A classical combinatorial theorem of Leighton says that if two finite graphs have the same universal covering, then they have a finite common covering. Put in a different way, if the Laplacian of two graphs with a common covering has zero eigenvalue, then they have a common cover with the same property. From this point of view we establish a generalization of Leighton's theorem for regular graphs: If the bottom of the Laplacian spectrum on two k-regular graphs are a,b, then for every \epsilon >0 they have a common covering for which the bottom of the spectrum is <
a+b+\epsilon
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