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We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random…

Let $\Gamma$ be a simple finite graph with vertex set $V(\Gamma)$ and edge set $E(\Gamma)$. Let $\mathcal{R}$ be an equivalence relation on $V(\Gamma)$. The $\mathcal{R}$-super $\Gamma$ graph $\Gamma^{\mathcal{R}}$ is a simple graph with…

We present an improved method for symbolic regression that seeks to fit data to formulas that are Pareto-optimal, in the sense of having the best accuracy for a given complexity. It improves on the previous state-of-the-art by typically…

For a finite simplicial graph $\Gamma$, let $A(\Gamma)$ denote the right-angled Artin group on $\Gamma$. Recently Kim and Koberda introduced the extension graph $\Gamma^e$ for $\Gamma$, and established the Extension Graph Theorem: for…

In a number of recent works [6, 7] the authors have introduced and studied a functor $\mathcal{F}_k$ which associates to each loose graph $\Gamma$ -which is similar to a graph, but where edges with $0$ or $1$ vertex are allowed - a…

To each multiquiver $\Gamma$ we attach a solution to the consistency equations associated to twisted generalized Weyl (TGW) algebras. This generalizes several previously obtained solutions in the literature. We show that the corresponding…

In this paper, we study positivity phenomena for the $e$-coefficients of Stanley's chromatic function of a graph. We introduce a new combinatorial object: the {\em correct} sequences of unit interval orders, and using these, in certain…

Let $\Gamma$ be a convex cocompact thin subgroup of an arithmetic lattice in $\operatorname{SO}(n, 1)$. We generalize Selberg's $\frac{3}{16}$ theorem in this setting, i.e., we prove uniform exponential mixing of the frame flow and obtain a…

Let $G_n=\mathbb{Z}_n\times \mathbb{Z}_n$ for $n\geq 4$ and $S=\{(i,0),(0,i),(i,i): 1\leq i \leq n-1\}\subset G_n$. Define $\Gamma(n)$ to be the Cayley graph of $G_n$ with respect to the connecting set $S$. It is known that $\Gamma(n)$ is a…

Let $G$ be $2$-generated group. The generating graph of $\Gamma(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G=\langle g,h\rangle$. This graph encodes the combinatorial…

We develop a new formalism to study nonlinear evolution in the growth of large-scale structure, by following the dynamics of gravitational clustering as it builds up in time. This approach is conveniently represented by Feynman diagrams…

A graph $\Gamma$ is $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. If $V(\Gamma)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such…

In the context of algebraic renormalization, the extended antifield formalism is used to derive the general forms of the anomaly consistency condition and of the Callan-Symanzik equation for generic gauge theories. A local version of the…

In an earlier paper, the authors considered three types of graphs, and three equivalence relations, defined on a group, viz.\ the power graph, enhanced power graph, and commuting graph, and the relations of equality, conjugacy, and same…

We generalize Kudryavtseva and Mazorchuk's concept of canonical form of elements in Kiselman's semigroups to the setting of a Hecke-Kiselman monoid $\mathbf{HK}_\Gamma$ associated with a simple oriented graph $\Gamma$. We use confluence…

In his seminal paper "Formality conjecture", M. Kontsevich introduced a graph complex $GC_{1ve}$ closely connected with the problem of constructing a formality quasi-isomorphism for Hochschild cochains. In this paper, we express the…

We discuss a partial normalisation of a finite graph of finite groups $(\Gamma(-), X)$ which leaves invariant the fundamental group. In conjunction with an easy graph-theoretic result, this provides a flexible and rather useful tool in the…

The (directed) Bruhat graph $\hat{\Gamma}(u,v)$ has the elements of the Bruhat interval $[u,v]$ as vertices, with directed edges given by multiplication by a reflection. Famously, $\hat{\Gamma}(e,v)$ is regular if and only if the Schubert…

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it…

Let $G$ be 2-generated group. The generating graph $\Gamma(G)$ of $G$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G = \langle g, h \rangle.$ This definition can be extended to a…