Topology of Cayley graphs applied to inverse additive problems

We present the basic isopermetric structure theory, obtaining some new simplified proofs. Let 1 ≤ r ≤ k be integers. As an application, we obtain simple descriptions for the subsets S of an abelian group with |kS| ≤ k|S|−k+1 or |kS−rS|−(k+r)|S|, where where S denotes as usual the Minkowski sum of copies of S. These results may be applied to several questions in Combinatorics and Additive Combinatorics including the Frobenius Problem, Waring’s problem in finite fields and the structure of abelian Cayley graphs with a big diameter.Peer Reviewe

On the connectivity of p-diamond-free vertex transitive graphs

AbstractLet G be a graph of order n(G), minimum degree δ(G) and connectivity κ(G). We call the graph G maximally connected when κ(G)=δ(G). The graph G is said to be superconnected if every minimum vertex cut isolates a vertex.For an integer p≥1, we define a p-diamond as the graph with p+2 vertices, where two adjacent vertices have exactly p common neighbors, and the graph contains no further edges. Usually, the 1-diamond is triangle and the 2-diamond is diamond. We call a graph p-diamond-free if it contains no p-diamond as a (not necessarily induced) subgraph. A graph is vertex transitive if its automorphism group acts transitively on its vertex set.In this paper, we give some sufficient conditions for vertex transitive graphs to be maximally connected. In addition, superconnected p-diamond-free (1≤p≤3) vertex transitive graphs are characterized

Further topics in connectivity

Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered. For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version

Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010

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