*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: hosoya@is.ocha.ac.jp

(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, Z_{G}, 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.

[Full text] (PDF 3.4 MB)