Eigenvalue bounds of the Kirchhoff Laplacian

05/23/2022

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by   Oliver Knill, et al.

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We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or quiver is bounded above by d(k)+d(k-1) for all k in 1,...,n. Here l(1),...,l(n) is a non-decreasing list of the eigenvalues of K and d(1),..,d(n) is a non-decreasing list of vertex degrees with the additional assumption d(0)=0. We also prove that in general the weak Brouwer-Haemers lower bound d(k) + (n-k) holds for all eigenvalues l(k) of the Kirchhoff matrix of a quiver.