A Geometric Approach to the Laplacian Matrix of a Graph

Let G be a finite undirected connected graph with n vertices. We assign to G an (n - 1)-simplex ∑(G) in the point Euclidean (n - 1)-space in such a way that the Laplacian L(G) of G is the Gram matrix of the outward normals of ∑(G). It is shown that the spectral properties of L(G) are reflected by the geometric shape of the Steiner circumscribed ellipsoid S of ∑(G) in a simple manner. In particular, the squares of the half-axes of S are proportional to the reciprocals of the eigenvalues of L(G). Also, a previously discovered relationship to resistive electrical circuits is mentioned.