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The eigenvalues of the Hessian matrices of the generating functions for trees with \(k\) components. (English) Zbl 1476.05129

Summary: Let us consider a truncated matroid \(M_{\Gamma}^r\) of rank \(r\) of a graphic matroid of a graph \(\Gamma\). The basis for \(M_{\Gamma}^r\) is the set of the forests with \(r\) edges in \(\Gamma\). We consider this basis generating function and compute its Hessian. In this paper, we show that the Hessian of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph does not vanish by calculating the eigenvalues of the Hessian matrix. Moreover, we show that the Hessian matrix of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph has exactly one positive eigenvalue. As an application, we show the strong Lefschetz property for the Artinian Gorenstein algebra associated to the truncated matroid.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05A15 Exact enumeration problems, generating functions
05C31 Graph polynomials
05C05 Trees
05B35 Combinatorial aspects of matroids and geometric lattices
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
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References:

[1] Harima, Tadahito; Maeno, Toshiaki; Morita, Hideaki; Numata, Yasuhide; Wachi, Akihito; Watanabe, Junzo, The Lefschetz Properties, Lecture Notes in Mathematics, vol. 2080 (2013), Springer: Springer Heidelberg, MR3112920 · Zbl 1284.13001
[2] Huh, June; Wang, Botong, Enumeration of points, lines, planes, etc., Acta Math., 218, 2, 297-317 (2017), MR3733101 · Zbl 1386.05021
[3] Maeno, Toshiaki; Numata, Yasuhide, On the Sperner property and Gorenstein algebras associated to matroids, (DMTCS Proceedings AR (2012)), 157-168 · Zbl 1412.05217
[4] Maeno, Toshiaki; Numata, Yasuhide, Sperner property, matroids and finite-dimensional Gorenstein algebras, (Tropical Geometry and Integrable Systems. Tropical Geometry and Integrable Systems, Contemp. Math., vol. 580 (2012), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 73-84, MR2985388 · Zbl 1317.13059
[5] Maeno, Toshiaki; Numata, Yasuhide, Sperner property and finite-dimensional Gorenstein algebras associated to matroids, J. Commut. Algebra, 8, 4, 549-570 (2016), MR3566530 · Zbl 1360.13048
[6] Maeno, Toshiaki; Watanabe, Junzo, Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials, Ill. J. Math., 53, 2, 591-603 (2009), MR2594646 · Zbl 1200.13031
[7] Moon, J. W., Enumerating labelled trees, (Graph Theory and Theoretical Physics (1967), Academic Press: Academic Press London), 261-272, MR0231755 · Zbl 0204.24502
[8] Murai, Satoshi; Nagaoka, Takahiro; Yazawa, Akiko, Strictness of the log-concavity of generating polynomials of matroids, J. Comb. Theory, Ser. A, 181, Article 105351 pp. (2021), MR4223331 · Zbl 1464.05035
[9] Nagaoka, Takahiro; Yazawa, Akiko, Strict log-concavity of the Kirchhoff polynomial and its applications to the strong Lefschetz property, J. Algebra, 577, 175-202 (2021), MR4234203 · Zbl 1460.05091
[10] Watanabe, Junzo, A remark on the Hessian of homogeneous polynomials, (The Curves Seminar at Queen’s, Vol. XIII. The Curves Seminar at Queen’s, Vol. XIII, Queen’s Papers in Pure and Appl. Math., vol. 119 (2000), Queen’s: Queen’s Univ., Kingston, ON), 171-178 · Zbl 1196.13009
[11] Yazawa, Akiko, The eigenvalues of Hessian matrices of the complete and complete bipartite graphs, J. Algebraic Comb. (2021) · Zbl 1460.05091
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