The Grone-merris Conjecture
نویسنده
چکیده
In spectral graph theory, the Grone-Merris Conjecture asserts that the spectrum of the Laplacian matrix of a finite graph is majorized by the conjugate degree sequence of this graph. We give a complete proof for this conjecture. The Laplacian of a simple graph G with n vertices is a positive semi-definite n×n matrix L(G) that mimics the geometric Laplacian of a Riemannian manifold; see §1 for definitions, and [2, 14] for comprehensive bibliographies on the graph Laplacian. The spectrum sequence λ(G) of L(G) can be listed in non-increasing order as λ1(G) ≥ λ2(G) ≥ · · · ≥ λn−1(G) ≥ λn(G) = 0. For two non-increasing real sequences x and y of length n, we say that x is majorized by y (denoted x y) if k ∑
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