The asymptotic number of spanning trees in circulant graphs

Let T ( G ) be the number of spanning trees in graph G . In this note, we explore the asymptotics of T ( G ) when G is a circulant graph with given jumps. rculant graph C n s 1 , s 2 , … , s k is the 2 k -regular graph with n vertices labeled 0 , 1 , 2 , … , n − 1 , where node i has the 2 k neighbors i ± s 1 , i ± s 2 , … , i ± s k where all the operations are ( mod n ) . We give a closed formula for the asymptotic limit lim n → ∞ T ( C n s 1 , s 2 , … , s k ) 1 n as a function of s 1 , s 2 , … , s k . We then extend this by permitting some of the jumps to be linear functions of n , i.e., letting s i , d i and e i be arbitrary integers, and examining lim n → ∞ T ( C n s 1 , s 2 , … , s k , ⌊ n d 1 ⌋ + e 1 , ⌊ n d 2 ⌋ + e 2 , … , ⌊ n d l ⌋ + e l ) 1 n . While this limit does not usually exist, we show that there is some p such that for 0 ≤ q < p , there exists c q such that limit (1) restricted to only n congruent to q modulo p does exist and is equal to c q . We also give a closed formula for c q . rther consequence of our derivation is that if s i go to infinity (in any arbitrary order), then lim s 1 , s 2 , … , s k → ∞ lim n → ∞ T ( C n s 1 , s 2 , … , s k ) 1 n = 4 exp [ ∫ 0 1 ∫ 0 1 ⋯ ∫ 0 1 ln ( ∑ i = 1 k sin 2 π x i ) d x 1 d x 2 ⋯ d x k ] . Interestingly, this value is the same as the asymptotic number of spanning trees in the k -dimensional square lattice recently obtained by Garcia, Noy and Tejel.