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

摘要

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,tau.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 tau.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.