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

Let $N\ge 2$ 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 $V_2=\{v\mid {\sigma }^i(v)\mathrm{\ne }vforany1<i<N\}$, 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 $V_1=\mathrm{\varnothing }$ 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 $C_n(s_1,s_2,\dots ,s_k)$ and $K_2\bigvee C_n(s_1,s_2,\dots ,s_k)$.