On the nullity of cycle-spliced T-gain graphs

Let Φ = (G, ϕ) be a T-gain (or complex unit gain) graph and A(Φ) be its adjacency matrix. The nullity of Φ, denoted by η(Φ), is the multiplicity of zero as an eigenvalue of A(Φ), and the cyclomatic number of Φ is defined by c(Φ) = e(Φ) − n(Φ) + κ(Φ), where n(Φ), e(Φ) and κ(Φ) are the number of vertices, edges and connected components of Φ, 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 Φ with c(Φ) cycles, we prove that 0 ≤ η(Φ) ≤ c(Φ) + 1. Moreover, we show that there is no cycle-spliced T-gain graph Φ of any order with η(Φ) = c(Φ) 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 Φ such that η(Φ) = c(Φ) − ε, ε ∈ {0, 1}. A characterization is given in case η(Φ) = c(Φ) when Φ is obtained by identifying a unique common vertex of 2 cycle-spliced T-gain graphs Φ1 and Φ2. Finally, we compute the nullity of all T-gain graphs Φ with c(Φ) = 2.