Associated Mersenne graphs

TL;DR

This paper introduces a new family of graphs called associated Mersenne graphs, extending Fibonacci-run graphs by using circular run constraints, and explores their structural and enumerative properties.

Contribution

The paper defines associated Mersenne graphs, investigates their properties, and connects their vertex count to associated Mersenne numbers, expanding the understanding of hypercube variants.

Findings

Vertices count equals the n-th associated Mersenne number

Derived structural properties including radius, diameter, and center

Established recursive relations for associated Mersenne graphs

Abstract

In this paper, a new sub-family of Hypercubes called the \textit{associated Mersenne graphs} $\mathcal{M}_{n}$ are introduced. The definition of associated Mersenne graphs is motivated from the Fibonacci-run graphs ({\"O}. E\v{g}ecio\v{g}lu, V. Ir\v{s}i\v{c}, 2021) by extending run-constrained strings to circularly-run-constrained strings. The name of this new family of graphs is identified with the interesting fact that $|V(\mathcal{M}_{n})|$ is equal to the $n$-th associated Mersenne number. Various interesting structural and enumerative properties of associated Mersenne graphs are investigated, including the analogue of the fundamental recursion, number of vertices and edges, radius, diameter, center, periphery and medianicity. Some future research directions and open problems concerning associated Mersenne graphs are also proposed.