Relation between the row left rank of a quaternion unit gain graph and the rank of its underlying graph

Let f = (G, U(Q), f) be a quaternion unit gain graph (or U(Q)-gain graph), where G is the underlying graph of-+- f, U(Q) = {z E Q : |z| = 1} is the circle group, and f : E-+ U(Q) is the gain function such that f(eij) = f(eji)-1 = f(eji). Let A(f) be the adjacency matrix of f and r(f) be the row left rank of f. In this paper, we prove that -2c(G) < r(f)-r(G) < 2c(G), where r(G) and c(G) are the rank and the dimension of cycle space of G, respectively. All corresponding extremal graphs are characterized. The results will generalize the corresponding results of signed graphs (Lu et al. [20] and Wang [33]), mixed graphs (Chen et al. [7]), and complex unit gain graphs (Lu et al. [21]).