On the degree distance matrix of connected graphs

Let $d_u$ denote the degree of vertex $u$ and $d_{uv}$ be the distance between vertices $u$ and $v$ in a connected graph $G$. We propose studying the degree distance matrix of a connected graph $G$, defined as $M_{DD}(G) = \left((d_u + d_v)d_{uv}\right)_{u,v \in V(G)}$. This study sheds new light on the spectra of degree and distance-based matrices. Some spectral properties of $M_{DD}(G)$ are given along with some open problems that can help to understand the degree distance matrix in depth. Furthermore, $M_{DD}$ spectra of some graphs are obtained. Moreover, an effort is made to get some sharp lower and upper bounds for the $M_{DD}$ spectral radius.