Counting independent sets in Riordan graphs

TL;DR

This paper investigates the enumeration and bounds of independent sets in Riordan graphs, a broad class generalizing Pascal and Toeplitz graphs, using diverse combinatorial methods.

Contribution

It provides the first exact counts and bounds for independent sets in various Riordan graph classes, introducing multiple novel analytical techniques.

Findings

Exact enumeration formulas for certain Riordan graphs

Lower and upper bounds for independent sets in Riordan graphs

Application of combinatorics on words to graph theory

Abstract

The notion of a Riordan graph was introduced recently, and it is a far-reaching generalization of the well-known Pascal graphs and Toeplitz graphs. However, apart from a certain subclass of Toeplitz graphs, nothing was known on independent sets in Riordan graphs. In this paper, we give exact enumeration and lower and upper bounds for the number of independent sets for various classes of Riordan graphs. Remarkably, we offer a variety of methods to solve the problems that range from the structural decomposition theorem to methods in combinatorics on words. Some of our results are valid for any graph.