Graph Theory, Lecture 30: Algebraic aspects of graphs II: duality of cycles and cuts

Cycle space and cut space are orthogonal.
Fundamental cycles and cuts, from 19:53.
From 35:30: Incidence matrix B. Interpretation as boundary map; its transpose as coboundary map.
Cycle and cut space as the kernel and image of these maps, and their LA duality.
Adjacency matrix A, diagonal matrix D, and Laplacian L. Relationship with B.
Role of the eigenvalues of L in clustering.
From 1:01: How to generate the cycle space. From induced cycles, from geodesic cycles, from fundamental cycles.
Face boundaries in plane graph generate the cycle space. Generalisation to non-planar graphs by
Tutte's Theorem 3.2.6, rebranding faces as non-separating induced cycles. Proof slowly developed.
Covers rest of Chapter 1.9, and whatever of Chapter 3.2 was not covered in Lecture 8.