The Merrifield-Simmons conjecture also holds for parity graphs

TL;DR

This paper proves the Merrifield-Simmons conjecture for parity graphs, establishing a relationship between vertex distances and independent set counts in these graphs, with implications for graph theory conjectures.

Contribution

The paper confirms the Merrifield-Simmons conjecture specifically for parity graphs, providing the first proof for this class and suggesting limitations for generalization.

Findings

The conjecture holds for parity graphs.

Evidence suggests the conjecture may not extend to other graph classes.

Provides a new proof for a longstanding conjecture in a specific graph class.

Abstract

The Merrifield-Simmons conjectures states a relation between the distance of vertices in a simple graph $G$ and the number of independent sets, denoted as $\sigma(G)$, in vertex-deleted subgraphs. Namely, that the sign of the term $\sigma(G_{-u}) \cdot \sigma(G_{-v}) - \sigma(G) \cdot \sigma(G_{-u-v})$ only depends on the parity of the distance of $u$ and $v$ in $G$. We prove this statement in the case of parity graphs and give some evidence that this result may not be further generalized to other classes of graphs.