On the nullity of cycle-spliced T-gain graphs

Let Phi = (G, phi) be a T-gain (or complex unit gain) graph and A(Phi) be its adjacency matrix. The nullity of Phi, denoted by eta(Phi), is the multiplicity of zero as an eigenvalue of A(Phi), and the cyclomatic number of Phi is defined by c(Phi) = e(Phi) - n(Phi) + kappa(Phi), where n(Phi), e(Phi) and kappa(Phi) are the number of vertices, edges and connected components of Phi, respectively. A connected graph is said to be cycle-spliced if every block in it is a cycle. We consider the nullity of cycle-spliced T-gain graphs. Given a cycle-spliced T-gain graph Phi with c(Phi) cycles, we prove that 0 < eta(Phi) < c(Phi) + 1. Moreover, we show that there is no cycle-spliced T-gain graph Phi of any order with eta(Phi) = c(Phi) whenever there are no odd cycles whose gain has real part 0. We give examples of cycle-spliced T-gain graphs whose nullity equals the cyclomatic number, and we show some properties of those graphs Phi such that eta(Phi) = c(Phi) - epsilon, epsilon is an element of {0, 1 }. A characterization is given in case eta(Phi) = c(Phi) when Phi is obtained by identifying a unique common vertex of 2 cycle-spliced T-gain graphs Phi( 1) and Phi (2) . Finally, we compute the nullity of all T-gain graphs Phi with c(Phi) = 2.