Infinite Graphs

The idea of metric dimension in graph theory was introduced by P J Slater in [2]. It has been found applications in optimization, navigation, network theory, image processing, pattern recognition etc.Several other authors have studied metric dimension of various standard graphs. In this paper we introduce a real valued function called generalized metric G X × X × X ® R+ d : where X = r(v /W) =
{(d(v,v1),d(v,v2 ),...,d(v,v ) / v V (G))} k Î , denoted d G and is used to study metric dimension of graphs. It has been proved that metric dimension of any connected finite simple graph remains constant if d G numbers of pendant edges are added to the non-basis vertices.

One of the natural topologies for infinite graphs with edge-ends is ETop. Also ETop is the coarest topology among other topologies for infinite graphs.
In this note, we characterize this topology with different methods and we show that it is always compact.

This note presents a new, elementary proof of a generalization of a theorem of Halin to graphs with unbounded degrees, which is then applied to show that every connected, countably infinite graph G with a subdegree-finite, infinite automorphism group whose cardinality is strictly less than continuum, has a finite set F of vertices that is setwise stabilized only by the identity automorphism. A bound on the size of such sets, which are called distinguishing, is also provided.
To put this theorem of Halin and its generalization into perspective, we also discuss several related non-elementary, independent results and their methods of proof.

A problem by Diestel is to extend algebraic flow theory of finite graphs to infinite graphs
with ends. In order to pursue this problem, we define an A-flow and non-elusive H-flow for
arbitrary graphs and for abelian topological Hausdorff groups H and compact subsets A ⊆ H.
We use these new definitions to extend several well-known theorems of flows in finite graphs
to infinite graphs