On Factoring of Characteristic Polynomial

Factoring Ch(x) of acyclic graphs can be often accomplished by expressing their characteristic polynomials in terms of the characteristic polynomials of the n-alkanes illustrated in Table 4.1, for which we will here use the notation L . For illustrations of characteristic polynomials of several families of branched alkanes, including characteristic polynomials of 35 isomers of -nonane as well as characteristic polynomials of 20 monocyclic structures with pending bonds, see ref. [45]. The multiplication table of polynomials (see Table 4.2) [49], which facilitates finding factors of characteristic polynomials when expressed in terms of L, is very simple. 

The diagonal elements are negative, and zeros are replaced by polynomials, where k indicates the characteristic polynomial of the fragment removed. Lj stands for vertices without pending bonds. 

Adjacent off-diagonal elements are the characteristic polynomial of the fragment of obtained by removing the vertex of attachment. In the case of vertices withont pending bonds, there remains the entry 1 from the adjacency matrix. 

Here Lj and Lj are the characteristic polynomials of the fragments of the graph obtained when the bond Cj-Cj has been deleted, and Ly and Ly are characteristic polynomials of the fragments of the graph obtained when the bond Q-Cj and vertices Ci, Cj have been deleted. Thus, for example, if in the case of 2,4-dimethylhexane, 

FIGURE 4.9 Construction of the reduced characteristic polynomial by elimination of terminal edges following the method of Balasubramanian L, = x, L2 = - 1, Lj = - 2x. 

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