Lifting Voltages in Graph Covers

Abstract

We consider voltage digraphs, here referred to as graphs, whose edges are labeled with elements from a given group, and explore their derived graphs. Given two voltage graphs, with voltages in abelian groups, we establish a necessary and sufficient condition for their two derived graphs to be isomorphic. This condition requires: (1) the existence of a voltage graph that covers both given graphs, and (2) when the two sets of voltages are lifted to the common cover, the correspondence between these sets of voltages determines an isomorphism between the groups generated by these voltages. We show that conditions (1) and (2) are decidable, and provide a method for constructing the common cover and for lifting the voltage assignments.

Figures (12)

• Figure 1: A periodic graph $\Gamma$ with a covering for a group that does not act freely.
• Figure 2: A lift graph constructed from the covering $\phi:\Gamma\to \Delta$. The lift graph $\Lambda$ is upper left; the unfilled dots are the boundary vertices, the black dots are interior.
• Figure 3: Two representations of $\Gamma$ left and right, with two representations of the quotient graph $\Delta$ in the middle. The middle graphs have different spanning tree sets indicated with red edges. The spanning tree set of the upper middle graph is lifted to the trees of the left depiction of $\Gamma$; the lower middle graph spanning tree set is lifted to the red edge set of $\Gamma$ to the right.
• Figure 4: The diagram for Proposition \ref{['prop:upwards']}. The dashed arrows indicate the maps that are asserted by the proposition's statement. Red maps are associated with the second half of the statement.
• Figure 5: Two illustrations of the situation in Propositon \ref{['prop:upwards']}. (a) The cover maps are depicted by setting vertex color to map to the same vertex color for covering $\Delta_1$, and vertex shape maps to the same vertex shape for covering $\Delta_2$. (b) The same color edges map to the respective color edges in $\Delta_1$ and $\Delta_2$. Vertical neighboring vertices map to vertices in $\Delta_1$ while horizontal neighboring vertices map to $\Delta_2$, white vertex to white and black vertex to black. In both examples $\Delta_{\top}$ is the minimal common cover graph.

Theorems & Definitions (44)

• Definition 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Example 3.4
• Definition 3.5
• Definition 3.6
• proof
• proof