Construction and studies of a new class of reciprocal trees: interknitting of the Pascal's triangle

Abstract

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 (1 ≤ m ≤ n − 1 except for n = 1). The trees are minimally Kekulenoid and hence contain reciprocal pairs of eigenvalues in their eigenspectra. The characteristic polynomial coefficients of these trees with given values of n and m are shown to be obtainable by appropriate use of the Pascal's triangle. A general formula for this purpose has been developed. An analytical formula for the Wiener indices of such trees in terms of m and n has been derived and some consequences of this formula are presented. The relevance of these trees to real molecular structures is discussed. The trees have been shown to be useful in observing the subspectrallity of two series of IPR fullerenes of formulae C50+10 n and C60+12 n (n is a positive integer).