Symmetric Function and Newton Identities

We end with a brief comment on getting the characteristic polynomial of chanical graphs nsing the symmetric function theory. R. Barakat in his paper has shown that the Frame s method is nothing but symmetric functions and Newton identities [52], In the view that a reference is made to Newton, this paper deserves to be included here if even at the very end of this section. Let c be the coefficients of the characteristic polynomial. They are the elementary symmetric fnnctions from the eigenvalues of the adjacency matrix (see, for example, Weyl [53] or other books on higher algebra). Thus, 

The hrst coefficient q = Tr(A), the trace of the adjacency matrix, is the sum of diagonal elements, and the last c = det A is the determinant of the adjacency matrix. For constrnction of the characteristic polynomial the power sums symmetric functions are reqnired. Newton s identities related the power sums symmetric functions (which can be calcnlated from known eigenvalues) to coefficients of the characteristic polynomial and allow one to calculate the coefficients. They can be written as 

The St are the traces of AL In the time of Newton, matrices were unknown, and the matrix notation adopted here was only subsequently developed by Newton s compatriot A. Cayley. Using these Newton s equations, one can express q in terms of the coefficients of the characteristic polynomial. By sequential manipulations, one can obtain explicit formulas. After relatively simple expression of smaller coefficients C2-C5 and even and o, the expressions become increasingly cumbersome and com- 

Observe that subscripts of the coefficients Cj where i can take values between 1 and k, in each term had to add to the subscript of s,. As one can see, the above equations no longer have a simple pattern seen in Newton s equations, which, in the case of Newton s equations, allows one to write the next line from the lines above. 

Or the same message from Hoffmann [56] in an alternative formnlation Even if the... achievements of modern computational chemistry are astounding, one still requires... a simple portable explanation.  

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