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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 com-binatorial 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… (More)

The chromatic symmetric function X G of a graph G was introduced by Stan-ley. In this paper we introduce a quasisymmetric generalization X k G called the k-chromatic quasisymmetric function of G and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of X G to χ G (λ), the chromatic polynomial, we… (More)

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