C:/Documents and Settings/golin/My Documents/Work/MyPapers/Working/Circulant4/Analco07_Final/Golin_Analco07_final_version.dvi

Let T (G) be the number of spanning trees in graph G. In this note we explore the asymptotics of T (G) for circulant graphs. The circulant graph Cs1,s2,···,sk n is the 2k regular graph with n vertices labelled 0, 1, 2, · · · , n − 1, where node i has the 2k neighbors, (0 ≤ i ≤ n − 1) adjacent to vertices i + s1, i + s2, · · · , i + sk mod n. In this note we give a closed formula for the asymptotic limit limn→∞ T (Cs1,s2,···,sk n ) 1 n as a function of s1, s2, . . . , sn. We then extend this by permitting the si to be linear functions of n, i.e., we give a closed formula for lim n→∞ T ( C s1,s2,...,sk,⌊ n d1 ⌋+e1,⌊ n d2 ⌋+e2,...,⌊ n dl ⌋+el n ) 1 n where the di and ei are arbitrary integers. One consequence of our derivation is that if we let the si go to infinity then lim s1,s2,···,sk→∞ lim m→∞ T (C12k n ) 1 n