Steiner radial number of generalized corona

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