Kirchhoff index, multiplicative degree-Kirchhoff index and spanning trees of the linear crossed polyomino chains

Let $G_n$ be a linear crossed polyomino chain with $n$ four-order complete graphs. In this paper, explicit formulas for the Kirchhoff index, the multiplicative degree-Kirchhoff index and the number of spanning trees of $G_n$ are determined, respectively. It is interesting to find that the Kirchhoff (resp. multiplicative degree-Kirchhoff) index of $G_n$ is approximately one quarter of its Wiener (resp. Gutman) index. More generally, let $\mathcal{G}^r_n$ be the set of subgraphs obtained by deleting $r$ vertical edges of $G_n$, where $0\leqslant r\leqslant n+1$. For any graph $G^r_n\in \mathcal{G}^r_{n}$, its Kirchhoff index and number of spanning trees are completely determined, respectively. Finally, we show that the Kirchhoff index of $G^r_n$ is approximately one quarter of its Wiener index.