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by FooBarBizBazz
635 days ago
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Algebraic graph theory Subfield: Spectral graph theory https://en.m.wikipedia.org/wiki/Algebraic_graph_theory As for adjacency matrices -- consider the (oriented) vertex-edge incidence matrix D (vertices x edges, each edge is a column, head is +1 and tail is -1, other entries are zero). Then D and D' (transpose) both have interpretations (stop and think about them; they're like gradient and divergence), as does L = D D' (graph Laplacian). This also generalizes from graphs to simplicial complexes (triangles, tetrahedra). And "connected components" of graphs generalize to higher homology groups (loops, voids): https://pi.math.cornell.edu/~hatcher/AT/AT.pdf |
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