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

Let G̃ = (G, U(ℚ), φ) be a quaternion unit gain graph (or U(ℚ)-gain graph), where G is the underlying graph of G̃, U(ℚ) = {q ∈ ℚ : |q| = 1} and φ: E⃗ → U(ℚ) is the gain function such that φ(e_(ij))=φ(e_(ji))⁻¹=φ(e_(ji))̄ for any adjacent vertices v_(i) and v_(j). Let A(G̃) be the adjacency matrix of G̃ and let r(G̃) be the row left rank of G̃. In this paper, we obtain lower bounds on the row left rank of U(ℚ)-gain graphs in terms of pendant vertices. All corresponding extremal graphs are characterized. The results of this paper are a generalization of those in paper from Ma et al. in 2016 and can be applied to signed graphs, mixed graphs and complex unit gain graphs.