Let $G$ be a finite group. We construct the prime graph of $ G $,which is denoted by $ Gamma(G) $ as follows: the vertex set of thisgraph is the prime divisors of $ |G| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ G $ contains anelement of order $ pq $.In this paper, we determine finite groups $ G $ with $ Gamma(G) =Gamma(L_3(q)) $, $2 leq q < 100 $ and prov...

We construct spanning trees in locally finite hyperbolic graphs whose boundary has finite (Assouad) dimension that represent their hyperbolic compactification in a good way: so that the tree has at least one but at most a bounded number of disjoint rays to each boundary point. As a corollary we extend a result of Gromov which says that from every hyperbolic graph with bounded degrees one can co...

Codewords in finite covers of a Tanner graph G are characterized. Since iterative, locally operating decoding algorithms cannot distinguish the underlying graph G from any covering graph, these codewords, dubbed pseudo-codewords are directly responible for sub-optimal behavior of iterative decoding algorithms. We give a simple characterization of pseudocodewords from finite covers and show that...

A triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. In 2000, Reay and Zamfirescu showed that all 2-connected, linearly convex triangular grid graphs (with the exception of one of them) are hamiltonian. The only exception is a graph D which is the linearlyconvex hull of the Star of David. We extend this result to a wider...

A Hamilton circle in an infinite graph is a homeomorphic copy of the unit circle S1 that contains all vertices and all ends precisely once. We prove that every connected, locally connected, locally finite, claw-free graph has such a Hamilton circle, extending a result of Oberly and Sumner to infinite graphs. Furthermore, we show that such graphs are Hamilton-connected if and only if they are 3-...

By a result of Gallai, every finite graph G has a vertex partition into two parts each inducing an element of its cycle space. This fails for infinite graphs if, as usual, the cycle space is defined as the span of the edge sets of finite cycles in G. However we show that, for the adaptation of the cycle space to infinite graphs recently introduced by Diestel and Kühn (which involves infinite cy...

we characterize the rational subsets of a finite group and discuss the relations to integral cayley graphs.

Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each k, there is a finite obstruction set Ok of graphs such that a graph G has linear rank-width at most k if and only if no vertex-minor of G is isomorphic to a graph in Ok. However, no attempts have been made to bound the number of gr...

We show that if E is an arbitrary acyclic graph then the Leavitt path algebra LK(E) is locally K-matricial; that is, LK(E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the fieldK. As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E: (1) LK (E) is von Neumann regu...