Abstract. By defining the non-adjacent number, p(G, k), and topological index, ZG, for a graph G, several sequences of graphs are shown to be closely related to the golden ratio, τ. Namely, the Z-values of the path and cycle graphs are Fibonacci, and Lucas numbers, respectively, and thus the ratio of consecutive terms of Z converges to τ. Several new sequences of graphs (golden family graphs) were found whose Z-values are either Fibonacci or Lucas numbers, or their multiples. Interesting mathematical relations among them are introduced and discussed.
It has already been shown by Lucas (1876) that the Fibonacci numbers can be obtained from the Pascal's triangle by rotation and addition in a certain way. The golden ratio can then be asymptotically obtained by taking the ratio of consecutive Fibonacci numbers, whose graph-theoretical aspects have been pointed out by the present author (Hosoya, 1971, 1973) in terms of the topological index, Z, which is the sum of the non-adjacent numbers for a given graph. Similar properties of the Lucas numbers related to the golden ratio have also been demonstrated.
Recently several new sequences of graphs were found whose Z-values are either Fibonacci or Lucas numbers, or their multiples. They are called golden family graphs, and their interesting mathematical structure will be presented in this paper. The methodology developed here can be applied to the problems of general recursive sequences, widening their field of algebraic analysis to geometrical or graph theoretical realms.
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