The degree-distance and transmission-adjacency matrices

Let G be a connected graph with adjacency matrix A(G) and distance matrix D(G). Let \({{\,\textrm{dist}\,}}(u,v)\) $\text{dist}(u,v)$ denote the distance between the pair of vertices \(u,v\in V(G)\) $u,v\in V(G)$ , then the transmission \({{\,\textrm{trs}\,}}(u)\) $\text{trs}(u)$ of vertex u is defined as \(\sum _{v\in V(G)}{{\,\textrm{dist}\,}}(u,v)\) ${\sum }_{v\in V(G)}\text{dist}(u,v)$ . Let \({{\,\textrm{trs}\,}}(G)\) $\text{trs}(G)$ be the diagonal matrix whose diagonal elements are the transmissions of the vertices of G. And, let \(\deg (G)\) $deg(G)$ be the diagonal matrix whose diagonal elements are the degrees of the vertices of G. In this paper we investigate the Smith normal form (SNF) and the spectrum of the matrices \(D^{\deg }_+(G):=\deg (G)+D(G)\) $D_{+}^{deg}(G):=deg(G)+D(G)$ , \(D^{\deg }(G):=\deg (G)-D(G)\) $D^{deg}(G):=deg(G)-D(G)$ , \(A^{{{\,\textrm{trs}\,}}}_+(G):={{\,\textrm{trs}\,}}(G)+A(G)\) $A_{+}^{\text{trs}}(G):=\text{trs}(G)+A(G)$ and \(A^{{{\,\textrm{trs}\,}}}(G):={{\,\textrm{trs}\,}}(G)-A(G)\) $A^{\text{trs}}(G):=\text{trs}(G)-A(G)$ . In particular, we explore how good the SNF and the spectrum of these matrices are for determining graphs up to isomorphism. We found that the SNF of \(A^{{{\,\textrm{trs}\,}}}\) $A^{\text{trs}}$ has an interesting behaviour when compared with other classical matrices. We note that the SNF of \(A^{{{\,\textrm{trs}\,}}}\) $A^{\text{trs}}$ can be used to compute the structure of the sandpile group of certain graphs. We compute the SNF of \(D^{\deg }_+\) $D_{+}^{deg}$ , \(D^{\deg }\) $D^{deg}$ , \(A^{{{\,\textrm{trs}\,}}}_+\) $A_{+}^{\text{trs}}$ and \(A^{{{\,\textrm{trs}\,}}}\) $A^{\text{trs}}$ for several graph families. We prove that the SNF of \(D^{\deg }_+\) $D_{+}^{deg}$ , \(D^{\deg }\) $D^{deg}$ , \(A^{{{\,\textrm{trs}\,}}}_+\) $A_{+}^{\text{trs}}$ and \(A^{{{\,\textrm{trs}\,}}}\) $A^{\text{trs}}$ determine complete graphs. Finally, we derive some results about the spectrum of \(D^{\deg }\) $D^{deg}$ and \(A^{{{\,\textrm{trs}\,}}}\) $A^{\text{trs}}$ .