Small-world graphs are examples of random graphs which mimic empirically observed features of social networks. We propose an intrinsic definition of small-world graphs, based on a probabilistic formulation of scaling properties of the graph, which does not rely on any particular construction. Our definition is shown to encompass existing models of small-world graphs, proposed by Watts (1999) and studied by Barbour and Reinert (2001), which are based on random perturbations of a regular lattice. We also propose alternative constructions of small-world graphs which are not based on lattices and study their scaling properties.

An S3-involution graph for a group G is a graph with vertex set a union of conjugacy classes of involutions of G such that two involutions are adjacent if they generate an S3-subgroup in a particular set of conjugacy classes. We investigate such graphs in general and also for the case where G=PSL(2,q).

We exhibit an interesting Cayley graph X of the elementary abelian group Z26 with the property that Aut(X) contains two regular subgroups, exactly one of which is normal. This demonstrates the existence of two subsets of Z26 that yield isomorphic Cayley graphs, even though the two subsets are not equivalent under the automorphism group of Z26.

The core of a graph Γ is the smallest graph Δ that is homomorphically equivalent to Γ (that is, there exist homomorphisms in both directions). The core of Γ is unique up to isomorphism and is an induced subgraph of Γ. We give a construction in some sense dual to the core. The hull of a graph Γ is a graph containing Γ as a spanning subgraph, admitting all the endomorphisms of Γ, and having as core a complete graph of the same order as the core of Γ. This construction is related to the notion of a synchronizing permutation group, which arises in semigroup theory; we provide some more insight by characterizing these permutation groups in terms of graphs. It is known that the core of a vertex-transitive graph is vertex-transitive. In some cases we can make stronger statements: for example, if Γ is a non-edge-transitive graph, we show that either the core of Γ is complete, or Γ is its own core. Rank-three graphs are non-edge-transitive. We examine some families of these to decide which of the two alternatives for the core actually holds. We will see that this question is very difficult, being equivalent in some cases to unsolved questions in finite geometry (for example, about spreads, ovoids and partitions into ovoids in polar spaces).

In this note we first prove that, for a positive integer n>1 with n≠p or p2 where p is a prime, there exists a transitive group of degree n without regular subgroups. Then we look at 2-closed transitive groups without regular subgroups, and pose two questions and a problem for further study.

A first kind Frobenius graph is a Cayley graph Cay(K,S) on the Frobenius kernel of a Frobenius group such that S=aH for some a∈K with 〈aH〉=K, where H is of even order or a is an involution. It is known that such graphs admit ‘perfect’ routing and gossiping schemes. A circulant graph is a Cayley graph on a cyclic group of order at least three. Since circulant graphs are widely used as models for interconnection networks, it is thus highly desirable to characterize those which are Frobenius of the first kind. In this paper we first give such a characterization for connected 4-valent circulant graphs, and then describe optimal routing and gossiping schemes for those which are first kind Frobenius graphs. Examples of such graphs include the 4-valent circulant graph with a given diameter and maximum possible order.

We investigate the average similarity of random strings as captured by the average number of ‘cousins’ in the underlying tree structures. Analytical techniques including poissonization and the Mellin transform are used for accurate calculation of the mean. The string alphabets we consider are m-ary, and the corresponding trees are m-ary trees. Certain analytic issues arise in the m-ary case that do not have an analog in the binary case.

In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.

A weighted graph G is a pair (V, ℰ) containing vertex set V and edge set ℰ, where each edge e ∈ ℰ is associated with a weight We . A subgraph of G is a forest if it has no cycles. All forests on the graph G form a probability space, where the probability of each forest is proportional to the product of the weights of its edges. This paper aims to simulate forests exactly from the target distribution. Methods based on coupling from the past (CFTP) and rejection sampling are presented. Comparisons of these methods are given theoretically and via simulation.

The hyper-Wiener index of a connected graph G is defined as , where V (G) is the set of all vertices of G and d(u,v) is the distance between the vertices u,v∈V (G). In this paper we find an exact expression for the hyper-Wiener index of TUHC6[2p,q], the zigzag polyhex nanotube.

We define the notion of minor for weighted graphs. We prove that with this minor relation, the set of weighted graphs is directed. We also prove that, given any two weights on a connected graph with the same total weight, we can transform one into the other using a sequence of edge subdivisions and edge contractions.

A simpleundirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be a prime. It was shown by Folkman [J. Folkman, ‘Regular line-symmetric graphs’, J. Combin. Theory3 (1967), 215–232] that a regular edge-transitive graph of order 2p or 2p2 is necessarily vertex-transitive. In this paper an extension of his result in the case of cubic graphs is given. It is proved that every cubic edge-transitive graph of order 8p2 is vertex-transitive.

Let G bea graph of order n. Let a, b and k be nonnegative integers such that 1≤a≤b. A graph G is called an (a,b,k)-critical graph if after deleting any k vertices of G the remaining graph of G has an [a,b]-factor. We provide a sufficient condition for a graph to be (a,b,k)-critical that extends a well-known sufficient condition for the existence of a k-factor.

Hu et al. [“A boundary problem for group testing”, SIAM J. Algebraic Discrete Meth.2 (1981), 81–87] conjectured that the minimax test number to find d defectives in 3d items is 3d−1, a surprisingly difficult combinatorial problem about which very little is known. In this article we state three more conjectures and prove that they are all equivalent to the conjecture of Hu et al. Notably, as a byproduct, we also obtain an interesting upper bound for M(d,n).

Let S 0 := 0 and Sk := ξ 1 + ··· + ξk for k ∈ ℕ := {1, 2, …}, where {ξk : k ∈ ℕ} are independent copies of a random variable ξ with values in ℕ and distribution pk := P{ξ = k}, k ∈ ℕ. We interpret the random walk {Sk : k = 0, 1, 2, …} as a particle jumping to the right through integer positions. Fix n ∈ ℕ and modify the process by requiring that the particle is bumped back to its current state each time a jump would bring the particle to a state larger than or equal to n. This constraint defines an increasing Markov chain {Rk (n) : k = 0, 1, 2, …} which never reaches the state n. We call this process a random walk with barrier n. Let Mn denote the number of jumps of the random walk with barrier n. This paper focuses on the asymptotics of Mn as n tends to ∞. A key observation is that, under p 1 > 0, {Mn : n ∈ ℕ} satisfies the distributional recursion M 1 = 0 and for n = 2, 3, …, where In is independent of M 2, …, M n−1 with distribution P{In = k} = pk / (p 1 + ··· + p n−1), k ∈ {1, …, n − 1}. Depending on the tail behavior of the distribution of ξ, several scalings for Mn and corresponding limiting distributions come into play, including stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is to compare (couple) the number of jumps, Mn , with the first time, Nn , when the unrestricted random walk {Sk : k = 0, 1, …} reaches a state larger than or equal to n. The results are applied to derive the asymptotics of the number of collision events (that take place until there is just a single block) for β(a, b)-coalescent processes with parameters 0 < a < 2 and b = 1.

In this paper we investigate the ‘local’ properties of a random mapping model, T n D̂, which maps the set {1, 2, …, n} into itself. The random mapping T n D̂ , which was introduced in a companion paper (Hansen and Jaworski (2008)), is constructed using a collection of exchangeable random variables D̂ 1, …, D̂ n which satisfy In the random digraph, G n D̂ , which represents the mapping T n D̂ , the in-degree sequence for the vertices is given by the variables D̂ 1, D̂ 2, …, D̂ n , and, in some sense, G n D̂ can be viewed as an analogue of the general independent degree models from random graph theory. By local properties we mean the distributions of random mapping characteristics related to a given vertex v of G n D̂ - for example, the numbers of predecessors and successors of v in G n D̂ . We show that the distribution of several variables associated with the local structure of G n D̂ can be expressed in terms of expectations of simple functions of D̂ 1, D̂ 2, …, D̂ n . We also consider two special examples of T n D̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine, for these examples, exact and asymptotic distributions for the local structure variables considered in this paper. These distributions are also of independent interest.

We define a subgraph of the zero divisor graph of a ring, associated to the ring idempotents. We study its properties and prove that for large classes of rings the connectedness of the graph is equivalent to the indecomposability of the ring and in those cases we also calculate the graph’s diameter.

We consider two different models of small-world graphs on nodes whose locations are modelled by a stochastic point process. In the first model each node is connected to a fixed number of its nearest neighbours, while in the second model each node is connected to all nodes located within some fixed distance. In both models, nodes are additionally connected via shortcuts to other nodes chosen uniformly at random. We obtain sufficient conditions for connectivity in the first model, and necessary conditions in the second model. Thereby, we show that connectivity is achieved at a smaller value of total degree (nearest neighbours plus shortcuts) in the first model. We also obtain bounds on the diameter of the graph in this model.

Consider a random graph, having a prespecified degree distribution F, but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R 0, the basic reproduction number, and τ0, the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine R v , the reproduction number, and τv , the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.

Large deviation estimates are derived for sums of random variables with certain dependence structures, including finite population statistics and random graphs. The argument is based on Stein's method, but with a novel modification of Stein's equation inspired by the Cramér transform.