Li et al. [‘On finite self-complementary metacirculants’, J. Algebraic Combin.40 (2014), 1135–1144] proved that the automorphism group of a self-complementary metacirculant is either soluble or has $\text{A}_{5}$ as the only insoluble composition factor, and gave a construction of such graphs with insoluble automorphism groups (which are the first examples of self-complementary graphs with this property). In this paper, we will prove that each simple group is a subgroup (so is a section) of the automorphism groups of infinitely many self-complementary vertex-transitive graphs. The proof involves a construction of such graphs. We will also determine all simple sections of the automorphism groups of self-complementary vertex-transitive graphs of $4$-power-free order.

For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r ⩾ 4 and 0 < p < 1. Suppose that H is an n-vertex triple system with r|n and the following two properties:

• • for every graph G with V(G) = V(H), at least p proportion of the triangles in G are also edges of H,

• • for every vertex x of H, the link graph of x is a quasirandom graph with density at least p.

For natural numbers $n,r\in \mathbb{N}$ with $n\geqslant r$, the Kneser graph $K(n,r)$ is the graph on the family of $r$-element subsets of $\{1,\ldots ,n\}$ in which two sets are adjacent if and only if they are disjoint. Delete the edges of $K(n,r)$ with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as $r/n$ is bounded away from $1/2$, even when the probability of retaining an edge of the Kneser graph is quite small. This gives us a random analogue of the Erdős–Ko–Rado theorem, since an independent set in the Kneser graph is the same as a uniform intersecting family. To prove our main result, we give some new estimates for the number of disjoint pairs in a family in terms of its distance from an intersecting family; these might be of independent interest.

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. The evolution of the rescaled size of the largest component in such variations of the Erdős–Rényi random graph process has recently received considerable attention, in particular for Bollobás's ‘product rule’. In this paper we establish the following result for rules such as the product rule: the limit of the rescaled size of the ‘giant’ component exists and is continuous provided that a certain system of differential equations has a unique solution. In fact, our result applies to a very large class of Achlioptas-like processes.

Our proof relies on a general idea which relates the evolution of stochastic processes to an associated system of differential equations. Provided that the latter has a unique solution, our approach shows that certain discrete quantities converge (after appropriate rescaling) to this solution.

Recently, settling a question of Erdős, Balogh, and Petříčková showed that there are at most $2^{n^{2}/8+o(n^{2})}$$n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here, we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph $G$ admits a vertex partition $X\cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set.

Our proof uses the Ruzsa–Szemerédi removal lemma, the Erdős–Simonovits stability theorem, and recent results of Balogh, Morris, and Samotij and Saxton and Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_{3}$s, which is of independent interest.

Let p 1, p 2, p 3 be three noncollinear points in the plane, and let P be a set of n other points in the plane. We show that the number of distinct distances between p 1, p 2, p 3 and the points of P is Ω(n 6/11), improving the lower bound Ω(n 0.502) of Elekes and Szabó [4] (and considerably simplifying the analysis).

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph Kn and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament Tk on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of Tk; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2 − o(1))log2n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2 − o(1))log2n Breaker can prevent the underlying graph of Maker's digraph from containing a k-clique. Moreover, the precise value of our lower bound differs from the upper bound only by an additive constant of 12.

We also discuss the question of whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two ‘clever’ players and the game played by two ‘random’ players, is supported by the tournament game. It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid.

Finally, we consider the orientation game version of the tournament game, where Maker wins the game if the final digraph – also containing the edges directed by Breaker – possesses a copy of Tk. We prove that in that game Breaker has a winning strategy for k = (4 + o(1))log2n.

Let $G$ be a finite group and ${\rm\Gamma}$ a $G$-symmetric graph. Suppose that $G$ is imprimitive on $V({\rm\Gamma})$ with $B$ a block of imprimitivity and ${\mathcal{B}}:=\{B^{g};g\in G\}$ a system of imprimitivity of $G$ on $V({\rm\Gamma})$. Define ${\rm\Gamma}_{{\mathcal{B}}}$ to be the graph with vertex set ${\mathcal{B}}$ such that two blocks $B,C\in {\mathcal{B}}$ are adjacent if and only if there exists at least one edge of ${\rm\Gamma}$ joining a vertex in $B$ and a vertex in $C$. Xu and Zhou [‘Symmetric graphs with 2-arc-transitive quotients’, J. Aust. Math. Soc. 96 (2014), 275–288] obtained necessary conditions under which the graph ${\rm\Gamma}_{{\mathcal{B}}}$ is 2-arc-transitive. In this paper, we completely settle one of the cases defined by certain parameters connected to ${\rm\Gamma}$ and ${\mathcal{B}}$ and show that there is a unique graph corresponding to this case.

A random k-out mapping (digraph) on [n] is generated by choosing k random images of each vertex one at a time, subject to a 'preferential attachment' rule: the current vertex selects an image i with probability proportional to a given parameter α = α(n) plus the number of times i has already been selected. Intuitively, the larger α becomes, the closer the resulting k-out mapping is to the uniformly random k-out mapping. We prove that α = Θ(n 1/2) is the threshold for α growing 'fast enough' to make the random digraph approach the uniformly random digraph in terms of the total variation distance. We also determine an exact limit for this distance for the α = βn 1/2 case.

We consider the number of spanning trees in circulant graphs of ${\it\beta}n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of ${\it\beta}n$ factors, while we derive a formula of ${\it\beta}-1$ factors. We also derive a formula for the number of spanning trees in discrete tori. Finally, we compare the spanning tree entropy of circulant graphs with fixed and nonfixed generators.

We give an easy method for constructing containers for simple hypergraphs. The method also has consequences for non-simple hypergraphs. Some applications are given; in particular, a very transparent calculation is offered for the number of H-free hypergraphs, where H is some fixed uniform hypergraph.

We study the isomorphism problem of vertex-transitive cubic graphs which have a transitive simple group of automorphisms.

We study 3-random-like graphs, that is, sequences of graphs in which the densities of triangles and anti-triangles converge to 1/8. Since the random graph $\mathcal{G}$ n,1/2 is, in particular, 3-random-like, this can be viewed as a weak version of quasi-randomness. We first show that 3-random-like graphs are 4-universal, that is, they contain induced copies of all 4-vertex graphs. This settles a question of Linial and Morgenstern [10]. We then show that for larger subgraphs, 3-random-like sequences demonstrate completely different behaviour. We prove that for every graph H on n ⩾ 13 vertices there exist 3-random-like graphs without an induced copy of H. Moreover, we prove that for every ℓ there are 3-random-like graphs which are ℓ-universal but not m-universal when m is sufficiently large compared to ℓ.

In 1981 Beck and Fiala proved an upper bound for the discrepancy of a set system of degree d that is independent of the size of the ground set. In the intervening years the bound has been decreased from 2d − 2 to 2d − 4. We improve the bound to 2d − log*d.

The Teter, Payne, and Allan “preconditioning” function plays a significant role in planewave DFT calculations. This function is often called the TPA preconditioner. We present a detailed study of this “preconditioning” function. We develop a general formula that can readily generate a class of “preconditioning” functions. These functions have higher order approximation accuracy and fulfill the two essential “preconditioning” purposes as required in planewave DFT calculations. Our general class of functions are expected to have applications in other areas.

The discrete Green's function (without boundary) $\mathbb{G}$ is a pseudo-inverse of the combinatorial Laplace operator of a graph G = (V, E). We reveal the intimate connection between Green's function and the theory of exact stopping rules for random walks on graphs. We give an elementary formula for Green's function in terms of state-to-state hitting times of the underlying graph. Namely, $\mathbb{G}(i,j) = \pi_j \bigl( H(\pi,j) - H(i,j) \bigr),$ where πi is the stationary distribution at vertex i, H(i, j) is the expected hitting time for a random walk starting from vertex i to first reach vertex j, and H(π, j) = ∑k∈V πk H(k, j). This formula also holds for the digraph Laplace operator.

The most important characteristics of a stopping rule are its exit frequencies, which are the expected number of exits of a given vertex before the rule halts the walk. We show that Green's function is, in fact, a matrix of exit frequencies plus a rank one matrix. In the undirected case, we derive spectral formulas for Green's function and for some mixing measures arising from stopping rules. Finally, we further explore the exit frequency matrix point of view, and discuss a natural generalization of Green's function for any distribution τ defined on the vertex set of the graph.

We consider large random graphs with prescribed degrees, as generated by the configuration model. In the regime where the empirical degree distribution approaches a limit μ with finite mean, we establish the systematic convergence of a broad class of graph parameters that includes the independence number, the maximum cut size, the logarithm of the Tutte polynomial, and the free energy of the anti-ferromagnetic Ising and Potts models. Contrary to previous works, our results are not a priori limited to the free energy of some prescribed graphical model. They apply more generally to any additive, Lipschitz and concave graph parameter. In addition, the corresponding limits are shown to be Lipschitz and concave in the degree distribution μ. This considerably extends the applicability of the celebrated interpolation method, introduced in the context of spin glasses, and recently related to the challenging question of right-convergence of sparse graphs.

We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graph Λ, this functor determines a category equivalence between the category of coverings of Λ and the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions for k-graphs: projective limits and crossed products by finitely generated free abelian groups.

The classical first and second Zagreb indices of a graph $G$ are defined as $M_{1}(G)=\sum _{v\in V(G)}d(v)^{2}$ and $M_{2}(G)=\sum _{e=uv\in E(G)}d(u)d(v),$ where $d(v)$ is the degree of the vertex $v$ of $G.$ Recently, Furtula et al. [‘On difference of Zagreb indices’, Discrete Appl. Math.178 (2014), 83–88] studied the difference of $M_{1}$ and $M_{2},$ and showed that this difference is closely related to the vertex-degree-based invariant $RM_{2}(G)=\sum _{e=uv\in E(G)}[d(u)-1][d(v)-1]$, the reduced second Zagreb index. In this paper, we present sharp bounds for the reduced second Zagreb index, given the matching number, independence number and vertex connectivity, and we also completely determine the extremal graphs.

We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degree d tends to some positive number c d > 0 almost surely as the number of steps goes to ∞, and c d ~ (eπ)1/2 d 1/4e-2√d holds as d → ∞.