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Characteristic Polynomial

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The characteristic polynomial is the polynomial left-hand side of the characteristic equation

det(A-lambdaI)=0,
(1)

where A is a square matrix and I is the identity matrix of identical dimension. Samuelson's formula allows the characteristic polynomial to be computed recursively without divisions. The characteristic polynomial of a matrix m may be computed in the Wolfram Language as CharacteristicPolynomial[m, x].

The characteristic polynomial of a 2×2 matrix

P_2(x)=(a_(11)a_(22)-a_(12)a_(21))-x(a_(11)+a_(22))+x^2
(2)

can be rewritten in the particularly nice form

P_2(x)=det(A)-xTr(A)+x^2,
(3)

where Tr(A) is the matrix trace of A and det(A) is its determinant.

Similarly, the characteristic polynomial of a 3×3 matrix is

P_3(x)=det(A)+1/2(a_(ij)a_(ji)-a_(ii)a_(jj))x+Tr(A)x^2-x^3,
(4)

where Einstein summation has been used, which can also be written explicitly in terms of traces as

P_3(x)=1/6[Tr^3(A)+2Tr(A^3)-3Tr(A)Tr(A^2)]-1/2[Tr^2(A)-Tr(A^2)]x+Tr(A)x^2-x^3,
(5)

In general, the characteristic polynomial has the form

f(lambda)=det(lambda1-A)
(6)
=lambda^n-a_1lambda^(n-1)+...+(-1)^na_n,
(7)

where a_1=suma_(ii) is the matrix trace Tr(A) of the matrix A, a_n=det(A), and a_i is the sum of the i-rowed diagonal minors of the matrix A (Jacobson 1974, p. 109).

Le Verrier's algorithm for computing the characteristic polynomial of a graph (Balasubramanian 1984; Trinajstić 1988; Ivanciuc and Balaban 2000, p. 89) can be formulated as the solution of the linear system

[1 0 ... 0 0; a_1 2 ... 0 0; a_2 a_1 ... 0 0; | ... ... ... |; a_(n-1) a_(n-2) ... a_1 n][c_1; c_2; c_3; |; c_n]=[a_1; a_2; a_3; |; a_n],
(8)

where

f(x)=sum_(k=0)^nc_kx^(n-k),
(9)

c_0=-1, and a_k=Tr(A^k).

An algorithm due to Balasubramanian computes c_k using the equation

c_k=1/kTr(B_(k-1)),
(10)

where

B_k=A(B_(k-1)-c_kI)
(11)

(Balasubramanian 1985, 1985, 1991; Ivanciuc and Balaban 2000, p. 90; typo corrected) with B_0=A and c_0=-1.

The characteristic polynomial of a graph g is defined as the characteristic polynomial of its adjacency matrix and can be computed in the Wolfram Language using CharacteristicPolynomial[AdjacencyMatrix[g], x]. The precomputed characteristic polynomial of a named graph in terms of a variable x can also be obtained using GraphData[graph, "CharacteristicPolynomial"][x].

CharacteristicPolynomialGraphs

Characteristic polynomials are not diagnostic for graph isomorphism, i.e., two nonisomorphic graphs may share the same characteristic polynomial. The smallest such example occurs for the two graphs on five nodes illustrated above, both of which have characteristic polynomial 4x^3-x^5. The number of distinct characteristic polynomials for simple undirected graphs on n=1, 2, ... nodes are 1, 2, 4, 11, 33, 151, 988, 11453, ... (OEIS A082104), giving the number of duplicated characteristic polynomials as 0, 0, 0, 0, 1, 5, 56, 893, 27311, ....

The following table summarizes the characteristic polynomials for some simple graphs.

graphcharacteristic polynomial
complete graph K_2(z-1)(z+1)
complete graph K_3-(z-2)(z+1)^2
complete graph K_4(z-3)(z+1)^3
cyclic graph C_3-(z-2)(z+1)^2
cyclic graph C_4(z-2)z^2(z+2)
cyclic graph C_5-(z-2)(z^2+z-1)^2
cubical graph(z-3)(z-1)^3(z+1)^3(z+3)
octahedral graph(z-4)z^3(z+2)^2
tetrahedral graph(z-3)(z+1)^3
wheel graph W_5-z^2(z+2)(z^2-2z-4)
wheel graph W_6(z^2-2z-5)(z^2+z+1)^2

See also

Cayley-Hamilton Theorem, Characteristic Equation, Eigenvalue, Graph Eigenvalue, Graph Spectrum, Matching Polynomial, Matrix Spectrum, Samuelson's Formula, Stability Index

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References

Balasubramanian, K. "Computer-Generation of the Characteristic-Polynomials of Chemical Graphs." J. Comput. Chem. 5, 387-394, 1984.Balasubramanian, K. "The Use of Frames Method for the Characteristic-Polynomials of Chemical Graphs." Theor. Chim. Acta 65, 49-58, 1984.Balasubramanian, K. "Computer-Assisted Enumeration of Walks and Self-Returning Walks on Chemical Graphs." Comput. Chem. 9, 43-52, 1985.Balasubramanian, K. "Comments on the Characteristic Polynomial of a Graph." J. Comput. Chem. 12, 248-253, 1991.Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 83-92, 2000.Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 310, 1996.Hagos, E. M. "The Characteristic Polynomial of a Graph is Reconstructible from the Characteristic Polynomials of its Vertex-Deleted Subgraphs and Their Complements." Elec. J. Combin. 7, No. 1, R12, 1-9, 2000. http://www.combinatorics.org/Volume_7/Abstracts/v7i1r12.html.Ivanciuc, P. "Chemical Graph Polynomials. Part 2. The Propagation Diagram Algorithm for the Computation of the Characteristic Polynomial of Molecular Graphs." Rev. Roumaine Chim. 37, 1341-134, 1992.Ivanciuc, O. and Balaban, A. T. "The Graph Description of Chemical Structures." Ch. 3 in Topological Indices and Related Descriptors in QSAR and QSPR (Ed. J. Devillers and A. T. Balaban). Amsterdam, Netherlands: Gordon and Breach, pp. 59-167, 2000.Jacobson, N. Basic Algebra I. San Francisco: W. H. Freeman, 1974.Krivka, P.; Jeričević, Ž.; and Trinajstić, N. "On the Computation of Characteristic Polynomial of a Chemical Graph." Int. J. Quant. Chem.: Quant. Chem. Symp. 19, 129-147, 1986.Sloane, N. J. A. Sequence A082104 in "The On-Line Encyclopedia of Integer Sequences."Trinajstić, N. "The Characteristic Polynomial of a Chemical Graph." J. Math. Chem. 2, 197-215, 1988.Zivković, T. "On the Evaluation of the Characteristic Polynomial of a Chemical Graph." J. Comput. Chem. 11, 217-222, 1990.

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Characteristic Polynomial

Cite this as:

Weisstein, Eric W. "Characteristic Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CharacteristicPolynomial.html

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