The Smith normal form of distance matrices of high dimensional trees

Abstract

Graham-Lov\'asz-Pollak \cite{GL,GP} obtained the celebrated formula $$\det({\sf D}(T_{n+1}))=(-1)^nn2^{n-1},$$ for the determinant of the distance matrix ${\sf D}(T_{n+1})$ for any tree $T_{n+1}$ with $n+1$ vertices. Later, Hou and Woo \cite{HW} extended this formula to the Smith normal form (SNF) obtaining that $\SNF({\sf D}(T_{n+1}))={\sf I}_2\oplus 2{\sf I}_{n-2}\oplus [2n]$, for any tree $T_{n+1}$ with $n+1$ vertices. A $k$-{\it tree} is either a complete graph on $k$ vertices or a graph obtained from a smaller $k$-tree by adjoining a new vertex together with $k$ edges connecting it to a $k$-clique. If $\tau$ and $\tau'$ are $d$-cliques in a $k$-tree $T$, a $d$-{\it walk} between $\tau$ and $\tau'$ is a finite sequence $\tau_1\sigma_1\tau_2\sigma_2\cdots\tau_l$, where $\tau_1=\tau$, $\tau_l=\tau'$, and the $d$-cliques $\tau_i$ and $\tau_{i+1}$ are incident to the same $(d+1)$-clique $\sigma_i$. For $d\in\{1,\dots,k\}$, the $d$-{\it distance} from the $d$-cliques $\tau$ and $\tau'$ is the number of $(d+1)$-cliques in a minimum $d$-walk from $\tau$ and $\tau'$, and is denoted by $\dist^d(\tau,\tau')$. Let $c_d$ denote the number of $d$-cliques in the $k$-tree $T$. Then the $d$-distance matrix ${\sf D}^d(T)$ of the $k$-tree $T$ is the $c_d\times c_d$ matrix, indexed by the $d$-cliques of $T$, such that the $(i,j)$-entry is $0$ if $i=j$, and $\dist^d(\tau_i,\tau_j)$ otherwise. Here, we show that, for $k$ and $n$ fixed, the SNF of the $k$-distance matrix is the same for any $k$-tree with $n$ vertices. Specifically, for any $k$-tree $T_{n}$ with $n$ vertices such that $n\geq k+2$, the Smith normal form of ${\sf D}^{k}(T_{n})$ is $${\sf I}_{(k-1)(n-k)+2}\oplus (k+1){\sf I}_{n-k-2}\oplus [k(k+1)(n-k)],$$ which extends Graham-Lov\'asz-Pollak and Hou-Woo results.