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Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2$ we canonically associate permanent of $M$ to the number of perfect matchings in the graph.

Is there a bipartite graph which is balanced we can associate to the square of the determinant or absolute value of the determinant in a canonical manner without computing the determinant explicitly?

Update Observe the construction of the graph is non-canonical in any known way. It has to be clever.

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  • $\begingroup$ $K_2$ is one example, though I assume you want a more robust categorization... $\endgroup$
    – Zach Hunter
    Commented Feb 20, 2021 at 2:14
  • $\begingroup$ @ZacharyHunter $n>1$ for general balanced bipartite graphs. $\endgroup$
    – Turbo
    Commented Feb 20, 2021 at 2:24
  • $\begingroup$ Slightly better: the determinant of the biadjacency matrix "counts" perfect matchings (there can be at most one) for any tree. $\endgroup$
    – lambda
    Commented Feb 20, 2021 at 7:00

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