The Aα-eigenvalues of the generalized subdivision graph

Let G = (V-G,E-G) be a graph with an adjacency matrix A(G )and a diagonal degree matrix DG. For any graph G and a real number alpha is an element of [0, 1], the A(alpha)-matrix of G, denoted by A(alpha)(G), is defined as A(alpha)(G) = alpha D-G + (1 - alpha)A(G ). The generalized subdivision graph S-G(n(1),m(1)), derived from the subdivision graph of G having the vertex set V-G boolean OR E-G, comprises a vertex set V(G )x{1, 2,& mldr;,n1}boolean OR E-G x{1, 2,& mldr;,m(1) }. This construction includes n1 replicas of V(G )and m(1) replicas of E-G, with edges established between vertices (v,i) and (e,j) where e is an element of E(G )is incident to v is an element of V(G )in G. In this paper, we derive the A(alpha)-characteristic polynomial of S-G((n(1),m(1)). We demonstrate that if G is a regular graph, then the A(alpha)-spectrum of S-G((n(1),m(1)) is completely determined by the Laplacian spectrum of G. Specifically, when n(1) = m(1), the A(alpha)-spectrum of S-G((n(1),m(1)) is completely determined by the Laplacian spectrum of the subdivision graph of G. In conclusion, as an application, we present the construction of infinite families of non-isomorphic graphs that are A(alpha)-cospectral.