For odd N with gcd(r,N)=1, the graphs G_{N,r} are isomorphic to G_{N,1} and admit explicit exponential formulas for effective resistance and spanning tree counts, with the ratio of spanning trees to those of K_N approaching e^{-2} as N grows.
Chair, The effective resistance of theN-cycle graph with four nearest neighbors, Journal of Statistical Physics154(2014), 1177–1190
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Effective resistance and spanning trees in complete graphs with distance-class deletions
For odd N with gcd(r,N)=1, the graphs G_{N,r} are isomorphic to G_{N,1} and admit explicit exponential formulas for effective resistance and spanning tree counts, with the ratio of spanning trees to those of K_N approaching e^{-2} as N grows.