Graph structure determines the spectrum of the adjacency and Laplacian matrices
Graph Structure
Adjacency Eigenvalues (Energy Levels)
Laplacian Eigenvalues
Graph: Path
n = 10, m = 9
λ_max(A): -
λ₂(L) (Fiedler): -
Cheeger bound h(G): -
Is bipartite: -
Fiedler Vector (v₂ of L)
Fiedler value λ₂(L)>0 iff graph is connected. Cheeger inequality: λ₂/2 ≤ h(G) ≤ √(2λ₂). Bipartite graphs have symmetric adjacency spectrum. The Petersen graph is a famous 3-regular graph with spectrum {3¹, 1⁵, −2⁵}.