Let G be a simple graph of order n. The Estrada index and Laplacian Estrada index of G are defined by $EE\left(G\right)={\sum }_{i=1}^n{e}^{{\lambda }_i\left(A\left(G\right)\right)}$ and $LEE\left(G\right)={\sum }_{i=1}^n{e}^{{\lambda }_i\left(L\left(G\right)\right)}$ , where ${\left\{{\lambda }_i\left(A\left(G\right)\right)\right\}}_{i=1}^n$ and ${\left\{{\lambda }_i\left(L\left(G\right)\right)\right\}}_{i=1}^n$ are the eigenvalues of its adjacency and Laplacian matrices, respectively. In this paper, we establish almost sure upper bounds and lower bounds for random interdependent graph model, which is fairly general encompassing Erdös-Rényi random graph, random multipartite graph, and even stochastic block model. Our results unravel the non-triviality of interdependent edges between different constituting subgraphs in spectral property of interdependent graphs.

中文翻译：

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随机相依图的Estrada指数和Laplacian Estrada指数

令G为n阶的简单图。G的Estrada指数和Laplacian Estrada指数定义为 $ËË（G)={\sum }_{i=1}^n{e}^{{\lambda }_i\left(A\left(G\right)\right)}$ 和 $LEE\left(G\right)={\sum }_{i=1}^n{e}^{{\lambda }_i\left(L\left(G\right)\right)}$ ，在哪里 ${\left\{{\lambda }_i\left(A\left(G\right)\right)\right\}}_{i=1}^n$ 和 ${\left\{{\lambda }_i\left(L\left(G\right)\right)\right\}}_{i=1}^n$ 分别是其邻接矩阵和Laplacian矩阵的特征值。在本文中，我们为随机相互依赖的图模型建立了几乎确定的上限和下限，该模型相当笼统地涵盖了Erdös-Rényi随机图，随机多部分图甚至随机块模型。我们的结果揭示了相互依赖图的光谱特性中不同构成子图之间相互依赖边的非平凡性。