Riordan graphs I: Structural properties

TL;DR

This paper introduces Riordan graphs, a broad class of graphs derived from Riordan matrices, and explores their structural properties, including decompositions and conditions for Eulerian and Hamiltonian cycles, with implications for network design and algorithms.

Contribution

It defines Riordan graphs using Riordan matrices and analyzes their structural properties, including decomposition and cycle conditions, expanding the understanding of graph classes related to Pascal and Toeplitz graphs.

Findings

Fundamental decomposition theorem for Riordan graphs

Conditions for Eulerian and Hamiltonian cycles in Riordan graphs

Riordan graphs exhibit interesting fractal properties

Abstract

In this paper, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The Riordan graphs are proved to have a number of interesting (fractal) properties, which can be useful in creating computer networks with certain desirable features, or in obtaining useful information when designing algorithms to compute values of graph invariants. The main focus in this paper is the study of structural properties of families of Riordan graphs obtained from infinite Riordan graphs, which includes a fundamental decomposition theorem and certain conditions on Riordan graphs to have an Eulerian trail/cycle or a Hamiltonian cycle. We will study spectral properties of the Riordan graphs in a follow up paper.