Continued fractions associated with the topological index of the caterpillar-bond graph

Abstract

In this paper, we give graphs whose topological index are exactly equal to the number $u_n$, satisfying the three term recurrence relation $$u_n=a{u}_{n-1}+b{u}_{n-2}(n\ge 2)u_0=0and{u}_1=u,$$ where $a$, $b$ and $u$ are positive integers. We show an interpretation from the continued fraction expansion in a more general case, so that the topological index can be computed easily. On the contrary, for any given positive integer $N$, we can find the graphs (trees) whose topological indices are exactly equal to $N$.