It is known from the algebraic graph theory that if $L$ is the Laplacian matrix of some tree $G$ with a vertex degree sequence $\vec{d}=(d_1, ..., d_n)^\top$ and $D$ is its distance matrix, then $LD+2I=(2\cdot\vec{1}-\vec{d})\vec{1}^\top$, where $\vec{1}$ is an all-ones column vector. We prove that if this matrix identity holds for the Laplacian matrix of some graph $G$ with a degree sequence $\vec{d}$ and for some matrix $D$, then $G$ is essentially a tree, and $D$ is its distance matrix. This result immediately generalizes to weighted graphs. If the matrix $D$ is symmetric, the lower triangular part of this matrix identity is redundant and can be omitted. Therefore, the above bilinear matrix equation in $L$, $D$, and $\vec{d}$ characterizes trees in terms of their Laplacian and distance matrices. Applications to the extremal graph theory (especially, to topological index optimization and to optimal tree problems) and to road topology design are discussed.

翻译：从代数图形理论中可以知道,如果美元是某些树的拉普拉西亚基质,如果美元是某种G$的拉普拉卡基质,带有顶点序列$\vec{d ⁇ (d_1,......,d_n) ⁇ top$和$D$是其距离矩阵,那么$LD+2I=(2\cdot\vec{1}-vec{d})\vec{1 ⁇ top$,其中${{{{{1}美元是一整块矢量矢量矢量矢量矢量矢量矢量矢量矢量矢量矢量矢量矢量。因此,如果这一矩阵身份的较低三角部分是多余的,可以省略。因此,如果这个基质量基质特性包含某种G$G$G$的拉利、$D$和$@vec{d}的拉普拉卡基质基质基质基质矩阵,以及一些D$=G$D$$,那么$G$美元是其基数的基数,然后是树的距离模型,然后将基质图状图状图状图状的图状,然后将Goralmabormmm 。