In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomials of trees are unimodal. Subsequently, many researchers proposed strengthening this conjecture to log-concavity. In 2023, Kadrawi, Levit, Yosef, and Mizrachi discovered independence polynomials of trees of order 26 that are not log-concave, which led them to construct two infinite families of such polynomials, denoted by $T_{3,m,n}$ and $T_{3,m,n}^*$. In this paper, we show that these two infinite families also satisfy the unimodal conjecture raised by Alavi, Malde, Schwenk, and Erdős.
Independence polynomials of well-covered graphs: generic counterexamples for the unimodality conjecture
一个图$G$是良好覆盖的，如果它的所有极大稳定集都具有相同的大小，记为alpha（G）（M）. D. Plummer，1970）。如果对任何$k$，多项式I（G;x）的第$k$个系数等于图$G$中基数$k$的稳定集的个数，则称它为$G$的独立多项式（Gutman and Harary，1983）. J. I.布朗，K。Dilcher和R. J. Nowakowski（2000）证明了对于任何覆盖良好的图$G$，I（G;x）是单峰的（即，存在一个索引$k$，使得从第一个到第$k$ -th的系数序列的一部分是非减的，而系数的另一部分是非增的）。T. S.迈克尔和W。N. Traves（2002）证明了这个断言对于alpha（G）< 4是正确的，而对于集合{4，5，6，7}中的alpha（G），他们提供了反例。本文证明了对任意整数$alpha$ > 7，存在一个（不）连通的良好覆盖图$G$，满足$alpha$ = alpha（G），且其独立多项式不是单峰的.此外，我们给出了α（G）< 7的图$G$具有单峰独立多项式的若干充分条件.
On Unimodality of Independence Polynomials of some Well-Covered Trees
图$G$中基数$k$的稳定集的个数是图$G$的独立多项式的第$k$个系数（I. Gutman和F. Harary，1983）。1990年，Y. O. Hamidoune证明了对于任何无爪图，它的独立多项式是单峰的，即，存在系数$k$，使得系数序列中从第一个到第$k$个的部分不减，而系数的第二部分不增。Y. Alavi，P. J. Malde，A. J. Schwenk和P. Erdös（1987）提出了一个问题：树（或者森林）的独立多项式是否是单峰的。J. I.布朗，K。Dilcher和R. J. Nowakowski（2000）证明了这对任何覆盖良好的图（所有最大独立集具有相同大小的图）都是正确的。V.E. Levit和E. Mandrescu（1999）证明了每一个良好覆盖的树都可以作为许多良好覆盖的蜘蛛的连接来获得，其中蜘蛛是一个最多有一个度至少为3的顶点的树。在本文中，我们证明了任何覆盖良好的蜘蛛的独立多项式是单峰的。此外，我们还介绍了一些关于独立多项式的图变换。它们使我们能够减少几种类型的覆盖良好的树无爪图，因此，证明他们的独立多项式是单峰的。
Unimodality of Independence Polynomials for Sufficiently Large Forests
Duke University（杜克大学）；Harvard T.H. Chan School of Public Health（哈佛陈曾熙公共卫生学院）；Hebrew University（希伯来大学）；Universidad de Valladolid（巴利亚多利德大学）；University of Hawai‘i at Mānoa（夏威夷大学马诺阿分校）
An independent set in a graph is a set of pairwise non-adjacent vertices. The independence number $α{(G)}$ is the size of a maximum independent set in the graph $G$. The independence polynomial of a graph is the generating function for the sequence of numbers of independent sets of each size. In other words, the $k$-th coefficient of the independence polynomial equals the number of independent sets comprised of $k$ vertices. For instance, the degree of the independence polynomial of the graph $G$ is equal to $α{(G)}$. In 1987, Alavi, Malde, Schwenk, and Erd{ö}s conjectured that the independenc…
We discuss two surprising properties of a family of polynomials that generalize the Mahonian $q$-Catalan polynomials, and more generally the $q$-Schröder polynomials. By interpreting them as $\mathfrak{sl}_2$-characters, we show that the rational $q$-Schröder polynomials are parity-unimodal, which means that the even- and odd-degree coefficients are separately unimodal. Second, we show that they exhibit a $q=-1$ phenomenon. This is a special case of a more general cyclic sieving phenomenon for certain transitive $S_n$-actions, deduced from Molien's formula.
In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of interlacing polynomials. Many of the open questions on stability and unimodality of polynomials pertain to the enumeration of faces of cell complexes. In this paper, we relate the theory of interlacing polynomials to the shellability of cell complexes. We first derive a sufficient condition for stability of the $h$-polynomial of a subdivision of a shellable comp…
Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove tha…
We consider a family of distance graphs in $\mathbb{R}^n$ and find its independent numbers in some cases. Define graph $J_{\pm}(n,k,t)$ in the following way: the vertex set consists of all vectors from $\{-1,0,1\}^n$ with $k$ nonzero coordinates; edges connect the pairs of vertices with scalar product $t$. We find the independence number of $J_{\pm}(n,k,t)$ for $n > n_0 (k,t)$ in the cases $t = 0$ and $t = -1$; these cases for $k = 3$ are solved completely. Also the independence number is found for negative odd $t$ and $n > n_0 (k,t)$.
Independence, induced subgraphs, and domination in $K_{1,r}$-free graphs
University of Haifa–Oranim（海法大学奥拉尼姆分校）；First Principles（第一原理）；Rice University（莱斯大学）；University of Johannesburg（约翰内斯堡大学）；University of Houston–Downtown（休斯顿城市大学）
The $k$-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than $k$. A graph is called $k$-partially walk-regular if the number of closed walks of a given length $l\le k$, rooted at a vertex $v$, only depends on $l$. In particular, a distance-regular graph is also $k$-partially walk-regular for any $k$. In this note, we introduce a new family of polynomials obtained from the spectrum of a graph. These polynomials, together with the interlacing technique, allow us to give tight spectral bounds on the $k$-independence number of a $k$-partially wa…
There is currently a growing interest in understanding which lattice simplices have unimodal local $h^\ast$-polynomials (sometimes called box polynomials); specifically in light of their potential applications to unimodality questions for Ehrhart $h^\ast$-polynomials. In this note, we compute a general form for the local $h^\ast$-polynomial of a well-studied family of lattice simplices whose associated toric varieties are weighted projective spaces. We then apply this formula to prove that certain such lattice simplices, whose combinatorics are naturally encoded using common systems of numerat…
Let $\mathcal{R}$ be a family of axis-parallel rectangles in the plane. The transversal number $τ(\mathcal{R})$ is the minimum number of points needed to pierce all the rectangles. The independence number $ν(\mathcal{R})$ is the maximum number of pairwise disjoint rectangles. Given a positive real number $r$, we say that $\mathcal{R}$ is an r-bounded family if, for any rectangle in $\mathcal{R}$, the aspect ratio of the longer side over the shorter side is at most $r$. Gyárfás and Lehel asked if it is possible to bound the transversal number $τ(\mathcal{R})$ with a linear function of the indep…
The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the polynomial is related to phase transitions, and plays an important role in the design of efficient algorithms to approximately compute evaluations of the polynomial. In this paper we directly relate the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. We do this by moreover relating the location of zeros to chaotic behaviour of a naturally associ…
The q-binomial coefficients were assumed to be unimodal as early as the 1850's, but it remained unproven until Sylvester's 1878 proof using invariant theory. In 1982, Proctor gave an "elementary" proof using linear algebra. Finally, in 1989, Kathy O'Hara provided a combinatorial proof of the unimodality of the q-binomial coefficients. Very soon thereafter, Doron Zeilberger translated the argument into an elegant recurrence. We introduce several perturbations to the recurrence to create a larger family of unimodal polynomials. We analyze how these perturbations affect the final polynomial and a…
