigns to a subset, η, of E, a probability which is proportional to p |η|(1-p)|E|-|η| q t(η), where t(η) is the number of triangles in η. It is shown that this model has maximum entropy in the class of distributions with given edge and triangle probabilities.

Using an analogue of the correspondence between the Fortuin-Kesteleyn random cluster model and the Potts model, the asymptotic behavior of the random triangle model on the complete graph is examined for p of order n −α, α > 0, and different values of q, where q is written in the form q = 1 + h(n) / n. It is shown that the model exhibits an explosive behavior in the sense that if h(n) ≤ c log n for c < 3α, then the edge probability and the triangle probability are asymptotically the same as for the ordinary G(n,p) model, whereas if h(n) ≥ c' log n for c' > 3α, then these quantities both tend to 1. For critical values, h(n) = 3α log n + o(log n), the probability mass divides between these two extremes.

Moreover, if h(n) is of higher order than log n, then the probability that η = E tends to 1, whereas if h(n) = o(log n) and α > 2/3, then, with a probability tending to 1, the resulting graph can be coupled with a graph resulting from the G(n,p) model. In particular these facts mean that for values of p in the range critical for the appearance of the giant component and the connectivity of the graph, the way in which triangles are rewarded can only have a degenerate influence.

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are N i (0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, ηi = N i (∞), as a function of N i (0).

We are able to obtain, for some special graphs, the limiting distribution of N i if the total number of particles N → ∞ in such a way that the fraction, N i (0)/S = ξi , at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S 2, the two-leaf star which has three vertices and two edges.