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We know that all adjacency matrices are square matrices. When our weighted directed graphs have loops or parallel edges, we obtain adjacency matrices that are, in general, asymmetric or non-Hermitian square matrices with complex-valued matrix elements. To me, this seems to be the most generic form of square matrices.

Do we have a constraint on drawing a (di)graph from any square matrix? In other words, do we have a class of square matrices that cannot form weighted digraphs (in their most generic definition with loops and parallel edges)?

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    $\begingroup$ It depends on your definitions: what do you make of negative entries? do you allow complex weights? $\endgroup$ Commented Jun 14 at 18:02
  • $\begingroup$ Yes, I allow having any complex number as a matrix element. $\endgroup$ Commented Jun 14 at 18:05
  • $\begingroup$ So you have a notion for a weighted digraph defined by an arbitrary square matrix. What is the question then? $\endgroup$ Commented Jun 14 at 18:42
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    $\begingroup$ No. Weighted digraphs that allow self-loops are "the same" as square matrices. (self loops correspond to the diagonal entries of the matrix) $\endgroup$ Commented Jun 14 at 18:52
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    $\begingroup$ Graphs can also have bands. There is a class of reversible digraphs where a loop may be reversed by an action on the graph, but where reversing the digraph may preserve or modify a loop. See groupoids.org.uk/pdffiles/graphmorphisms-v15i1a1.pdf $\endgroup$ Commented Jun 15 at 7:57

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