# Diameter of io-decomposable Riordan graphs of the Bell type

## TL;DR

This paper investigates the diameter of io-decomposable Riordan graphs of the Bell type, providing counterexamples, proofs for specific cases, and proposing new conjectures to advance understanding of their properties.

## Contribution

It refutes an existing conjecture with a counterexample, proves the conjecture for certain graph sizes, and introduces a new conjecture for these graph classes.

## Key findings

- Counterexample disproves the first conjecture.
- First conjecture holds for particular graph sizes.
- Second conjecture is valid for some special graphs.

## Abstract

Recently, in the paper \cite{CJKM1} we suggested the two conjectures about the diameter of io-decomposable Riordan graphs of the Bell type. In this paper, we give a counterexample for the first conjecture. Then we prove that the first conjecture is true for the graphs of some particular size and propose a new conjecture. Finally, we show that the second conjecture is true for some special io-decomposable Riordan graphs.