on the nullity of cycle-spliced $mathbb{t}$-gain graphs

Let $phi=(g,varphi)$ be a $mathbb{t}$-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 $mathbb{t}$-gain graphs. given a cycle-spliced $mathbb{t}$-gain graph $phi$ with $c(phi)$ cycles, we prove that $0 leq eta(phi)leq c(phi)+1$. moreover, we show that there is no cycle-spliced  $mathbb{t}$-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  $mathbb{t}$-gain graphs whose nullity equals the cyclomatic number, and we show some properties of those graphs $phi$ such that $eta(phi)=c(phi)-varepsilon$, $varepsilon in {0,1}$. a characterization is given in case $eta(phi)=c(phi)$ when $phi$ is obtained by identifying a unique common vertex of $2$ cycle-spliced $mathbb{t}$-gain graphs $phi_1$ and $phi_2$. finally, we compute the nullity of all $mathbb{t}$-gain graphs $phi$ with $c(phi)=2$.