Vertex-transitive graphs attaining the radius bound

Classify the vertex-transitive graphs G whose radius capture number equals rad(G) − 1.

Background

The paper proves that several important vertex-transitive families—including generously transitive, distance-transitive, and generalized Johnson graphs—satisfy rc(G) = rad(G) − 1. It also exhibits vertex-transitive graphs for which the inequality is strict, showing that the property does not hold universally. The authors therefore leave open the classification of vertex-transitive graphs attaining equality; this is identified as a special case of Problem 4.