\(D_{\alpha }\)-spectrum of subdivision join of graphs

For a connected graph G, let \( D_{\alpha }\left( G\right) \) $D_{\alpha }\left(G\right)$ be the generalized distance matrix and \(0 \le \alpha \le 1 \) $0\le \alpha \le 1$ . This matrix is of the form \( D_{\alpha }\left( G\right) = \alpha Tr\left( G\right) + \left( 1 - \alpha \right) D\left( G\right) \) $D_{\alpha }\left(G\right)=\alpha Tr\left(G\right)+\left(1,-,\alpha \right)D\left(G\right)$ , where \( Tr\left( G\right) \) $Tr\left(G\right)$ is the diagonal matrix whose elements are its vertex transmissions and \( D\left( G\right) \) $D\left(G\right)$ is the distance matrix of G. In this paper, we obtain the generalized distance spectra of Subdivision vertex-join and Subdivision edge-join of two regular graphs.