In this note, we bound the metric dimension of the circulant graphs $C_n(1,2,\ldots ,t)$. We shall prove that if $n=2tk+t$ and if t is odd, then $\dim (C_n(1,2,\ldots ,t))=t+1$, which confirms Conjecture 4.1.1 in Chau and Gosselin (2017, Opuscula Mathematica 37, 509–534). In Vetrík (2017, Canadian Mathematical Bulletin 60, 206–216; 2020, Discussiones Mathematicae. Graph Theory 40, 67–76), the author has shown that $\dim (C_n(1,2,\ldots ,t))\leq t+\left \lceil \frac {p}{2}\right \rceil $ for $n=2tk+t+p$, where $t\geq 4$ is even, $1\leq p\leq t+1$, and $k\geq 1$. Inspired by his work, we show that $\dim (C_n(1,2,\ldots ,t))\leq t+\left \lfloor \frac {p}{2}\right \rfloor $ for $n=2tk+t+p$, where $t\geq 5$ is odd, $2\leq p\leq t+1$, and $k\geq 2$.

Resolving a problem of Conlon, Fox and Rödl, we construct a family of hypergraphs with arbitrarily large tower height separation between their $2$-colour and q-colour Ramsey numbers. The main lemma underlying this construction is a new variant of the Erdős–Hajnal stepping-up lemma for a generalized Ramsey number $r_k(t;q,p)$, which we define as the smallest integer n such that every q-colouring of the k-sets on n vertices contains a set of t vertices spanning fewer than p colours. Our results provide the first tower-type lower bounds on these numbers.

We study the locations of complex zeroes of independence polynomials of bounded-degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer’s result on the optimal zero-free disc, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $\Delta$ have a zero-free disc almost as large as the optimal disc for graphs of maximum degree $\Delta$ established by Shearer (of radius $\sim 1/(e \Delta )$). Up to logarithmic factors in $\Delta$ this is optimal, even for hypergraphs with all edge sizes strictly greater than $2$. We conjecture that for $k\ge 3$, $k$-uniform linear hypergraphs have a much larger zero-free disc of radius $\Omega (\Delta ^{- \frac{1}{k-1}} )$. We establish this in the case of linear hypertrees.

It is consistent relative to an inaccessible cardinal that ZF+DC holds, and the hypergraph of isosceles triangles on $\mathbb {R}^2$ has countable chromatic number while the hypergraph of isosceles triangles on $\mathbb {R}^3$ has uncountable chromatic number.

Let $\ell $ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell $-towers of multigraphs. In this context, growth patterns are realized by certain analogs of Iwasawa invariants, which depend on the prime $\ell $ and the abelian $\ell $-tower of multigraphs. We formulate and study statistical questions about the behavior of the Iwasawa $\mu $ and $\lambda $ invariants.

We show that an $n$-uniform maximal intersecting family has size at most $e^{-n^{0.5+o(1)}}n^n$. This improves a recent bound by Frankl ((2019) Comb. Probab. Comput. 28(5) 733–739.). The Spread Lemma of Alweiss et al. ((2020) Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing.) plays an important role in the proof.

We make precise and prove a conjecture of Klivans about actions of the sandpile group on spanning trees. More specifically, the conjecture states that there exists a unique ‘suitably nice’ sandpile torsor structure on plane graphs which is induced by rotor-routing.

First, we rigorously define a sandpile torsor algorithm (on plane graphs) to be a map which associates each plane graph (i.e., planar graph with an appropriate ribbon structure) with a free transitive action of its sandpile group on its spanning trees. Then, we define a notion of consistency, which requires a torsor algorithm to be preserved with respect to a certain class of contractions and deletions. Using these definitions, we show that the rotor-routing sandpile torsor algorithm is consistent. Furthermore, we demonstrate that there are only three other consistent algorithms on plane graphs, which all have the same structure as rotor-routing.

We also define sandpile torsor algorithms on regular matroids and suggest a notion of consistency in this context. We conjecture that the Backman-Baker-Yuen algorithm is consistent, and that there are only three other consistent sandpile torsor algorithms on regular matroids, all with the same structure.

We study generalised quasirandom graphs whose vertex set consists of $q$ parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lovász and Sós showed that the structure of such graphs is forced by homomorphism densities of graphs with at most $(10q)^q+q$ vertices; subsequently, Lovász refined the argument to show that graphs with $4(2q+3)^8$ vertices suffice. Our results imply that the structure of generalised quasirandom graphs with $q\ge 2$ parts is forced by homomorphism densities of graphs with at most $4q^2-q$ vertices, and, if vertices in distinct parts have distinct degrees, then $2q+1$ vertices suffice. The latter improves the bound of $8q-4$ due to Spencer.

We study Cayley graphs of abelian groups from the perspective of quantum symmetries. We develop a general strategy for determining the quantum automorphism groups of such graphs. Applying this procedure, we find the quantum symmetries of the halved cube graph, the folded cube graph, and the Hamming graphs.

Given a connected graph $H$ which is not a star, we show that the number of copies of $H$ in a dense uniformly random regular graph is asymptotically Gaussian, which was not known even for $H$ being a triangle. This addresses a question of McKay from the 2010 International Congress of Mathematicians. In fact, we prove that the behavior of the variance of the number of copies of $H$ depends in a delicate manner on the occurrence and number of cycles of $3,4,5$ edges as well as paths of $3$ edges in $H$. More generally, we provide control of the asymptotic distribution of certain statistics of bounded degree which are invariant under vertex permutations, including moments of the spectrum of a random regular graph. Our techniques are based on combining complex-analytic methods due to McKay and Wormald used to enumerate regular graphs with the notion of graph factors developed by Janson in the context of studying subgraph counts in $\mathbb {G}(n,p)$.

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size $C\log _2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of ‘random-like’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables.

The proof proceeds via an ‘additive structure’ dichotomy on the degree sequence and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics and low-rank approximation. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes.

Real networks often exhibit clustering, the tendency to form relatively small groups of nodes with high edge densities. This clustering property can cause large numbers of small and dense subgraphs to emerge in otherwise sparse networks. Subgraph counts are an important and commonly used source of information about the network structure and function. We study probability distributions of subgraph counts in a community affiliation graph. This is a random graph generated as an overlay of m partly overlapping independent Bernoulli random graphs (layers) $G_1,\dots,G_m$ with variable sizes and densities. The model is parameterised by a joint distribution of layer sizes and densities. When m grows linearly in the number of nodes n, the model generates sparse random graphs with a rich statistical structure, admitting a nonvanishing clustering coefficient and a power-law limiting degree distribution. In this paper we establish the normal and $\alpha$-stable approximations to the numbers of small cliques, cycles, and more general 2-connected subgraphs of a community affiliation graph.

If ${\mathbf v} \in {\mathbb R}^{V(X)}$ is an eigenvector for eigenvalue $\lambda $ of a graph X and $\alpha $ is an automorphism of X, then $\alpha ({\mathbf v})$ is also an eigenvector for $\lambda $. Thus, it is rather exceptional for an eigenvalue of a vertex-transitive graph to have multiplicity one. We study cubic vertex-transitive graphs with a nontrivial simple eigenvalue, and discover remarkable connections to arc-transitivity, regular maps, and number theory.

Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that the only volume-preserving B-series method is the exact flow of the differential equation. The question was raised whether there exists a volume-preserving integrator that can be expanded as an aromatic B-series. In this work, we introduce a new algebraic tool, called the aromatic bicomplex, similar to the variational bicomplex in variational calculus. We prove the exactness of this bicomplex and use it to describe explicitly the key object in the study of volume-preserving integrators: the aromatic forms of vanishing divergence. The analysis provides us with a handful of new tools to study aromatic B-series, gives insights on the process of integration by parts of trees, and allows to describe explicitly the aromatic B-series of a volume-preserving integrator. In particular, we conclude that an aromatic Runge–Kutta method cannot preserve volume.

We propose a modification to the random destruction of graphs: given a finite network with a distinguished set of sources and targets, remove (cut) vertices at random, discarding components that do not contain a source node. We investigate the number of cuts required until all targets are removed, and the size of the remaining graph. This model interpolates between the random cutting model going back to Meir and Moon (J. Austral. Math. Soc. 11, 1970) and site percolation. We prove several general results, including that the size of the remaining graph is a tight family of random variables for compatible sequences of expander-type graphs, and determine limiting distributions for binary caterpillar trees and complete binary trees.

A bootstrap percolation process on a graph with n vertices is an ‘infection’ process evolving in rounds. Let $r \ge 2$ be fixed. Initially, there is a subset of infected vertices. In each subsequent round, every uninfected vertex that has at least r infected neighbors becomes infected as well and remains so forever.

We consider this process in the case where the underlying graph is an inhomogeneous random graph whose kernel is of rank one. Assuming that initially every vertex is infected independently with probability $p \in (0,1]$, we provide a law of large numbers for the size of the set of vertices that are infected by the end of the process. Moreover, we investigate the case $p = p(n) = o(1)$, and we focus on the important case of inhomogeneous random graphs exhibiting a power-law degree distribution with exponent $\beta \in (2,3)$. The first two authors have shown in this setting the existence of a critical $p_c =o(1)$ such that, with high probability, if $p =o(p_c)$, then the process does not evolve at all, whereas if $p = \omega(p_c)$, then the final set of infected vertices has size $\Omega(n)$. In this work we determine the asymptotic fraction of vertices that will eventually be infected and show that it also satisfies a law of large numbers.

The celebrated Erdős–Ko–Rado (EKR) theorem for Paley graphs of square order states that all maximum cliques are canonical in the sense that each maximum clique arises from the subfield construction. Recently, Asgarli and Yip extended this result to Peisert graphs and other Cayley graphs which are Peisert-type graphs with nice algebraic properties on the connection set. On the other hand, there are Peisert-type graphs for which the EKR theorem fails to hold. In this article, we show that the EKR theorem of Paley graphs extends to almost all pseudo-Paley graphs of Peisert-type. Furthermore, we establish the stability results of the same flavor.

In this note, we give a precise description of the limiting empirical spectral distribution for the non-backtracking matrices for an Erdős-Rényi graph $G(n,p)$ assuming $np/\log n$ tends to infinity. We show that derandomizing part of the non-backtracking random matrix simplifies the spectrum considerably, and then, we use Tao and Vu’s replacement principle and the Bauer-Fike theorem to show that the partly derandomized spectrum is, in fact, very close to the original spectrum.