Let Γ be a graph and let G be a vertex-transitive subgroup of the full automorphism group Aut(Γ) of Γ. The graph Γ is called G-normal if G is normal in Aut(Γ). In particular, a Cayley graph Cay(G, S) on a group G with respect to S is normal if the Cayley graph is R(G)-normal, where R(G) is the right regular representation of G. Let T be a non-abelian simple group and let G = Tℓ with ℓ ≥ 1. We prove that if every connected T-vertex-transitive cubic symmetric graph is T-normal, then every connected G-vertex-transitive cubic symmetric graph is G-normal. This result, among others, implies that a connected cubic symmetric Cayley graph on G is normal except for T ≅ A47 and a connected cubic G-symmetric graph is G-normal except for T ≅ A7, A15 or PSL(4, 2).

An interval in a combinatorial structure R is a set I of points that are related to every point in R∖I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes—this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: an arbitrary structure S of size n belonging to a class 𝒞 can be embedded into a simple structure from 𝒞 by adding at most f(n) elements. We prove such results when 𝒞 is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than two. The functions f(n) in these cases are 2, ⌈log 2(n+1)⌉, ⌈(n+1)/2⌉, ⌈(n+1)/2⌉, ⌈log 4(n+1)⌉, ⌈log 3(n+1)⌉ and 1, respectively. In each case these bounds are the best possible.

In this paper we study the size of the largest clique ω(G(n, α)) in a random graph G(n, α) on n vertices which has power-law degree distribution with exponent α. We show that, for ‘flat’ degree sequences with α > 2, with high probability, the largest clique in G(n, α) is of a constant size, while, for the heavy tail distribution, when 0 < α < 2, ω(G(n, α)) grows as a power of n. Moreover, we show that a natural simple algorithm with high probability finds in G(n, α) a large clique of size (1 − o(1))ω(G(n, α)) in polynomial time.

We investigate the final size distribution of the SIR (susceptible-infected-recovered) epidemic model in the critical regime. Using the integral representation of Martin-Löf (1998) for the hitting time of a Brownian motion with parabolic drift, we derive asymptotic expressions for the final size distribution that capture the effect of the initial number of infectives and the closeness of the reproduction number to zero. These asymptotics shed light on the bimodularity of the limiting density of the final size observed in Martin-Löf (1998). We also discuss the connection to the largest component in the Erdős-Rényi random graph, and, using this connection, find an integral expression of the Laplace transform of the normalized Brownian excursion area in terms of Airy functions.

A dynamic model for a random network evolving in continuous time is defined, where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is connected to an existing vertex with probability proportional to a function b of the fitness of the existing vertex. Furthermore, a vertex dies at a rate given by a function d of its fitness. Using results from the theory of general branching processes, an expression for the asymptotic empirical fitness distribution {p k } is derived and analyzed for a number of specific choices of b and d. When b(i) = i + α and d(i) = β, that is, linear preferential attachment for the newborn and random deaths, then p k ∼ k -(2+α). When b(i) = i + 1 and d(i) = β(i + 1), with β < 1, then p k ∼ (1 + β)−k , that is, if the death rate is also proportional to the fitness, then the power-law distribution is lost. Furthermore, when b(i) = i + 1 and d(i) = β(i + 1)γ, with β, γ < 1, then logp k ∼ -k γ, a stretched exponential distribution. The momentaneous in-degrees are also studied and simulations suggest that their behaviour is qualitatively similar to that of the fitnesses.

The expansion constant of a simple graph G of order n is defined as where denotes the set of edges in G between the vertex subset S and its complement . An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.

Let n points be placed independently in d-dimensional space according to the density f(x) = A d e−λ||x||α , λ, α > 0, x ∈ ℝd , d ≥ 2. Let d n be the longest edge length of the nearest-neighbor graph on these points. We show that (λ−1 log n)1−1/α dn - b n converges weakly to the Gumbel distribution, where b n ∼ ((d − 1)/λα) log log n. We also prove the following strong law for the normalized nearest-neighbor distance d̃ n = (λ−1 log n)1−1/α dn / log log n: (d − 1)/αλ ≤ lim infn→∞ d̃ n ≤ lim supn→∞ d̃ n ≤ d/αλ almost surely. Thus, the exponential rate of decay α = 1 is critical, in the sense that, for α > 1, d n → 0, whereas, for α ≤ 1, d n → ∞ almost surely as n → ∞.

We study first passage percolation (FPP) on the configuration model (CM) having power-law degrees with exponent τ ∈ [1, 2) and exponential edge weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of edges on the minimal-weight path, both of which can be computed in terms of the Poisson-Dirichlet distribution. We explicitly describe these limits via construction of infinite limiting objects describing the FPP problem in the densely connected core of the network. We consider two separate cases, the original CM, in which each edge, regardless of its multiplicity, receives an independent exponential weight, and the erased CM, for which there is an independent exponential weight between any pair of direct neighbors. While the results are qualitatively similar, surprisingly, the limiting random variables are quite different. Our results imply that the flow carrying properties of the network are markedly different from either the mean-field setting or the locally tree-like setting, which occurs as τ > 2, and for which the hopcount between typical vertices scales as log n. In our setting the hopcount is tight and has an explicit limiting distribution, showing that information can be transferred remarkably quickly between different vertices in the network. This efficiency has a down side in that such networks are remarkably fragile to directed attacks. These results continue a general program by the authors to obtain a complete picture of how random disorder changes the inherent geometry of various random network models; see Aldous and Bhamidi (2010), Bhamidi (2008), and Bhamidi, van der Hofstad and Hooghiemstra (2009).

A random intersection graph G(n, m, p) is defined on a set V of n vertices. There is an auxiliary set W consisting of m objects, and each vertex v ∈ V is assigned a random subset of objects W v ⊆ W such that w ∈ W v with probability p, independently for all v ∈ V and all w ∈ W . Given two vertices v 1, v 2 ∈ V , we set v 1 ∼ v 2 if and only if W v1 ∩ W v2 ≠ ∅. We use Stein's method to obtain an upper bound on the total variation distance between the distribution of the number of h-cliques in G(n, m, p) and a related Poisson distribution for any fixed integer h.

Gibbs fields are constructed and studied which correspond to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs of a certain type, for which the Gaussian Gibbs fields need not be existing. In these graphs, the vertex degree growth is controlled by a summability requirement formulated with the help of a generalized Randić index. In particular, it is proven that the Gibbs fields obey uniform integrability estimates, which are then used in the study of the topological properties of the set of Gibbs fields. In the second part, a class of graphs is introduced in which the mentioned summability is obtained by assuming that the vertices of large degree are located at large distances from each other. This is a stronger version of the metric property employed in Bassalygo and Dobrushin (1986).

Suppose that, under the action of gravity, liquid drains through the unit d-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal basis directions of R d , d ≥ 2. The resulting network is a version of the so-called minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the large-sample asymptotic behaviour of the total power-weighted edge length of the network on uniform random points in (0, 1)d . The distributional results exhibit a weight-dependent phase transition between Gaussian and boundary-effect-derived distributions. These boundary contributions are characterized in terms of limits of the so-called on-line nearest-neighbour graph, a natural model of spatial network evolution, for which we also present some new results. Also, we give a convergence in distribution result for the length of the longest edge in the drainage network; when d = 2, the limit is expressed in terms of Dickman-type variables.

Clark et al. [‘The axiomatizability of topological prevarieties’, Adv. Math.218 (2008), 1604–1653] have shown that, for k≥2, there exists a Boolean topological graph that is k-colourable but not topologically k-colourable; that is, for every ϵ>0, it cannot be coloured by a paintbrush of width ϵ. We generalize this result to show that, for k≥2, there is a Boolean topological graph that is 2-colourable but not topologically k-colourable. This graph is an inverse limit of finite graphs which are shown to exist by an Erdős-style probabilistic argument of Hell and Nešetřil [‘The core of a graph’, Discrete Math.109 (1992), 117–126]. We use the fact that there exists a Boolean topological graph that is 2-colourable but not k-colourable, and some other results (some new and some previously known), to answer the question of which finitely generated topological residual classes of graphs are axiomatizable by universal Horn sentences. A more general version of this question was raised in the above-mentioned paper by Clark et al., and has been investigated by various authors for other structures.

We determine the minimal density of triangles in a tripartite graph with prescribed edge densities. This extends a previous result of Bondy, Shen, Thomassé and Thomassen characterizing those edge densities guaranteeing the existence of a triangle in a tripartite graph. To be precise we show that a suitably weighted copy of the graph formed by deleting a certain 9-cycle from K3,3,3 has minimal triangle density among all weighted tripartite graphs with prescribed edge densities.

Supplementary materials are available with this article.

An (r,r+1)-factor of a graph G is a spanning subgraph H such that dH(v)∈{r,r+1} for all vertices v∈𝒱(G). If G is expressed as the union of edge-disjoint (r,r+1)-factors, then this expression is an (r,r+1)-factorization of G. Let μ(r) be the smallest integer with the property that if G is a regular loopless multigraph of degree d with d≥μ(r), then G has an (r,r+1)-factorization. It is shown that if r is even. The proof employs a novel list-coloring approach. Together with known results, this shows that μ(r)=r2+1 if r is odd and if r is even.

We answer a recent question posed by Li et al. [‘Imprimitive symmetric graphs with cyclic blocks’, European J. Combin.31 (2010), 362–367] regarding a family of imprimitive symmetric graphs.

We consider a preferential duplication model for growing random graphs, extending previous models of duplication graphs by selecting the vertex to be duplicated with probability proportional to its degree. We show that a special case of this model can be analysed using the same stochastic approximation as for vertex-reinforced random walks, and show that ‘trapping’ behaviour can occur, such that the descendants of a particular group of initial vertices come to dominate the graph.

We investigate the degree sequence of the geometric preferential attachment model of Flaxman, Frieze and Vera (2006), (2007) in the case where the self-loop parameter α is set to 0. We show that, given certain conditions on the attractiveness function F, the degree sequence converges to the same sequence as found for standard preferential attachment in Bollobás et al. (2001). We also apply our method to the extended model introduced in van der Esker (2008) which allows for an initial attractiveness term, proving similar results.

In Reinert and Röllin (2009) a new approach - called the ‘embedding method’ - was introduced, which allows us to make use of exchangeable pairs for normal and multivariate normal approximations with Stein's method in cases where the corresponding couplings do not satisfy a certain linearity condition. The key idea is to embed the problem into a higher-dimensional space in such a way that the linearity condition is then satisfied. Here we apply the embedding to U-statistics as well as to subgraph counts in random graphs.

A graph is s-transitive if its automorphism group acts transitively on s-arcs but not on (s+1)-arcs in the graph. Let X be a connected tetravalent s-transitive graph of order twice a prime power. In this paper it is shown that s=1,2,3 or 4. Furthermore, if s=2, then X is a normal cover of one of the following graphs: the 4-cube, the complete graph of order 5, the complete bipartite graph K5,5 minus a 1-factor, or K7,7 minus a point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,2); if s=3, then X is a normal cover of the complete bipartite graph of order 4; if s=4, then X is a normal cover of the point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,3). As an application, we classify the tetravalent s-transitive graphs of order 2p2 for prime p.

We consider a variety of subtrees of various shapes lying on the fringe of a recursive tree. We prove that (under suitable normalization) the number of isomorphic images of a given fixed tree shape on the fringe of the recursive tree is asymptotically Gaussian. The parameters of the asymptotic normal distribution involve the shape functional of the given tree. The proof uses the contraction method.