Given a connected graph, in many cases it is possible to construct a structure tree that provides information about the ends of the graph or its connectivity. For example Stallings’ theorem on the structure of groups with more than one end can be proved by analyzing the action of the group on a structure tree and Tutte used a structure tree to investigate finite 2-connected graphs, that are not...

Let $G$ be a finite group. The graph $D(G)$ is a divisibility graph of $G$. Its vertex set is the non-central conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. In this paper, we investigate the structure of the divisibility graph $D(G)$ for a non-solvable group with $sigma^{ast}(G)=2$, a finite simple group $G$ that satisfies the one-p...

We consider tilings of the plane whose graphs are locally finite and 3-connected. We show that if the automorphism group of the graph has finitely many orbits, then there is an isomorphic tiling in either the Euclidean or hyperbolic plane such that the group of automorphisms acts as a group of isometries. We apply this fact to the classification of self-dual doubly periodic tilings via Coxeter’...

We discuss the finite-size scaling of ferromagnetic Ising model on random regular graphs. These graphs are locally tree-like, and in limit large graphs, Bethe approximation gives exact free energy per site. In thermodynamic limit, these show a phase transition. This transition is rounded off for finite verify theory prediction that this rounding described terms variable $[T/T_c -1] S^{1/2}$ (wh...

An isogeny graph is a graph whose vertices are principally polarized abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel described the structure of isogeny graphs for elliptic curves and showed that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth first search algorithm in the graph. In dimension 2,...