Let n points be placed independently in ν-dimensional space according to the standard ν-dimensional normal distribution. Let M n be the longest edge-length of the minimal spanning tree on these points; equivalently let M n be the infimum of those r such that the union of balls of radius r/2 centred at the points is connected. We show that the distribution of (2 log n)1/2M n - b n converges weakly to the Gumbel (double exponential) distribution, where b n are explicit constants with b n ~ (ν - 1)log log n. We also show the same result holds if M n is the longest edge-length for the nearest neighbour graph on the points.

We develop a technique for establishing statistical tests with precise confidence levels for upper bounds on the critical probability in oriented percolation. We use it to give pc < 0.647 with a 99.999967% confidence. As Monte Carlo simulations suggest that pc ≈ 0.6445, this bound is fairly tight.

This article continues an investigation begun in [2]. A random graph Gn (x) is constructed on independent random points U 1, · ··, Un distributed uniformly on [0, 1]d , d ≧ 1, in which two distinct such points are joined by an edge if the l ∞-distance between them is at most some prescribed value 0 < x < 1.

Almost-sure asymptotic results are obtained for the convergence/divergence of the minimum vertex degree of the random graph, as the number n of points becomes large and the edge distance x is allowed to vary with n. The largest nearest neighbor link dn, the smallest x such that Gn (x) has no vertices of degree zero, is shown to satisfy Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be complete, a.s. These criteria imply a.s. limiting behavior of the diameter of the vertex set.

On independent random points U 1 ,· ··,Un distributed uniformly on [0, 1]d , a random graph Gn (x) is constructed in which two distinct such points are joined by an edge if the l ∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn } are provided which guarantee the random graph to be empty of edges, a.s.

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree Tn with n labeled nodes is a recursive tree if n = 1, or n > 1 and Tn can be constructed by joining node n to a node of some recursive tree Tn– 1. For arbitrary nodes i < n in a random recursive tree we give the exact distribution of Xi,n , the distance between nodes i and n. We characterize this distribution as the convolution of the law of Xi,j+ 1 and n – i – 1 Bernoulli distributions. We further characterize the law of Xi,j+ 1 as a mixture of sums of Bernoullis. For i = in growing as a function of n, we show that is asymptotically normal in several settings.

A graph H decomposes into a graph G if one can write H as an edge-disjoint union of graphs isomorphic to G. H decomposes into D, where D is a family of graphs, when H can be written as a union of graphs each isomorphic to some member of D, and every member of D is represented at least once. In this paper it is shown that the d-dimensional cube Qd decomposes into any graph G of size d each of whose blocks is either an even cycle or an edge. Furthennore, Qd decomposes into D, where D is any set of six trees of size d.

Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any non-negative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random p-vector, which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of p colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve for example, martingale, Markov chain or various mixing assumptions.

In this paper we obtain asymptotics for the number of rooted 3-connected maps on an arbitrary surface and use them to prove that almost all rooted 3-connected maps on any fixed surface have large edge-width and large face-width. It then follows from the result of Roberston and Vitray [10] that almost all rooted 3-connected maps on any fixed surface are minimum genus embeddings and their underlying graphs are uniquely embeddable on the surface.

A number of constructions are given for arc-transitive digraphs, based on modifications of permutation representations of finite groups. In particular, it is shown that for every positive integer s and for any transitive permutation group p of degree k, there are infinitely many examples of a finite k-regular digraph with a group of automorphisms acting transitively on s-arcs (but not on (s + 1)-arcs), such that the stabilizer of a vertex induces the action of P on the out-neighbour set.

In the paper we study the asymptotic behaviour of the number of trees with n vertices and diameter k = k(n), where k/√n → ∞ as n → ∞ but k = o(n).

Regular maps of type {p, q}r and the associated groups Gp,q,r are considered for small values for p, q and r. In particular, it is shown that the groups G4,6,6 and G5,5,6 are Abelian-by-infinite, and there are infinitely many regular maps of each of the types {4, 6}6, {5, 5}6, {5, 6}6 and {6, 6}6.

An infinite family of vertex-and edge-transitive, but not arc-transitive, graphs of degree 4 is constructed.

In this paper, we first give a characterization of a regular endomorphism of a graph. Then, we explicitly describe its inverses through idempotents.

The behaviour of the size of the largest complete subgraph contained in an overlap graph of a random acyclic digraph is presented.

Consider a forest of maple trees in autumn, with leaves falling on the ground. Those coming late cover the others below, so eventually the fallen leaves form a statistically homogeneous spatial pattern. In particular, the uncovered leaf boundaries form a mosaic. We formulate a mathematical model to describe this mosaic, firstly in the case where the leaves are polygonal and later for leaves with curved boundaries. Mean values of certain statistics of the mosaic are derived.

Let fn be a sequence of nonnegative integers and let f(x): = Σn≥0 fn xn be its generating function. Assume f(x) has the following properties: it has radius of convergence r, 0 < r < 1, with its only singualarity on the circle of convergence at x = r and f(r) = s; y = f(x) satisfies an analytic identity F(x, y) = 0 near (r, s); for some k ≥ 2 F0.j = 0, 0 ≤ j < k, F0.k ≠ 0 where Fi is the value at (r, s) of the ith partial derivative with respect to x and the jth partial derivative with respect to y of F. These assumptions form the basis of what we call the typical and general cases. In both cases we show how to obtain an asymptotic expansion of fn. We apply our technique to produce several terms in the asymptotic expansion of combinatorial sequences for which previously only the first term was known.

The Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square.

We establish a characterization of certain trees of polygons similar to that of n-gon-trees given by Chao and Li.

A simple identity for the incomplete factorial of sums of zero-one variables is exploited to provide the factorial moments of the number of components and the number of cyclical elements of the random mapping (T, {pi }) considered by Ross (1981).

A three-parameter model of a random directed graph (digraph) is specified by the probability of ‘up arrows' from vertex i to vertex j where i < j, the probability of ‘down arrows' from i to j where i ≥ j, and the probability of bidirectional arrows between i and j. In this model, a phase transition—the abrupt appearance of a giant strongly connected component—takes place as the parameters cross a critical surface. The critical surface is determined explicitly. Before the giant component appears, almost surely all non-trivial components are small cycles. The asymptotic probability that the digraph contains no cycles of length 3 or more is computed explicitly. This model and its analysis are motivated by the theory of food webs in ecology.