Estrada Index of Random Bipartite Graphs

The topological structures of many social, biological, and technological systems can be characterized by the connectivity properties of the interaction pathways (edges) between system components (vertices) [1]. Starting with the Königsberg seven-bridge problem in 1736, graphs with bidirectional or symmetric edges have ideally epitomized structures of various complex systems, and have developed into one of the mainstays of the modern discrete mathematics and network theory. Formally, a simple graph G consists of a vertex set V = {1, 2, · · · , n} and an edge set E ⊆ V × V . The adjacency matrix of G is a symmetric (0, 1)-matrix A(G) = (aij) ∈ Rn×n, where aij = aji = 1 if vertices i and j are adjacent, and aij = aji = 0 otherwise. It is well-known in algebraic graph theory that A(G) has exactly n real eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn due to its symmetry. They are usually called the spectrum (eigenvalues) of G itself [2]. A spectral graph invariant, the Estrada index EE(G) of G, is defined as