Isomorphisms of Some Cyclic Abelian Covers of Symmetric Digraphs

$D$ if $D^{j}$ is a g-cyclic A-cover of $D$ for some $g\in A$ . In the case that $A$ is abelian, . then $D_{g}(\alpha)$ is simply called a cyclic abelian cover. Let $\alpha$ and $\beta$ be two alternating functions $homA(D)$ into $A$ , and let $\Gamma$ be a subgroup of the automorphism group Aut $D$ of $D$ , denoted $\Gamma\leq$ Aut $D$ . Let $g,$ $h\in A$ . Then two cyclic A-covers $D_{g}(\alpha)$ and $D_{h}(\beta)$ are called $\Gamma$ -isomorphic, denoted $D_{g}(\alpha)\cong\Gamma D_{h}(\beta)$ , if there exi$st$ an isomorphism $\Phi$ : $D_{g}(\alpha)\rightarrow D_{h}(\beta)$ and a $\gamma\in\Gamma$ such that $\pi\Phi=\gamma\pi$ , i.e., the diagram