A new result similar to the Graham-Pollak theorem

Abstract: 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.