On the topology of vertex-transitive graphs

The undirected graphs atoms were introduced independently Mader[21] and Watkins [28] in order to show that the connectivity of a connected undirected vertex-transitive is large. The directed graphs atoms were introduced by Chaty in [3]. The author obtained a structure Theorem for the atoms in the last case [9], showing that the connectivity of a connected vertex-transitive directed is large. The isoperimetric connectivity and isoperimetric atoms introduced by the author [7] in connection with problems in Additive Combinatorics. They form also a tool in investigation of the superconnectivity and its generalizations [19, 15]. Some applications of these concepts in Additive Combinatorics are mentionned in one of the survey papers [27, 10], where the relation with additive results like the Cauchy-Davenport Theorem [2, 4] is explained. A classical result of Kemperman [18] classifying the inequality |A+B| ≤ |A|+|B|−1, may be used to calculate all the small cutsets in abelian Cayley graphs. Kemperman’s result was used by the authors of [16] to obtain a characterization of superconnected abelian Cayley digraphs. The 2-atoms are not the appropriate object to simplify the classical complicated proof of Kemperman’s result. Another object, introduced in [11] and called the hyper-atom, may do the last job. In the abelian case, there is a 2-atom which is a subgroup (with few exceptions related to arithmetic progressions), if the second connectivity is not large [8]. This is not valid in the non-abelian case, but another object (the second antiatom) is sometimes a subgroup allowing a description for the inequality |AB| ≤ |A| + |B| − 1, in the non-abelian case [9]. In a recent paper, Balandraud [1] introduced another object called a cell to give a new proof of Kneser’s Theorem [23]. More recently, the author introduced the molecules [12], in order to generalize the Scherk-Kemperman Theorem [17]. In the first part of this talk, I plane to give a survey trying to unify these objects and mentioning some application.