A random matrix theory result showing that the combinatorial Laplacian L = D - A of an Erdos-Renyi random graph G(n,p) has, with very high probability, a simple spectrum (all eigenvalues distinct) together with quantitatively effective lower bounds on the gaps between consecutive eigenvalues. Because the Laplacian's entries are dependent - each edge contributes to two diagonal degrees - the proof develops new eigenvector-delocalization, eigenvalue-overcrowding, and small-entry estimates rather than importing Wigner-matrix results directly.
The combinatorial Laplacian L = D - A of an Erdos-Renyi random graph G(n,p) is a symmetric random matrix whose entries are dependent (each off-diagonal edge also enters two diagonal degrees), which p…