0

Let $A \in \mathbb{R}^{m \times m}$ be a nonsymmetric zero diagonal matrix such that $A^k, k=2n+1, n\in\mathbb{N}_0$ has also zero diagonal.

Is $A$ always the adjacency matrix of some possibly directed, weighted (allowing negative weights) bipartite graph as well?

Separate question related to this post.

Astor
  • 424
  • 1
    Are you asking if the diagonal entries are necessarily zero? Of course not. For instance, take $$ A = \pmatrix{1&1\0&-1} $$ – Ben Grossmann Aug 15 '17 at 21:47
  • No, maybe I am not making myself clear again. I am asking: provided that the diagonal entries are zero and that the diagonal entries of $A^k, k=2n+1, n\in\mathbb{N}_0$ are also zero, is the matrix and adjacency matrix? I'll modify the post to make this clear. – Astor Aug 15 '17 at 21:49
  • 1
    but isn't every matrix with zeros on the diagonal an adjacency matrix of a directed graph with certain weights? – Ben Grossmann Aug 15 '17 at 21:51
  • It is, but does the periodicity imply that the graph is bipartite? – Astor Aug 15 '17 at 21:51
  • 1
    aha. You said bipartite in the title, but not the body, of the question – Ben Grossmann Aug 15 '17 at 21:52
  • Thanks! :P I am writing and modifying a million times, sorry! – Astor Aug 15 '17 at 21:53

0 Answers0