Homogeneity in graphs and digraphs Seminar

We discuss several symmetry conditions on graphs and directed graphs. There is a wide range between two classes that may be seen as the extremal ones: the transitive graphs and the homogeneous graphs. Having a particular notion of symmetry, the question of a classification of all graphs with that symmetry condition arises. In this talk we will discuss this question for some particular classes of graphs. We will focus on connected-homogeneous graphs; their definition is a natural weakening of homogeneity. We will describe the state of the art of the classification of the connected-homogeneous directed graphs and give arguments that show that their classification is on the one hand complete but on the other hand still incomplete.