Can someone please clarify how the spectral gap of a graph relates to its algebraic connectivity (aka Fiedler value) and whether these use the adjacency matrix or laplacian matrix?
2026-03-26 14:16:07.1774534567
Spectral gap vs. algebraic connectivity
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If the distinct eigenvalues of the adjacency matrix of a $k$-regular graph are $k=\theta_1\ge\cdots\ge\theta_m$, the eigenvalues of its Laplacian in non-decreasing order are \[ 0, k-\theta_2,\ldots,k-\theta_m. \] So in this case the algebraic connectivity and the spectral gap coincide.
If the graph is not regular then, in general, there is no simple or useful relation between the eigenvalue of the adjacency matrix and the eigenvalues of the Laplacian.