Bilinear matrix equation characterizes Laplacian and distance matrices of weighted trees

TL;DR

This paper establishes a bilinear matrix equation that uniquely characterizes trees through their Laplacian and distance matrices, extending known algebraic graph theory results to weighted graphs and applications in topology optimization.

Contribution

It proves that the matrix identity involving Laplacian and distance matrices characterizes trees and extends to weighted graphs, providing a new algebraic tool for graph analysis.

Findings

The matrix identity characterizes trees uniquely.

Extension of the result to weighted graphs.

Applications in extremal graph theory and topology design.

Abstract

It is known from the algebraic graph theory that if $L$ is the Laplacian matrix of some tree $G$ with a vertex degree sequence $\mathbf{d}=(d_1, ..., d_n)^\top$ and $D$ is its distance matrix, then $LD+2I=(2\cdot\mathbf{1}-\mathbf{d})\mathbf{1}^\top$, where $\mathbf{1}$ 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 $\mathbf{d}$ 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 $\mathbf{d}$ 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.