  Spectrum graph theory not only facilitate comprehensively reflect the
topological structure and dynamic characteristics of networks, but also offer
significant and noteworthy applications in theoretical chemistry, network
science and other fields. Let $L_{n}^{8,4}$ represent a linear
octagonal-quadrilateral network, consisting of $n$ eight-member ring and $n$
four-member ring. The M\"{o}bius graph $Q_{n}(8,4)$ is constructed by reverse
identifying the opposite edges, whereas cylinder graph $Q'_{n}(8,4)$ identifies
the opposite edges by order. In this paper, the explicit formulas of Kirchhoff
indices and complexity of $Q_{n}(8,4)$ and $Q'_{n}(8,4)$ are demonstrated by
Laplacian characteristic polynomials according to decomposition theorem and
Vieta's theorem. In surprise, the Kirchhoff index of
$Q_{n}(8,4)$($Q'_{n}(8,4)$) is approximately one-third half of its Wiener index
as $n\to\infty$.
