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

## TL;DR

This paper establishes a connection between the topological index of caterpillar-bond graphs and continued fractions, providing a method to compute these indices and construct graphs with specified topological index values.

## Contribution

It introduces a novel interpretation of the topological index via continued fractions and offers a way to generate graphs with a given topological index.

## Key findings

- Topological indices correspond to numbers satisfying a three-term recurrence relation.
- Continued fraction expansion facilitates easy computation of the topological index.
- Graphs with a specified topological index can be explicitly constructed.

## Abstract

In this paper, we give graphs whose topological index are exactly equal to the number $u_n$, satisfying the three term recurrence relation $$ u_n=a u_{n-1}+b u_{n-2}\quad(n\ge 2)\quad u_0=0\quad\hbox{and}\quad u_1=u\,, $$ 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 $N$.