Bilinear matrix equation characterizes Laplacian and distance matrices of weighted trees

It is known from algebraic graph theory that if L the Laplacian matrix of some tree G with a vertex degree sequence d=(δ1,…,δn)⊤ and D its distance matrix, then LD+2I=(2⋅1−d)1⊤, where 1 an all-ones column vector. We prove converse proposition: this identity holds for d D, essentially tree, matrix. This result immediately generalizes to weighted graphs. Therefore, above bilinear equation in L, characterizes trees terms their matrices, so it can be used as constraint mixed-integer formulations distance-related topology design problems (e.g., optimum communication spanning or hop-constrained minimum problems). If symmetric, lower triangular part redundant omitted, which halves number constraints optimization problem. Applications extremal (especially, topological index optimal problems) road are discussed.