Results 1  10
of
87
Rotation distance, triangulations, and hyperbolic geometry
 J. Amer. Math. Soc
, 1988
"... A rotation in a binary tree is a local restructuring of the tree that changes it into another tree. One can execute a rotation by collapsing an internal edge of the tree to a point, thereby obtaining a node with three children, and then reexpanding the node of order three in the alternative way int ..."
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Cited by 112 (4 self)
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A rotation in a binary tree is a local restructuring of the tree that changes it into another tree. One can execute a rotation by collapsing an internal edge of the tree to a point, thereby obtaining a node with three children, and then reexpanding the node of order three in the alternative way into two nodes of
Haplotyping as Perfect Phylogeny: Conceptual Framework and Efficient Solutions (Extended Abstract)
, 2002
"... The next highpriority phase of human genomics will involve the development of a full Haplotype Map of the human genome [12]. It will be used in largescale screens of populations to associate specific haplotypes with specific complex geneticinfluenced diseases. A prototype Haplotype Mapping strat ..."
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Cited by 109 (10 self)
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The next highpriority phase of human genomics will involve the development of a full Haplotype Map of the human genome [12]. It will be used in largescale screens of populations to associate specific haplotypes with specific complex geneticinfluenced diseases. A prototype Haplotype Mapping strategy is presently being finalized by an NIH workinggroup. The biological key to that strategy is the surprising fact that genomic DNA can be partitioned into long blocks where genetic recombination has been rare, leading to strikingly fewer distinct haplotypes in the population than previously expected [12, 6, 21, 7]. In this paper
Reflection positivity, rank connectivity, and homomorphism of graphs
 Journal of the American Mathematical Society
"... It is shown that a graph parameter can be realized as the number of homomorphisms into a fixed (weighted) graph if and only if it satisfies two linear algebraic conditions: reflection positivity and exponential rankconnectivity. In terms of statistical physics, this can be viewed as a characterizat ..."
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Cited by 60 (23 self)
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It is shown that a graph parameter can be realized as the number of homomorphisms into a fixed (weighted) graph if and only if it satisfies two linear algebraic conditions: reflection positivity and exponential rankconnectivity. In terms of statistical physics, this can be viewed as a characterization of partition functions of vertex models. 1
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros ..."
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Cited by 47 (11 self)
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I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
Chromatic roots are dense in the whole complex plane
 In preparation
, 2000
"... to appear in Combinatorics, Probability and Computing I show that the zeros of the chromatic polynomials PG(q) for the generalized theta graphs Θ (s,p) are, taken together, dense in the whole complex plane with the possible exception of the disc q − 1  < 1. The same holds for their dichromatic pol ..."
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Cited by 37 (14 self)
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to appear in Combinatorics, Probability and Computing I show that the zeros of the chromatic polynomials PG(q) for the generalized theta graphs Θ (s,p) are, taken together, dense in the whole complex plane with the possible exception of the disc q − 1  < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Pottsmodel partition functions) ZG(q,v) outside the disc q + v  < v. An immediate corollary is that the chromatic roots of notnecessarilyplanar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha–Kahane–Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof. KEY WORDS: Graph, chromatic polynomial, dichromatic polynomial, Whitney rank function, Tutte polynomial, Potts model, Fortuin–Kasteleyn representation,
MODULAR CONSTRUCTIONS FOR COMBINATORIAL GEOMETRIES
, 1975
"... R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 12 (1971), 214—217), proved that the characteristic polynomial xW of a modular flat x divides the characteristic polynomial x(G) of a geometry G. In this paper we identify the quotient: THEOREM. / / x is a modular ..."
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Cited by 33 (2 self)
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R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 12 (1971), 214—217), proved that the characteristic polynomial xW of a modular flat x divides the characteristic polynomial x(G) of a geometry G. In this paper we identify the quotient: THEOREM. / / x is a modular flat of G, x(G)/x(x) = X(7^(G))/( \ 1), where TX{G) is the complete Brown truncation of G by x. (The lattice of TX(G) consists of all flats containing x and all flats disjoint from x, with the induced order from G.) We give many characterizations of modular flats in terms of their lattice properties axiom and a modular version of the MacLaneSteinitz as well as by means of a shortcircuit Modular flats are shown to have many of the useful properties exchange axiom. of points and distributive flats (separators) in addition to being much more prevalent. The theorem relating the chromatic polynomials of two graphs and the polynomial of their vertex join across a common clique generalizes to geometries: THEOREM. Given geometries G and H, if x is a modular flat of G as well as a subgeometry of H, then there exists a geometry P = PX(G, H) which is a pushout in the category of injective strong maps and such that x(P) = X(G)x(H)lx(x) The closed set structure, rank function, independent sets, and lattice properties of P are characterized. After proving a modular extension theorem we give applications of our results to Crapo's single element extension theorem, Crapo's join operation, chain groups, unimodular geometries, transversal geometries, and graphs.
Planar Minimally Rigid Graphs and PseudoTriangulations
, 2003
"... Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under cer ..."
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Cited by 30 (14 self)
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Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide—to the best of our knowledge—the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.
Zeros of chromatic and flow polynomials of graphs
 J. Geometry
, 2003
"... We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids. 1 ..."
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Cited by 21 (4 self)
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We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids. 1
On the Enumeration of Inscribable Graphs
 Manuscript, NEC Research Institute
, 1991
"... We explore the question of counting, and estimating the number and the fraction of, inscribable graphs. In particular we will concern ourselves with the number of inscribable and circumscribable maximal planar graphs (synonym: simplicial polyhedra) on V vertices, or, dually, the number of circumscr ..."
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Cited by 21 (4 self)
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We explore the question of counting, and estimating the number and the fraction of, inscribable graphs. In particular we will concern ourselves with the number of inscribable and circumscribable maximal planar graphs (synonym: simplicial polyhedra) on V vertices, or, dually, the number of circumscribable and inscribable trivalent (synonyms: 3regular, simple) polyhedra. For small V we provide computergenerated tables. Asymptotically for large V we will prove bounds showing that these graphs are exponentially numerous, but, viewed as a fraction of all maximal planar graphs, they are exponentially rare. Many of our results are based on a lemma, the "strong 01 law for maximal planar graphs," of independent interest. This is part of a series of TMs exploring graphtheoretic consequences of the recent RivinSmith characterization of "inscribable graphs." (A graph is a set of "vertices," some pairs of which are joined by "edges." A graph is "inscribable" if it is the 1skeleton of a conve...