Counting spanning trees of (1, [formula omitted])-periodic graphs

Abstract

Let N≥2 be an integer, a (1, N)-periodic graph G is a periodic graph whose vertices can be partitioned into two sets V1={v∣σ(v)=v} and V2={v∣σi(v)≠vfor any1<i<N}, where σ is an automorphism with order N of G. The subgraph of G induced by V1 is called a fixed subgraph. Yan and Zhang (2011) studied the enumeration of spanning trees of a special type of (1, N)-periodic graphs with V1=0̸ for any non-trivial automorphism with order N. In this paper, we obtain a concise formula for the number of spanning trees of (1, N)-periodic graphs. Our result can reduce to Yan and Zhang’s when V1 is empty. As applications, we give a new closed formula for the spanning tree generating function of cobweb lattices, and obtain formulae for the number of spanning trees of circulant graphs Cn(s1,s2,…,sk), K1⋁Cn(s1,s2,…,sk) and K2⋁Cn(s1,s2,…,sk).