Forma, Vol. 19 (No. 4), pp. 389-403, 2004
Original Paper

Some Graph-Theoretical Aspects of the Golden Ratio: Topological Index, Isomatching Graphs, and Golden Family Graphs

Haruo Hosoya

Ochanomizu University, Bunkyo-ku, Tokyo 112-8610, Japan
E-mail address:

(Received February 15, 2005; Accepted March 15, 2005)

Keywords: Fibonacci Numbers, Lucas Numbers, Golden Family Graph, Topological Index, Non-adjacent number

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, t. 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 t. 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.

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