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The permanent can be interpreted as the number of perfect matchings in bipartite graphs.

Is there a similar graph-theoretic interpretation of the determinant?

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Frank Harary has written an article about this: "Determinants, Permanents and Bipartite graphs" jstor.org/pss/2689132 –  Timothy Wagner Dec 10 '10 at 3:44
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up vote 12 down vote accepted

I'm aware of a few. There is the Lindström-Gessel-Viennot lemma, and there is also the matrix-tree theorem. If $A$ is the adjacency matrix of a finite graph $G$ then $\frac{1}{\det(I - At)}$ describes a kind of "zeta function" of $G$. I describe some of how this works in this blog post.

You may also be interested in Kuperberg's An exploration of the permanent-determinant method.

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A picky remark: the guy's name was Lindström, not Lindstrom. –  Hans Lundmark Dec 10 '10 at 7:31
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Corrected. Thanks! –  Qiaochu Yuan Dec 10 '10 at 7:50
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You're welcome! Those dots are a matter of Swedish national pride, you see. ;) –  Hans Lundmark Dec 10 '10 at 11:13
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