Related papers: Two generalisations of Leighton's Theorem

We prove that every connected locally finite regular graph has a double cover which is isomorphic to a Schreier graph.

Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric…

Generalizations of Redfield's master theorem and superposition theorem are proved by using decomposition of the tensor product of several induced monomial representations of the symmetric group $S_d$ into transitive constituents. As direct…

A recent development in graph-minor theory is to study local separators, vertex-sets that separate graphs locally but not necessarily globally. The local separators of a graph roughly correspond to the genuine separators of its local…

For a finite group $G$, let $\mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $\mathrm{diam}(G)$ is bounded by ${(\log_{2} |G|)}^{O(1)}$ in case $G$ is a…

We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can…

The closure of a graph $G$ is the graph $G^*$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n$, where $n$ is the number of vertices of $G$. The well-known Closure Lemma…

There are many results asserting the existence of tree-decompositions of minimal width which still represent local connectivity properties of the underlying graph, perhaps the best-known being Thomas' theorem that proves for every graph $G$…

We show that the groupoids of two directed graphs are isomorphic if and only if the two graphs are orbit equivalent by an orbit equivalence that preserves isolated eventually periodic points. We also give a complete description of the…

We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon on a unique metric space and we relate the dimension of this…

We investigate how to find generic and globally rigid realizations of graphs in $\mathbb{R}^d$ based on elementary geometric observations. Our arguments lead to new proofs of a combinatorial characterization of the global rigidity of graphs…

We present a rigorous proof of the convergence theorem for the Feynman graphs in arbitrary massive Euclidean quantum field theories on non-commutative R^d (NQFT). We give a detailed classification of divergent graphs in some massive NQFT…

We introduce and study asymptotically rigid mapping class groups of certain infinite graphs. We determine their finiteness properties and show that these depend on the number of ends of the underlying graph. In a special case where the…

We prove that every vertex transitive, planar, 1-ended, graph covers every graph whose balls of radius r are isomorphic to the ball of radius r in G for a sufficiently large r. We ask whether this is a general property of finitely presented…

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs. We also give an alternative proof of an extension to locally finite graphs…

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple…

A foundational theorem of Laman provides a counting characterisation of the finite simple graphs whose generic bar-joint frameworks in two dimensions are infinitesimally rigid. Recently a Laman-type characterisation was obtained for…

Random matrix models generalize to Group Field Theories (GFT) whose Feynman graphs are dual to gluings of higher dimensional simplices. It is generally assumed that GFT graphs are always dual to pseudo manifolds. In this paper we prove that…

This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with…

We introduce a new graph invariant of finite groups that provides a complete characterization of the splitting types of unramified prime ideals in normal number field extensions entirely in terms of the Galois group. In particular, each…