Nonisomorphic graphs do not necessarily have distinct idiosyncratic polynomials. For example, the Harries graph and Harries-Wong
graph share the same polynomial. The smallest nonisomorphic graphs sharing an
idiosyncratic polynomial occur for graphs on seven vertices.

The idiosyncratic polynomial is not multiplicative with respect to graph disjoint
unions.

Ellis-Monaghan, J. A. and Merino, C. "Graph Polynomials and Their Applications II: Interrelations and Interpretations." 28 Jun 2008.
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