Is there any connection known between the two?
One can naturally define lifts of graphs by groups like $\mathbb{Z}_k$ and hence I wonder if representation theoretic properties can be used to say something about graph lifts.
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2 Answers
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There's MR1186756: Godsil, C. D.; Hensel, A. D. Distance regular covers of the complete graph. J. Combin. Theory Ser. B 56 (1992), no. 2, 205–238. This only considers covers of complete graphs, but much of the theory sketched extends without problems. It seems hard to do anything useful is the group is not abelian.
answered Jan 22, 2015 at 2:01
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One other thing to be aware of is that since every connected even-degree regular graph is a Schreier coset graph for some group [1], we can reason about a graph in terms of the corresponding permutation representation of that group, and then take quotients of the graph (thereby obtaining graphs covered by it) by taking quotients of the representation.
[1]: Gross, Jonathan L. "Every connected regular graph of even degree is a Schreier coset graph." Journal of Combinatorial Theory, Series B 22.3 (1977): 227-232.
