Let X be an in®nite, locally ®nite, almost transitive graph with polynomial growth. We show that such a graph X is the inverse limit of an in®nite sequence of ®nite graphs satisfying growth conditions which are closely related to growth properties of the in®nite graph X. 1991 Mathematics Subject Classi®cation. Primary 05C25, Secondary 20F18 1. Introduction and statement of main result. We think...

The usual deenition of hyperbolicity of a group G demands that all geodesic triangles in the Cayley graph of G should be thin. Using the theorem that a subquadratic isoperimetric inequality implies a linear one, we show that it is in fact only necessary for all triangles from a given combing to be thin, thus giving a new criterion for hyperbolicity of nitely presented groups. Given a group G th...

Let p be a prime. It was shown by Folkman (J. Combin. Theory 3 (1967) 215) that a regular edge-transitive graph of order 2p or 2p is necessarily vertex-transitive. In this paper an extension of his result in the case of cubic graphs is given. It is proved that, with the exception of the Gray graph on 54 vertices, every cubic edge-transitive graph of order 2p is vertex-transitive. c © 2003 Elsev...

Graphs form a powerful modeling tool to represent complex relationships among objects in an effective manner. Graph pattern matching is one of the areas of data mining where the data is stored in the form of graphs and the set of tuples that match a user-given graph pattern are extracted. For finding the set of matching tuples faster, all the possible paths in the large directed graph, i.e., tr...

Let $G$ be a transitive permutation group on finite set $\Omega$ and recall that base for is subset of with trivial pointwise stabiliser. The size $G$, denoted $b(G)$, the minimal base. If $b(G)=2$ then we can study Saxl graph $\Sigma(G)$ which has vertex two vertices are adjacent if they form This vertex-transitive graph, conjectured to connected diameter at most $2$ when primitive. In this pa...

The problem of interactive foreground/background segmentation in still images is of great practical importance in image editing. They avoid the boundary-length bias of graph-cut methods and results in increased sensitivity to seed placement. A new proposed method of fully automatic processing frameworks is given based on Graph-cut and Geodesic Graph cut algorithms. This paper addresses the prob...

For two vertices X,Y ∈ V (G), a cycle is called a geodesic cycle with X and Y if a shortest path joining X and Y lies on the cycle. A graph G is called to be geodesic k-pancyclic if any two vertices X,Y on G have such geodesic cycle of length l that 2dG(X,Y ) + k ≤ l ≤ |V (G)|. In this paper, we show that the n-dimensional Möbius cube MQn is geodesic 2-pancyclic for n ≥ 3. This result is optima...

We call an undirected graph X half-transitive if the automorphism group Aut X of X acts transitively on the vertex set and edge set but not on the set of ordered pairs of adjacent vertices of X. In this paper we determine all half-transitive graphs of order p3 and degree 4, where p is an odd prime; namely, we prove that all such graphs are Cayley graphs on the non-Abelian group of order p3 and ...

In this paper a theorem is derived in order to provide a wide sufficient condition for an orthogonally transitive cylindrical spacetime to be singularity-free. The applicability of the theorem is tested on examples provided by the literature that are known to have regular curvature invariants.

A transitive decomposition is a pair (Γ,P) where Γ is a graph and P is a partition of the arc set of Γ such that there is a subgroup of automorphisms of Γ which leaves P invariant and transitively permutes the parts in P. In an earlier paper we gave a characterisation of G-transitive decompositions where Γ is the graph product Km×Km and G is a rank 3 group of product action type. This character...