Cliques in exact distance powers of graphs of given maximum degree

The exact distance p-power of a graph G, denoted G[#p], is on vertex set V(G) in which two vertices are adjacent if they at exactly p G. Given integers k and p, we define f(k, p) to be the maximum possible order clique p-powers graphs with degree + 1. It easily observed that 2) ≤ k2 We prove equality may only hold connected component G isomorphic member class Pk incidence finite projective k-geometries. (These famous combinatorial structures known exist when prime power, conjectured not for other values k.) then study case 1 number k. One way obtain such remove from P k; call Pk' all resulting graphs. any whose square has (k2 k)-clique, either subgraph P’k, or (k 1)-regular bipartite 2(k2 k). Ok graphs, their structural properties. These properties imply (if exist) must highly symmetric. Using this information, show O2 contains one graph, as Franklin graph. O3 also consists single build. Furthermore, O4 O5 empty. For general 1)k[p/2] 1, bound tight every odd integer ≥ 3. This implies = 3) whenever there exists k-geometry, however, case, could reached by symmetric built