The vertex distance complement spectrum of subdivision vertex join and subdivision edge join of two regular graphs

The vertex distance complement (VDC) matrix $C$, of a connected graph $G$ with vertex set consisting of $n$ vertices, is a real symmetric matrix $[c_{ij}]$ that takes the value $n - d_{ij}$ where $d_{ij}$ is the distance between the vertices $v_i$ and $v_j$ of $G$ for $i \neq j$ and 0 otherwise. The vertex distance complement spectrum of the subdivision vertex join, $G_1 \dot{\bigvee} G_2$ and the subdivision edge join $G_1 \underline{\bigvee} G_2$ of regular graphs $G_1$ and $G_2$ in terms of the adjacency spectrum are determined in this paper.