The row left rank of a quaternion unit gain graph in terms of maximum degree

TL;DR

This work studies the row left rank $r(G^{φ})$ of quaternion unit gain graphs $G^{φ}$ of order $n$ with maximum degree $Δ$, establishing sharp lower bounds $r(G^{φ})≥\frac{n}{Δ}$ and, for connected graphs, $r(G^{φ})≥\frac{n-2}{Δ-1}$. The authors derive these results using subgraph decompositions, rank-inequality lemmas, and edge-counting arguments, and they completely characterize the extremal structures achieving equality. In the general case, equality occurs precisely when $G^{φ}$ is a disjoint union of copies of the complete bipartite gain graph $K^{φ}_{Δ,Δ}$ arranged to yield $r(K^{φ}_{Δ,Δ})=2$ with Type 1 4-cycles; in the connected case, equality forces specific configurations: either a Type 1 cycle when $Δ=2$, or a complete bipartite gain graph $K^{φ}_{n/2,n/2}$ with Type 1 4-cycles when $Δ≥3$. These results extend spectral-type bounds for gain graphs to the quaternion unit gain setting and illuminate the extremal structures under degree constraints.

Abstract

Let $Φ=(G,U(\mathbb{Q}),\varphi)$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph) of order $n$, $A(Φ)$ be the adjacency matrix of $Φ$ and $r(Φ)$ be the row left rank of $Φ$. Let $Δ$ be the maximum degree of $Φ$. In this paper, we prove that $r(Φ)\geq\frac{n}Δ$. Moreover, if $Φ$ is connected, we obtain that $r(Φ)\geq\frac{n-2}{Δ-1}$. All the corresponding extremal graphs are characterized.

Theorems & Definitions (19)

• Lemma 1.1
• Example 1.2
• Lemma 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8