On smallest regular graphs with a given isopart

For a nonempty graph, G, we define p(G) and r(G) to be respectively the minimum order and minimum degree of regularity among all connected regular graphs H having a nontrivial decomposition into subgraphs isomorphic to G. By f(G), we denote the least integer t for which there is a connected regular graph H having a decomposition into t subgraphs isomorphic to G. In this article, the values of these parameters are determined for complete graphs, cycles, and stars. Furthermore, we show that Δ( T ) ⩽ r ( T ) ⩽ δ ( T ) + 1 for every tree T . and r ( T ) Δ( T ) if the maximum degree Δ( T ) is even.