Pbib-designs and association schemes via minimum nonsplit dominating sets of some circulant graphs

A dominating set D of a graph G = (V, E) is a nonsplit dominating set if the induced subgraph ⟨V – D⟩ is connected. The nonsplit domination number γ_ns (G) is the G is the minimum cardinality of a nonsplit dominating set of G. The set of vertices is a γ_ns-set if it is nonsplit dominating set with γ_ns (G). In this paper, we obtain the total number of γ_ns-sets, the Partially Balanced Incomplete Block (PBIB)-Designs on minimum γ_ns-sets of Circulant graphs with m-association schemes for 1 ≤ m ≤ p/2 .