ABSTRACT
Both polynomials can be unified to yield /<(í7«A;t), provided the cyclic contributions in X{G, A) are multiplied by t. For t = 0 and t = 1 the following equations hold:
Let ^ be a simple graph, (a, b) S 6 aa edge of G and {Zab} the set of all cycles in G comprising the edge (a,(); (a, 6) € Zat E G-From eq.(57) it follows that:
K S t A; t ) = n{G - (o, b). A; t) - f i { G - a -J>, A; i) - 2i ^ fi{G - Z«*, A; t) {z.*y
(570 where (7 — (a, 6), G — a — b, G — Zat are the g r^ h s obtmned from G by removing respectivdy the edge (a, b), the vertices a and b or the cycle Zab’
- i T “ (a) < A,- < + íT “ (a), (58)
where g’^ { G ) is the highest vertex degree occurring ulG-If £7 is a line graph, then its eigenvalues lie in the range:
- 2 < Ai < + lT “ (í7). (580
Eigenvalues of graphs are of major importance in specific quantum chemical methods (see Section 13).
