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This document has mistakes.

The consequences of Theorem 2.2 are wrong as stated. Substitute k = n, and count the number of eigenvalues they say exist.

Caught another mistake:

Section 3.2: First, an observation: the column sums of Q(G) are all zero is wrong. Contradicts their definition of Q(G).

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"Caught another mistake: Section 3.2: First, an observation: the column sums of Q(G) are all zero is wrong. Contradicts their definition of Q(G)." True, but it's pretty obvious he was referring to a directed graph G where matrix Q(G) is defined as Qij = 1 if vertex vi has an outgoing edge ej; -1 if vertex vi has an incoming edge ej; 0 otherwise.

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