Steiner radial number of generalized corona

The Steiner n-radial graph associated with a graph G having p vertices is represented as SR_n(G). In this graph, the vertex set remains identical to that of G. Vertices are mutually adjacent in SR_n(G) if they satisfy the condition of being n-radial in G, where n satisfies 2 ≤ n ≤ p. Regarding the edge set of SR_n(G), a complete graph Kn is constructed for every subset of n vertices that are n-radial in G. Let us consider two graphs G and H, each with p vertices. If there exists a positive integer n such that SR_n(G) ≅ H, the graph H is termed a Steiner completion of G. A specific case arises when H = Kp, and the corresponding Steiner completion number of G relative to H is referred to as the Steiner radial number of G, denoted as rS(G). Explicitly, rS(G) is the smallest positive integer n for which the Steiner n-radial of G forms a complete graph. In this work, we compute the Steiner radial number for generalized vertex corona constructions of arbitrary graphs H and for generalized edge coronae of trees. Additionally, we demonstrate the existence of distinct, non-isomorphic graphs with p ≥ 2 vertices that share the Steiner radial number p.