1,716 research outputs found
Set maps, umbral calculus, and the chromatic polynomial
Some important properties of the chromatic polynomial also hold for any
polynomial set map satisfying p_S(x+y)=\sum_{T\uplus U=S}p_T(x)p_U(y). Using
umbral calculus, we give a formula for the expansion of such a set map in terms
of any polynomial sequence of binomial type. This leads to some new expansions
of the chromatic polynomial. We also describe a set map generalization of Abel
polynomials.Comment: 20 page
T=0 Partition Functions for Potts Antiferromagnets on Moebius Strips and Effects of Graph Topology
We present exact calculations of the zero-temperature partition function of
the -state Potts antiferromagnet (equivalently the chromatic polynomial) for
Moebius strips, with width or 3, of regular lattices and homeomorphic
expansions thereof. These are compared with the corresponding partition
functions for strip graphs with (untwisted) periodic longitudinal boundary
conditions.Comment: 9 pages, Latex, Phys. Lett. A, in pres
An Abstraction of Whitney's Broken Circuit Theorem
We establish a broad generalization of Whitney's broken circuit theorem on
the chromatic polynomial of a graph to sums of type
where is a finite set and is a mapping from the power set of into
an abelian group. We give applications to the domination polynomial and the
subgraph component polynomial of a graph, the chromatic polynomial of a
hypergraph, the characteristic polynomial and Crapo's beta invariant of a
matroid, and the principle of inclusion-exclusion. Thus, we discover several
known and new results in a concise and unified way. As further applications of
our main result, we derive a new generalization of the maximums-minimums
identity and of a theorem due to Blass and Sagan on the M\"obius function of a
finite lattice, which generalizes Rota's crosscut theorem. For the classical
M\"obius function, both Euler's totient function and its Dirichlet inverse, and
the reciprocal of the Riemann zeta function we obtain new expansions involving
the greatest common divisor resp. least common multiple. We finally establish
an even broader generalization of Whitney's broken circuit theorem in the
context of convex geometries (antimatroids).Comment: 18 page
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