108 research outputs found

    A Tutte polynomial inequality for lattice path matroids

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    Let MM be a matroid without loops or coloops and let T(M;x,y)T(M;x,y) be its Tutte polynomial. In 1999 Merino and Welsh conjectured that max(T(M;2,0),T(M;0,2))T(M;1,1)\max(T(M;2,0), T(M;0,2))\geq T(M;1,1) holds for graphic matroids. Ten years later, Conde and Merino proposed a multiplicative version of the conjecture which implies the original one. In this paper we prove the multiplicative conjecture for the family of lattice path matroids (generalizing earlier results on uniform and Catalan matroids). In order to do this, we introduce and study particular lattice path matroids, called snakes, used as building bricks to indeed establish a strengthening of the multiplicative conjecture as well as a complete characterization of the cases in which equality holds.Comment: 17 pages, 9 figures, improved exposition/minor correction

    Some inequalities for the Tutte polynomial

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    We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases of M*. For this latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a >= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same class of matroids. We also prove this conjecture for some families of graphs and matroids.Comment: 17 page

    Aspects of the Tutte polynomial

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    The Merino--Welsh conjecture is false for matroids

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    The matroidal version of the Merino--Welsh conjecture states that the Tutte polynomial TM(x,y)T_M(x,y) of any matroid MM without loops and coloops satisfies that max(TM(2,0),TM(0,2))TM(1,1).\max(T_M(2,0),T_M(0,2))\geq T_M(1,1). Equivalently, if the Merino--Welsh conjecture is true for all matroids without loops and coloops, then the following inequalities are also satisfied for all matroids without loops and coloops: TM(2,0)+TM(0,2)2TM(1,1),T_M(2,0)+T_M(0,2)\geq 2T_M(1,1), and TM(2,0)TM(0,2)TM(1,1)2.T_M(2,0)T_M(0,2)\geq T_M(1,1)^2. We show a counter-example for these inequalities.Comment: 6 page

    Order Quasisymmetric Functions Distinguish Rooted Trees

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    Richard P. Stanley conjectured that finite trees can be distinguished by their chromatic symmetric functions. In this paper, we prove an analogous statement for posets: Finite rooted trees can be distinguished by their order quasisymmetric functions.Comment: 16 pages, 5 figures, referees' suggestions incorporate

    On the number of spanning trees in random regular graphs

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    Let d3d \geq 3 be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random dd-regular graph with nn vertices. (The asymptotics are as nn\to\infty, restricted to even nn if dd is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) dd. Numerical evidence is presented which supports our conjecture.Comment: 26 pages, 1 figure. To appear in the Electronic Journal of Combinatorics. This version addresses referee's comment
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