The idiosyncratic polynomial is the bivariate graph polynomial defined as the characteristic polynomial in 
of 
, where 
is the adjacency matrix,
is the unit matrix, and 
is the identity matrix.
Here, 
is the adjacency matrix of the graph
complement of the graph with adjacency matrix 
(Ellis-Monaghan and Merino 2008).
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.
