The Aα-eigenvalues of the generalized subdivision graph

Let [Formula: see text] be a graph with an adjacency matrix [Formula: see text] and a diagonal degree matrix [Formula: see text]. For any graph [Formula: see text] and a real number [Formula: see text], the [Formula: see text]-matrix of [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text]. The generalized subdivision graph [Formula: see text], derived from the subdivision graph of [Formula: see text] having the vertex set [Formula: see text], comprises a vertex set [Formula: see text]. This construction includes [Formula: see text] replicas of [Formula: see text] and [Formula: see text] replicas of [Formula: see text], with edges established between vertices [Formula: see text] and [Formula: see text] where [Formula: see text] is incident to [Formula: see text] in [Formula: see text]. In this paper, we derive the [Formula: see text]-characteristic polynomial of [Formula: see text]. We demonstrate that if [Formula: see text] is a regular graph, then the [Formula: see text]-spectrum of [Formula: see text] is completely determined by the Laplacian spectrum of [Formula: see text]. Specifically, when [Formula: see text], the [Formula: see text]-spectrum of [Formula: see text] is completely determined by the Laplacian spectrum of the subdivision graph of [Formula: see text]. In conclusion, as an application, we present the construction of infinite families of non-isomorphic graphs that are [Formula: see text]-cospectral.