Some properties of Cayley signed graphs on finite abelian groups

Abstract

Let $\Sigma=(\Gamma, \sigma)$ is a signed graph(or sigraph in short), where $\Gamma$ is a underlying graph of $\Sigma$ and $\sigma:E\longrightarrow \{+, -\}$ is a function. Consider $\Gamma=Cay(\mathbb{Z}_{p_{1}}\times \mathbb{Z}_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}} \ldots p_{k}^{\alpha_{k}}}, \Phi)$, where all $p_{1}, p_{2}, \ldots, p_{k}$ are distinct prime factors and $\Phi=\varphi_{p_{1}}\times\varphi_{p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}} \ldots p_{k}^{\alpha_{k}}}$. For any positive integer $n$, $\varphi_{n}=\{\ell| 1\leq \ell<n, \gcd(\ell, n)=1\}$. Motivated by \cite{s14}, we will investigate balancing in $\Sigma$ and $L(\Sigma)$, clusterability and sign-compatibility of $\Sigma$.