Bilinear matrix equation characterizes Laplacian and distance matrices of weighted trees
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It is known from the algebraic graph theory that if L is the Laplacian matrix of some tree G with a vertex degree sequence d⃗=(d_1, ..., d_n)^⊤ and D is its distance matrix, then LD+2I=(2·1⃗-d⃗)1⃗^⊤, where 1⃗ 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 d⃗ 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 d⃗ 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.
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