Idiosyncratic Polynomial

The idiosyncratic polynomial is the bivariate graph polynomial defined as the characteristic polynomial in x of A+y(J-I-A), where A is the adjacency matrix, J is the unit matrix, and I is the identity matrix. Here, A^_=J-I-A is the adjacency matrix of the graph complement of the graph with adjacency matrix A (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.

See also

Adjacency Matrix, Characteristic Polynomial

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Ellis-Monaghan, J. A. and Merino, C. "Graph Polynomials and Their Applications II: Interrelations and Interpretations." 28 Jun 2008., W. T. "All the King's Horses." In Graph Theory and Related Topics (Ed. J. A. Bondy and U. R. S. Murty). New York: Academic Press, pp. 15-33, 1979.van Dam, E. R. "Cospectral Graphs and the Generalized Adjacency Matrix." Linear Alg. Appl. 423, 33-41, 2007.

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Idiosyncratic Polynomial

Cite this as:

Weisstein, Eric W. "Idiosyncratic Polynomial." From MathWorld--A Wolfram Web Resource.

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