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Graph Transformers, which incorporate self-attention and positional encoding, have recently emerged as a powerful architecture for various graph learning tasks. Despite their impressive performance, the complex non-convex interactions…
The graph of overlapping permutations is a directed graph that is an analogue to the De Bruijn graph. It consists of vertices that are permutations of length $n$ and edges that are permutations of length $n+1$ in which an edge $a_1\cdots…
In order to have a compact visualization of the order type of a given point set S, we are interested in geometric graphs on S with few edges that unambiguously display the order type of S. We introduce the concept of exit edges, which…
The widely studied edge modification problems ask how to minimally alter a graph to satisfy certain structural properties. In this paper, we introduce and study a new edge modification problem centered around transforming a given graph into…
We define the $k$-cut complex of a graph $G$ with vertex set $V(G)$ to be the simplicial complex whose facets are the complements of sets of size $k$ in $V(G)$ inducing disconnected subgraphs of $G$. This generalizes the Alexander dual of a…
The problem of finding graph structure of functions commuting with a given function in terms of their functional graphs is considered. Structure of functional graphs of commuting functions is described. The problem is reduced to describing…
Motivated by a fundamental geometrical object, the cut locus, we introduce and study a new combinatorial structure on graphs.
We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on $H$-free graphs, that is, graphs that do not contain some fixed graph $H$ as an induced subgraph. In particular, we prove that for every $s\geq 1$, both…
It is a well known result in the covering groups that a subgroup $G$ of the fundamental group at the identity of a semi-locally simply connected topological group determines a covering morphism of topological groups with characteristic…
The purpose of this paper is to survey the structure of closed and transitive transformation groups acting on a closed surface. In particular, we prove a number of relations between groups acting on the sphere that contain the rotation…
In this article we continue the study of automorphism groups of constant length substitution shifts and also their topological factors. We show that up to conjugacy, all roots of the identity map are letter exchanging maps, and all other…
Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple…
We present a new general theory of function-based hypergraph transformations on finite families of finite hypergraphs. A function-based hypergraph transformation formalises the action of structurally modifying hypergraphs from a family in a…
In this paper we study for the incompressible Euler equations the global structure of the bifurcation diagram for the rotating doubly connected patches near the degenerate case. We show that the branches with the same symmetry merge forming…
We present a strong connection between quantum information and quantum permutation groups. Specifically, we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups, and then show…
We study some percolation problems on the complete graph over $\mathbf N$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the…
Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabiliser in $G$ is trivial. In this paper we introduce and study an associated graph $\Sigma(G)$, which we call the Saxl graph of…
Let $G$ be a finite group. A number of graphs with the vertex set $G$ have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful…
We generalize the enhanced power graph by replacing elements with classes under automorphisms. We show that the connectivity and diameter of this graph is similar to that of the enhanced power graph. We consider the universal vertices of…
Orbits of graphs under local complementation (LC) and edge local complementation (ELC) have been studied in several different contexts. For instance, there are connections between orbits of graphs and error-correcting codes. We define a new…