A Family of Simultaneously Cospectral Trees for Degree-Distance Matrices

Spectral characterization of graphs for various graph matrices constitutes a central topic in spectral graph theory. Let $G$ be a graph with adjacency matrix $A(G)$, diagonal degree matrix $\text{Deg}(G)$, distance matrix $D(G)$, and transmission matrix $\text{Trs}(G)$, respectively. Recently, Alfaro and Zapata (2024) introduced the degree-distance matrices $\text{Ddegp}(G)=\text{Deg}(G)+D(G)$ and $\text{Ddeg}(G)=\text{Deg}(G)-D(G)$, together with the transmission-adjacency matrices $\text{Atrsp}(G)=\text{Trs}(G)+A(G)$ and $\text{Atrs}(G)=\text{Trs}(G)-A(G)$. Based on computational evidence for trees on at most $20$ vertices, they conjectured that all trees are determined by the spectra of $\text{Ddegp}$ as well as $\text{Ddeg}$. In this paper, we disprove these conjectures by constructing an infinite family of pairs of non-isomorphic trees. More precisely, for each integer $r\ge 3$, we construct a pair of trees on $17r-15$ vertices which are simultaneously cospectral with respect to the following six matrices $$A,L,Q,D,\text{Ddegp},\text{Ddeg}\mathrm{.}$$ The construction is based on an $r$-regularized leaf extension and an equitable-partition reduction. We also record a simple sign-switching observation for transmission-adjacency matrices: if $G$ is bipartite, then $\text{Atrs}(G)$ and $\text{Atrsp}(G)$ are similar via a diagonal $\{\pm 1\}$-matrix and have the same Smith normal form. Consequently, for trees, the spectral and Smith normal form problems for $\text{Atrs}$ and $\text{Atrsp}$ are equivalent.