Further analysis of the number of spanning trees in circulant graphs

Abstract

Let 1 ⩽ s 1 < s 2 < ⋯ < s k ⩽ ⌊ n / 2 ⌋ be given integers. An undirected even-valent circulant graph, C n s 1 , s 2 , … , s k , has n vertices 0 , 1 , 2 , … , n - 1 , and for each s i ( 1 ⩽ i ⩽ k ) and j ( 0 ⩽ j ⩽ n - 1 ) there is an edge between j and j + s i ( mod n ) . Let T ( C n s 1 , s 2 , … , s k ) stand for the number of spanning trees of C n s 1 , s 2 , … , s k . For this special class of graphs, a general and most recent result, which is obtained in [Y.P. Zhang, X. Yong, M. Golin, [The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337–350]], is that T ( C n s 1 , s 2 , … , s k ) = na n 2 where a n satisfies a linear recurrence relation of order 2 s k - 1 . And, most recently, for odd-valent circulant graphs, a nice investigation on the number a n is [X. Chen, Q. Lin, F. Zhang, The number of spanning trees in odd-valent circulant graphs, Discrete Math. 282 (2004) 69–79]. In this paper, we explore further properties of the numbers a n from their combinatorial structures. Comparing with the previous work, the differences are that (1) in finding the coefficients of recurrence formulas for a n , we avoid solving a system of linear equations with exponential size, but instead, we give explicit formulas; (2) we find the asymptotic functions and therefore we ‘answer’ the open problem posed in the conclusion of [Y.P. Zhang, X. Yong, M. Golin, The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337–350]. As examples, we describe our technique and the asymptotics of the numbers.