Abstract
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.
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© 1993 Springer-Verlag New York, Inc.
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Fiedler, M. (1993). A Geometric Approach to the Laplacian Matrix of a Graph. In: Brualdi, R.A., Friedland, S., Klee, V. (eds) Combinatorial and Graph-Theoretical Problems in Linear Algebra. The IMA Volumes in Mathematics and its Applications, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8354-3_3
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DOI: https://doi.org/10.1007/978-1-4613-8354-3_3
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