The row left rank of quaternion unit gain graphs in terms of pendant vertices

TL;DR

The study establishes sharp lower bounds for the row left rank $r(\widetilde{G})$ of quaternion unit gain graphs in terms of the cyclomatic number $c(\widetilde{G})$ and pendant count $p(\widetilde{G})$, using switching invariance and detailed structural analysis. By combining induction on order with a careful case split on pendant vertices and cycle connectivity, the authors derive unified bounds and then completely characterize the extremal graphs attaining equality. The extremal structures are described in terms of tree-based constructions with Type 1 cycles attached at leaves (and related two-block configurations), providing a comprehensive taxonomy of when $r(\widetilde{G})$ reaches its theoretical minimum given $c$ and $p$. These results generalize earlier work on complex and signed gain graphs and deepen the spectral understanding of quaternion unit gain graphs by linking rank to precise cycle and pendant-vertex configurations.

Abstract

Let $\widetilde{G}=(G,U(\mathbb{Q}),\varphi)$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph), where $G$ is the underlying graph of $\widetilde{G}$, $U(\mathbb{Q})=\{q\in \mathbb{Q}: |q|=1\}$ and $\varphi:\overrightarrow{E}\rightarrow U(\mathbb{Q})$ is the gain function such that $\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$ for any adjacent vertices $v_{i}$ and $v_{j}$. Let $A(\widetilde{G})$ be the adjacency matrix of $\widetilde{G}$ and let $r(\widetilde{G})$ be the row left rank of $\widetilde{G}$. In this paper, we prove some lower bounds on the row left rank of $U(\mathbb{Q})$-gain graphs in terms of pendant vertices. All corresponding extremal graphs are characterized.

Figures (8)

• Figure 1: $\widetilde{G}$.
• Figure 2: $\infty(p,1,q)$, $\infty(p,l,q)$ and $\theta(p,l,q)$.
• Figure 3: $\widetilde{G}_{1}$ and $\widetilde{G}_{2}$.
• Figure 4: $\widetilde{G}-y$.
• Figure 5: $\widetilde{G}$.

Theorems & Definitions (31)

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