Nested (2,r)-regular graphs and their network properties.

A graph G is a (t, r)-regular graph if every collection of t independent vertices is collectively adjacent to exactly r vertices. If a graph G is (2, r)-regular where p, s, and m are positive integers, and m ≥ 2, then when n is sufficiently large, then G is isomorphic to G = Ks+mKp, where 2(p-1)+s = r. A nested (2,r)-regular graph is constructed by replacing selected cliques with a (2,r)-regular graph and joining the vertices of the peripheral cliques. For example, in a nested \u27s\u27 graph when n = s + mp, we obtain n = s1+m1p1+mp. The nested \u27s\u27 graph is now of the form Gs = Ks1+m1Kp1+mKp. We examine the network properties such as the average path length, clustering coefficient, and the spectrum of these nested graphs