The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and $2n$ squares, denoted by $L_n^{6,4,4}$. In this paper, by using the normalized Laplacian polynomial decomposition theorem, we have investigated the normalized Laplacian spectrum of $L_n^{6,4,4}$ consisting of the eigenvalues of symmetric tri-diagonal matrices ${\mathcal{L}}_A$ and ${\mathcal{L}}_S$ of order $4n+1$. As an application, the significant formula is obtained to calculate the multiplicative degree-Kirchhoff index and the number of spanning trees of generalized phenylene network based on the relationships between the coefficients and roots.

中文翻译：

---

归一化拉普拉斯算子、广义亚苯基的度-基尔霍夫指数和生成树

归一化拉普拉斯算子对于分析非正则图的结构性质极其重要。广义亚苯基的分子图由n 个六边形和$2n$ 正方形，表示为 ${升}_n^{6,4,4}$. 在本文中，通过使用归一化拉普拉斯多项式分解定理，我们研究了归一化拉普拉斯谱${升}_n^{6,4,4}$ 由对称三对角矩阵的特征值组成 ${\mathcal{升}}_{\mathrm{一种}}$ 和 ${\mathcal{升}}_{秒}$ 按顺序 $4n+1$. 作为应用，根据系数与根的关系，得到了计算乘法度-Kirchhoff指数和广义亚苯基网络生成树数的显着公式。