Separation of a large eigenvalue from the bulk of the spectrum

Eigenvalue spectrum of adjacency matrices of many complex networks reveals that a large real eigenvalue separate from the bulk of the population of eigenvalues. A related theorem in this observation is that of Perron-Frobenius, which states that ifˆA is a positive matrix, then there exists a unique eigenvalue ofˆA, which has the greatest absolute value, and its associated eigenvector may be taken as positive. In this letter we show, analytically and numerically, that regardless of the matrix being positive or non-negative if the mean of the matrix entries is greater than the variance then a single eigenvalue which has the greatest value separates from the rest. The corresponding eigenvector may be positive if the matrix is positive, however the numerical results show that for the given mean value µ of order O(1/N) in no cases the corresponding eigenvector is localised.