Roundness properties of graphs

Abstract

The notion of the roundness of a metric space was introduced by Per Enflo as a tool to study geometric properties of Banach spaces. Recently, roundness and generalized roundness have been used in the context of group theory to investigate relationships between the geometry of a Cayley graph of a group and the algebraic properties of the group. In this paper, we study roundness properties of connected graphs in general. We explicitly calculate the roundness of members of two classes of graphs and we give results of computer calculations of the roundness of all connected graphs on $7$, $8$ and $9$ vertices. We also show that no connected graph can have roundness between ${log}_{2}3$ and $2$.