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Algebraic methods in Graph Theory; the linear algebra method, graph homomorphisms, group theoretic methods (for example Cayley graphs), and graph invariants. For graph eigenvalue problems use the spectral-graph-theory tag. For strongly regular graphs use the strongly-regular-graph tag. For Kneser graphs use the kneser-graph tag.
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Bicycles and spanning trees of graphs
The number of spanning trees of a plane(ly embedded) graph can be obtained from the Tutte polynomial.
(Pardon the proof via knot theory: the Jones polynomial of an alternating knot diagram is a spec …