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Spectrum graph theory not only facilitate comprehensively reflect thetopological structure and dynamic characteristics of networks, but also offersignificant and noteworthy applications in theoretical chemistry, networkscience and other fields. Let $L_{n}^{8,4}$ represent a linearoctagonal-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 reverseidentifying the opposite edges, whereas cylinder graph $Q'_{n}(8,4)$ identifiesthe opposite edges by order. In this paper, the explicit formulas of Kirchhoffindices and complexity of $Q_{n}(8,4)$ and $Q'_{n}(8,4)$ are demonstrated byLaplacian characteristic polynomials according to decomposition theorem andVieta's theorem. In surprise, the Kirchhoff index of$Q_{n}(8,4)$($Q'_{n}(8,4)$) is approximately one-third half of its Wiener indexas $n\to\infty$.

The Laplacians, Kirchhoff index and complexity of linear M\"obius and cylinder octagonal-quadrilateral networks