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Paper IPM / M / 15539  


Abstract:  
Given a symmetric matrix M = [m_{ij}], or equivalently
a weighted graph ∧G whose edge ij has the
weight m_{ij}, let μ be its eigenvalue of multiplicity k ≥ 1. Let M_{i} be the principal submatrix of M obtained by deleting
both ith row and ith column from M. Then i is a
downer, or neutral, or Parter vertex of M and/or
∧G, depending whether the multiplicity of μ in M_{i} or
∧G−i is k−1, or k, or k+1, respectively. We consider
vertex types in the sense of downer, neutral and Parter vertices
in threshold and chain graphs.
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