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$D_n=Cay(D_n,b^{\pm {\beta }_1},b^{\pm {\beta }_2},\dots ,b^{\pm {\beta }_s},a{b}^{{\gamma }_1},a{b}^{{\gamma }_2},\dots ,a{b}^{{\gamma }_t})$ on the dihedral group $D_n=〈a,b|a^2=1,b^n=1,{\left(ab\right)}^2=1〉$ of order 2n.

We obtain a closed formula for the number $\tau \left(n\right)$ of spanning trees in $D_n$ 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 $F\left(x\right)=\sum _{n=1}^{\infty }\tau \left(n\right)x^n$ is a rational function with integer coefficients.