On the evaluation of the characteristic polynomial via symmetric function theory

@article{Randic1987OnTE,
  title={On the evaluation of the characteristic polynomial via symmetric function theory},
  author={Milan Randic},
  journal={Journal of Mathematical Chemistry},
  year={1987},
  volume={1},
  pages={145-152}
}
  • M. Randic
  • Published 1 March 1987
  • Mathematics
  • Journal of Mathematical Chemistry
The use of power sum symmetric functions leads to Newton's identities, which relate the traces of various powers ofA, the adjacency matrix of a graph, and the coefficients of the characteristic polynomials. While it is possible to solve Newton's identities and generate the coefficients by recursion or, alternatively, to derive them by sequential manipulations (yielding the explicit formulas), we show how the results can be expressed using a combinatorial approach and relate the evaluation of… 
A note on some coefficients of the Chebyshev polynomial form of the characteristic polynomial
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On Evaluating the Characteristic Polynomial through Symmetric Functions
TLDR
The method suggested by Barakat is to compute the powers of adjacency matrix and compute their traces, thereby getting the power sums of eigenvalues, and work recursively through Newton’s identities to obtain elementary symmetric functions of the eigen values which are coefficients of the characteristic polynomial.
Construction of characteristic polynomial of reciprocal graphs from the number of pendant vertices
A method for construction of the characteristic polynomial (CP) coefficients of the three classes of reciprocal graphs, viz., Ln + n(p), Cn + n(p), and K1,n−1 + n(p), has been developed that requires
Method for construction of characteristic polynomials via graph linearization
A new method for construction of characteristic polynomials (CP) of complicated graphs having arbitrary edge and vertex weights has been developed. The method first converts the graph into
A Pascal's triangle-like approach for the determination of characteristic polynomial coefficients of reciprocal graphs
Characteristic polynomial coefficients of three classes of graph, namely Ln + n(p), Cn + n(p) and K 1,n-1 + n(p), which are known to have reciprocal pairs of eigenvalues, have been shown to be
The factorisation of chemical graphs and their polynomials: A systematic study of certain trees
Characteristic polynomials of the set of 284 trees which have 1 – 12 vertices of valency 1 – 3 have been examined for possible factors (divisors) using polynomial division. Twenty of these trees are
The characteristic polynomial of a chemical graph
We list uses of, and the computational methods for the characteristic polynomial of a (chemical) graph. Pour computational methods are singled out for more detailed presentation. These are the
Formulas for the characteristic polynomial coefficients of the pendant graphs of linear chains, cycles and stars
Formulas for the characteristic polynomial (CP) coefficients of three classes of (n + p)-vertex graphs, i.e. linear chains, cycles and stars where p pendant vertices are attached to n base vertices
Construction and studies of a new class of reciprocal trees: interknitting of the Pascal's triangle
A method of construction of a new class of trees with reciprocal pairs of eigenvalues (λ, 1/λ) has been developed. They are derived from star graphs and can be symbolized as K 1, n −1 + n(p) + mK 2
Comments on the characteristic polynomial of a graph
Several unique advantages of the Le Verrier–Fadeev–Frame method for the characteristic polynomials of graphs over the method proposed by Zivković recently based on the Givens–Householder method are
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