567 research outputs found
A unified approach to polynomial sequences with only real zeros
We give new sufficient conditions for a sequence of polynomials to have only
real zeros based on the method of interlacing zeros. As applications we derive
several well-known facts, including the reality of zeros of orthogonal
polynomials, matching polynomials, Narayana polynomials and Eulerian
polynomials. We also settle certain conjectures of Stahl on genus polynomials
by proving them for certain classes of graphs, while showing that they are
false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres
Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
The chromatic polynomial of a graph G counts the number of proper colorings
of G. We give an affirmative answer to the conjecture of Read and
Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic
polynomial form a log-concave sequence. We define a sequence of numerical
invariants of projective hypersurfaces analogous to the Milnor number of local
analytic hypersurfaces. Then we give a characterization of correspondences
between projective spaces up to a positive integer multiple which includes the
conjecture on the chromatic polynomial as a special case. As a byproduct of our
approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor
number with the Newton polytope.Comment: Improved readability. Final version, to appear in J. Amer. Math. So
h-Vectors of matroids and logarithmic concavity
Let M be a matroid on E, representable over a field of characteristic zero.
We show that h-vectors of the following simplicial complexes are log-concave:
1. The matroid complex of independent subsets of E. 2. The broken circuit
complex of relative to an ordering of E. The first implies a conjecture of
Colbourn on the reliability polynomial of a graph, and the second implies a
conjecture of Hoggar on the chromatic polynomial of a graph. The proof is based
on the geometric formula for the characteristic polynomial of Denham,
Garrousian, and Schulze.Comment: Major revision, 9 page
Construction numbers: How to build a graph?
Counting the number of linear extensions of a partial order was considered by
Stanley about 50 years ago. For the partial order on the vertices and edges of
a graph determined by inclusion, we call such linear extensions {\it
construction sequences} for the graph as each edge follows both of its
endpoints. The number of such sequences for paths, cycles, stars, double-stars,
and complete graphs is found. For paths, we agree with Stanley (the Tangent
numbers) and get formulas for the other classes. Structure and applications are
also studied.Comment: 18 page
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