On the complexity of Cayley graphs on a dihedral group

Abstract

In this paper, we investigate the complexity of an infinite family of Cayley graphs $\mathcal{D}_{n}=Cay(\mathbb{D}_{n}, b^{\pm\beta_1},b^{\pm\beta_2},\ldots,b^{\pm\beta_s}, a b^{\gamma_1}, a b^{\gamma_2},\ldots, a b^{\gamma_t} )$ on the dihedral group $\mathbb{D}_{n}=\langle a,b| a^2=1, b^n=1,(a\,b)^2=1\rangle$ of order $2n.$ We obtain a closed formula for the number $\tau(n)$ of spanning trees in $\mathcal{D}_{n}$ in terms of Chebyshev polynomials, investigate some arithmetical properties of this function, and find its asymptotics as $n\to\infty.$ Moreover, we show that the generating function $F(x)=\sum\limits_{n=1}^\infty\tau(n)x^n$ is a rational function with integer coefficients.