Revisiting the Characteristic Polynomial

FIGURE 4.7 Count of Hosoya s nonadjacent edges for 3-methylhexane. The first two rows show the count of a single disconnected edge, p(G, 1) = 6 the next three rows count two disjointed edges p(G, 2) = 9 the last row counts three disjointed edges p(G, 3) = 3, which give for the characteristic polynomial x - 6x + 9x - 3x. 

FIGURE 4.8 The four symmetry unrelated Ulam s graphs for benzocyclobutadiene. 

When these are added and multiplied by two, one obtains for the derivative of the characteristic polynomials 

Because benzocyclobutadiene is not a benzenoid molecule (having a four-member ring), one has to find the so-called algebraic Kekule structures introduced by Wilcox [46,47]. It is based on generalizing the concept of parity of Kekule valence structures of Dewar and Longuet-Higgins [48] to nonbenzenoid systems, such as benzocyclobutadiene. We are not to elaborate on the parity concept here, and interested readers should consult literature. It turns out that the algebraic Kekule structures for benzocyclobutadiene equal 1, and the constant in the above expression is 1. 

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