Munarini graphs: a generalization of Fibonacci cubes and Pell graphs. Part I

TL;DR

This paper introduces Munarini graphs, a new generalization of Fibonacci and Pell graphs, highlighting their structure and properties, and establishing their status as daisy cubes.

Contribution

It presents Munarini graphs as an alternative generalization that are daisy cubes, unlike previous models, and explores their fundamental properties.

Findings

Munarini graphs are daisy cubes, like Fibonacci and Pell graphs.

The paper analyzes size, recursive structure, and polynomial invariants of Munarini graphs.

Fundamental properties of Munarini graphs are established in this initial study.

Abstract

The Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive $1$s. Munarini introduced Pell graphs, a variation of Fibonacci cubes defined on ternary strings. A generalization of Pell graphs to $(k+1)$-ary strings has recently been proposed. In this paper we introduce Munarini graphs, which constitute an alternative generalization of Fibonacci cubes and Pell graphs. One of the main advantages of Munarini graphs is that, unlike previously proposed generalization, they are daisy cubes, as are Fibonacci cubes and Pell graphs. In this first article, we study some of their fundamental properties including the size, the recursive structure, the cube and maximal cube polynomials.