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The Incidence Hopf Algebra of Graphs
Humpert, Brandon Eugene Martin, Jeremy L.
Humpert, Brandon Eugene
Martin, Jeremy L.
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Abstract
The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite graphs, and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Takeuchi's and Schmitt's more general formulas for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial.
Description
This is the published version, also available here: http://dx.doi.org/10.1137/110820075.
Date
2012-05-03
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Society for Industrial and Applied Mathematics
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Keywords
combinatiorial Hopf algebra, graph, chromatic polynomial, tutte polynomial, acyclic orientation
Citation
Humpert, Brandon & Martin, Jeremy L. "The Incidence Hopf Algebra of Graphs." (2012) SIAM J. Discrete Math., 26(2), 555–570. (16 pages). http://dx.doi.org/10.1137/110820075.