A Tutte polynomial inequality for lattice path matroids

Knauer, Kolja; Martínez-Sandoval, Leonardo; Alfonsín, Jorge Luis Ramírez

doi:10.1016/j.aam.2016.11.008

Cited by 13 publications

(20 citation statements)

References 17 publications

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“…As observed in [71], the inequality (7.31) implies the inequality (7.30). Some works related to these conjectures include [72]- [76]. In particular, the Merino-Welsh conjecture has been proved for wheel graphs W h n , complete graphs K n , and complete bipartite graphs K r,s with r ≥ s ≥ 2 in [72], and for series-parallel graphs in [75].…”

Section: B Comparison With Spanning Treesmentioning

confidence: 99%

Asymptotic behavior of acyclic and cyclic orientations of directed lattice graphs

Chang

Shrock

2020

Physica A: Statistical Mechanics and its Applications

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We calculate exponential growth constants describing the asymptotic behavior of several quantities enumerating classes of orientations of arrow variables on the bonds of several types of directed lattice strip graphs G of finite width and arbitrarily great length, in the infinite-length limit, denoted {G}. Specifically, we calculate the exponential growth constants for (i) acyclic orientations, α({G}), (ii) acyclic orientations with a single source vertex, α 0 ({G}), and (iii) totally cyclic orientations, β({G}). We consider several lattices, including square (sq), triangular (tri), and honeycomb (hc). From our calculations, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. To our knowledge, these are the best current bounds on these quantities. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for the exponential growth constants, with fractional uncertainties ranging from O(10 −4 ) to O(10 −2 ). Further, we present exact values of α(tri), α 0 (tri), and β(hc) and use them to show that our lower and upper bounds on these quantities are very close to these exact values, even for modest strip widths. Results are also given for a nonplanar lattice denoted sq d . We show that α({G}), α 0 ({G}), and β({G}) are monotonically increasing functions of vertex degree for these lattices. We also study the asymptotic behavior of the ratios of the quantities (i)-(iii) divided by the total number of edge orientations as the number of vertices goes to infinity. A comparison is given of these exponential growth constants with the corresponding exponential growth constant τ ({G}) for spanning trees. Our results are in agreement with inequalities following from the Merino-Welsh and Conde-Merino conjectures.

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Section: B Comparison With Spanning Treesmentioning

confidence: 99%

Asymptotic behavior of acyclic and cyclic orientations of directed lattice graphs

Chang

Shrock

2020

Physica A: Statistical Mechanics and its Applications

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“…From this graphical realization of the minimal connected matroid of rank k and cardinality n, it is possible to verify that it is a Lattice Path Matroid [3] with a nice structure. Using [16,Theorem 3.2] and all the terminology defined within that article, we can see that T k,n coincides with the snake S(n − k, k).…”

Section: The Matroid Polytope Of Minimal Matroidsmentioning

confidence: 99%

On the Ehrhart Polynomial of Minimal Matroids

Ferroni

2020

Preprint

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We provide a formula for the Ehrhart polynomial of the connected matroid of size n and rank k with the least number of bases, also known as a minimal matroid [10]. We prove that their polytopes are Ehrhart positive and h * -real-rooted (and hence unimodal). We use our formula for these Ehrhart polynomials to prove that the operation of circuit-hyperplane relaxation of a matroid preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are h * -real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.

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“…5) where our abbreviation CMC stands for Conde-Merino conjecture. Some relevant related papers include[35]-[38]. For our purposes, we first observe that the Merino-Welsh and Conde-Merino conjectures imply the following inequalities on exponential growth constants, where {G} is the n(G) → ∞ limit of graphs G that satisfy the premise of the MWC and CMC:…”

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confidence: 99%

Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices

Chang

Shrock

2019

J Stat Phys

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We infer upper and lower bounds on the exponential growth constants α(Λ), α 0 (Λ), and β(Λ) describing the large-n behavior of, respectively, the number of acyclic orientations, acyclic orientations with a unique source vertex, and totally cyclic orientations of arrows on bonds of several n-vertex heteropolygonal Archimedean lattices Λ. These are, to our knowledge, the best bounds on these growth constants. The inferred upper and lower bounds on the growth constants are quite close to each other, which enables us to derive rather accurate values for the actual exponential growth constants. Combining our new results for heteropolygonal Archimedean lattices with our recent results for homopolygonal Archimedean lattices, we show that the exponential growth constants α(Λ), α 0 (Λ), and β(Λ) on these lattices are monotonically increasing functions of the lattice coordination number. Comparisons are made with the corresponding growth constants for spanning trees on these lattices. Our findings provide further support for the Merino-Welsh and Conde-Merino conjectures.

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A Tutte polynomial inequality for lattice path matroids

Cited by 13 publications

References 17 publications

Asymptotic behavior of acyclic and cyclic orientations of directed lattice graphs

Asymptotic behavior of acyclic and cyclic orientations of directed lattice graphs

On the Ehrhart Polynomial of Minimal Matroids

Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices

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