The degree-distance and transmission-adjacency matrices

Let $G$ be a connected graph with adjacency matrix $A(G)$. The distance matrix $D(G)$ of $G$ has rows and columns indexed by $V(G)$ with $uv$-entry equal to the distance $\mathrm{dist}(u,v)$ which is the number of edges in a shortest path between the vertices $u$ and $v$. The transmission $\mathrm{trs}(u)$ of $u$ is defined as $\sum_{v\in V(G)}\mathrm{dist}(u,v)$. Let $\mathrm{trs}(G)$ be the diagonal matrix with the transmissions of the vertices of $G$ in the diagonal, and $\mathrm{deg}(G)$ the diagonal matrix with the degrees of the vertices in the diagonal. In this paper we investigate the Smith normal form (SNF) and the spectrum of the matrices $D^{\mathrm{deg}}_+(G):=\mathrm{deg}(G)+D(G)$, $D^{\mathrm{deg}}(G):=\mathrm{deg}(G)-D(G)$, $A^{\mathrm{trs}}_+(G):=\mathrm{trs}(G)+A(G)$ and $A^{\mathrm{trs}}(G):=\mathrm{trs}(G)-A(G)$. In particular, we explore how good the spectrum and the SNF of these matrices are for determining graphs up to isomorphism. We found that the SNF of $A^{\mathrm{trs}}$ has an interesting behaviour when compared with other classical matrices. We note that the SNF of $A^{\mathrm{trs}}$ can be used to compute the structure of the sandpile group of certain graphs. We compute the SNF of $D^{\mathrm{deg}}_+$, $D^{\mathrm{deg}}$, $A^{\mathrm{trs}}_+$ and $A^{\mathrm{trs}}$ for several graph families. We prove that complete graphs are determined by the SNF of $D^{\mathrm{deg}}_+$, $D^{\mathrm{deg}}$, $A^{\mathrm{trs}}_+$ and $A^{\mathrm{trs}}$. Finally, we derive some results about the spectrum of $D^{\mathrm{deg}}$ and $A^{\mathrm{trs}}$.