Spectra of Graphs Obtained by \(S - N \) Partial Cartesian Product

Let G be a graph on p vertices and q edges of diameter d. Consider \( S \subset V \left( G \right) \) with \( |S| = m \) . In this paper, a new graph operation on G called \( S-N \) Partial Cartesian product of G is introduced. The \( S-N \) Partial Cartesian product of G is defined as the Cartesian product of G taken N times, i.e., \( \underbrace{G \Box _{S} G \Box _{S}\dots \Box _{S} G}_{[N - times]} \) with respect to the subset S and the resultant graph is denoted by \( H_S^N\left( G\right) \) . We derive the eigenvalues of \( H_S^N\left( G\right) \) . In particular, we consider G as the Generalized wheel graph \( W_{m, n}, \ m\ge 2, n\ge 3 \) which is the join of an empty graph and a cycle graph on m and n vertices, respectively. Using \( S-N \) Partial Cartesian product on the Generalized wheel graph, we construct a family of graphs called the Hyper-Dumbbell graph, denoted by \( H_S^N\left( W_{m, n}\right) \) . Furthermore, the spectrum, (signless) Laplacian spectrum of the Generalized wheel graph and \( H_S^N\left( W_{m, n}\right) \) are obtained.