Counting spanning trees of (1, N)-periodic graphs

Abstract: Let $N\geq 2$ be an integer, a (1, $N$)-periodic graph $G$ is a periodic graph whose vertices can be partitioned into two sets $V_1={v\mid\sigma(v)=v}$ and $V_2={v\mid\sigma^i(v)\neq v\ \mbox{for any}\ 1<i<N}$, where $\sigma$ is an automorphism with order $N$ of $G$. The subgraph of $G$ induced by $V_1$ is called a fixed subgraph. Yan and Zhang [Enumeration of spanning trees of graphs with rotational symmetry, J. Comb. Theory Ser. A, 118(2011): 1270-1290] studied the enumeration of spanning trees of a special type of (1, $N$)-periodic graphs with $V_1=\emptyset$ 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 $V_1$ 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 $C_n(s_1,s_2,\ldots,s_k)$ and $K_2\bigvee C_n(s_1,s_2,\ldots,s_k)$.