Distance distribution of nodes in star graphs

The purpose of the work is to provide a solution to the long-standing problem of computing the distance distribution among the nodes in a star graph, i.e., computing the exact number of nodes at a distance kk from the identity node in a star graph where kk varies from 0 to the diameter of the graph. A star graph is a Cayley graph like a hypercube; for a hypercube QnQn, there are exactly nr nodes at a distance rr from the identity node where rr varies from 0 to nn.