  Let $n>1$ be an integer, and let $T$ be a tree with $n+1$ vertices
$v_1,\ldots,v_{n+1}$, where $v_1$ and $v_{n+1}$ are two leaves of $T$. For each
edge $e$ of $T$, assign a complex number $w(e)$ as its weight. We obtain that
$$\det[x+d(v_{j+1},v_k)]_{1\le j,k\le n}=2^{n-2}\prod_{e\in E(T)}w(e),$$ where
$d(v_{j+1},v_k)$ is the weighted distance between $v_{j+1}$ and $v_k$ in the
tree $T$. This is similar to the celebrated Graham-Pollak theorem on
determinants of distance matrices for trees. Actually, a more general result is
deduced in this paper.
