Characteristic Polynomial
The characteristic polynomial is the polynomial left-hand side of the characteristic equation
 |
(1)
|
where 
is a square matrix and 
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 
may be computed in the Wolfram Language as CharacteristicPolynomial[m, x].
The characteristic polynomial of a 
matrix
 |
(2)
|
can be rewritten in the particularly nice form
 |
(3)
|
where 
is the matrix trace of 
and 
is its determinant.
Similarly, the characteristic polynomial of a 
matrix is
 |
(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,](https://mathworld.wolfram.com/images/equations/CharacteristicPolynomial/NumberedEquation5.svg) |
(5)
|
In general, the characteristic polynomial has the form
where 
is the matrix trace 
of the matrix 
, 
, and 
is the sum of the 
-rowed diagonal minors of the matrix 
(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],](https://mathworld.wolfram.com/images/equations/CharacteristicPolynomial/NumberedEquation6.svg) |
(8)
|
where
 |
(9)
|
, and 
.
An algorithm due to Balasubramanian computes 
using the equation
 |
(10)
|
where
 |
(11)
|
(Balasubramanian 1985, 1985, 1991; Ivanciuc and Balaban 2000, p. 90; typo corrected) with 
and 
.
The characteristic polynomial of a graph 
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 
can also be obtained using GraphData[graph, "CharacteristicPolynomial"][x].
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 
. The number of distinct characteristic polynomials for simple undirected graphs on 
, 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.
| graph |
characteristic polynomial |
complete graph  |
 |
complete graph  |
 |
complete graph  |
 |
cyclic graph  |
 |
cyclic graph  |
 |
cyclic graph  |
 |
| cubical graph |
 |
| octahedral graph |
 |
| tetrahedral graph |
 |
wheel graph  |
 |
wheel graph  |
 |
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 A. T. Balaban (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.
