Abstract
We continue our series of papers on the graph theoretic spectral theory of matrices. Let A be an M-matrix. We introduce the concepts of combinatorial vectors and proper combinatorial vectors in the generalized nullspace E(A) of A. We explore the properties of combinatorial bases for E(A) and Jordan bases for E(A) derived from proper combinatorial sets of vectors. We use properties of these bases to prove additional new conditions for the equality of the (spectral) height (or Weyr) characteristic and the (graph theoretic) level characteristic of A. We and DMS-also explore the role of the Hall Marriage Condition, well structured graphs and their anchored chain decompositions in the study of the equality of the two characteristics.
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