A bound on the girth of quaternion unit gain graphs in terms of the rank

TL;DR

This paper establishes an upper bound on the girth of quaternion unit gain graphs based on the rank of their adjacency matrix, extending known results from other graph types.

Contribution

It introduces a bound g <= r + 2 for quaternion unit gain graphs and characterizes cases where equality holds, extending prior graph theory results.

Findings

Girth g is at most rank r plus 2.

Characterization of graphs where g equals r + 2.

Extension of known bounds to quaternion unit gain graphs.

Abstract

We obtain a bound on the girth g of a quaternion unit gain graph in terms of the rank r of its adjacency matrix. In particular, we show that g <= r + 2 and characterize all quaternion unit gain graphs for which g = r+2. This extends corresponding results for (ordinary) graphs, signed graphs, and complex unit gain graphs.