| Summary: | Let <i>G</i> be a simple graph of order <i>n</i>. The Estrada index and Laplacian Estrada index of <i>G</i> are defined by <inline-formula> <math display="inline"> <semantics> <mrow> <mi>E</mi> <mi>E</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msup> <mi>e</mi> <mrow> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mi>E</mi> <mi>E</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msup> <mi>e</mi> <mrow> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>L</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <msubsup> <mrow> <mo>{</mo> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>}</mo> </mrow> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msubsup> <mrow> <mo>{</mo> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>L</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>}</mo> </mrow> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> </semantics> </math> </inline-formula> 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.
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