In this paper, we prove that if $f(x)=\sum_{k=0}^n{n\choose k}a_kx^k$ is a polynomial with real zeros only, then the sequence $\{a_k\}_{k=0}^n$ satisfies the following inequalities $a_{k+1}^2(1-\sqrt{1-c_k})^2/a_k^2 \leq(a_{k+1}^2-a_ka_{k+2})/(a_k^2-a_{k-1}a_{k+1}) \leq a_{k+1}^2(1+\sqrt{1-c_k})^2/a_k^2$, where $c_k=a_ka_{k+2}/a_{k+1}^2$. This inequality holds for the coefficients of the Riemann $ξ$-function, the ultraspherical, Laguerre and Hermite polynomials, and the partition function. Moreover, as a corollary, for the partition function $p(n)$, we prove that $p(n)^2-p(n-1)p(n+1)$ is increasing for $n\geq 55$. We also find that for a positive and log-concave sequence $\{a_k\}_{k\geq 0}$, the inequality $a_{k+2}/a_k\leq (a_{k+1}^2-a_ka_{k+2})/(a_k^2-a_{k-1}a_{k+1}) \leq a_{k+1}/a_{k-1}$ is the sufficient condition for both the $2$-log-concavity and the higher order Tur{á}n inequalities of $\{a_k\}_{k\geq 0}$. It is easy to verify that if $a_k^2\geq ra_{k+1}a_{k-1}$, where $r\geq 2$, then the sequence $\{a_k\}_{k\geq 0}$ satisfies this inequality.
In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted $U$-polynomial for every $k$; by this we mean that the polynomials agree on terms with degree at most $k+1$. The main tool for this construction is a generalization of the $U$-polynomial to rooted graphs, which we introduce and study in this article. Most notably we show that rooted trees can be reconstructed from its rooted $U$-polynomial.
A polynomial is real-rooted if all of its roots are real. This note gives a simple proof of the Hermite-Sylvester theorem that a polynomial $f(x) \in {\mathbf R}[x]$ is real-rooted if and only if an associated quadratic form is positive semidefinite.
The independence polynomial of a graph $G$ is \[I(G,x)=\sum\limits_{k\ge 0}i_k(G)x^k,\] where $i_k(G)$ denotes the number of independent sets of $G$ of size $k$ (note that $i_0(G)=1$). In this paper we show a new method to prove real-rootedness of the independence polynomials of certain families of trees. In particular we will give a new proof of the real-rootedness of the independence polynomials of centipedes (Zhu's theorem), caterpillars (Wang and Zhu's theorem), and we will prove a conjecture of Galvin and Hilyard about the real-rootedness of the independence polynomial of the so-called Fi…
In this paper, we prove the real-rootedness of two classes of generalized Narayana polynomials: one arising as the $h$-polynomials of the generalized associahedron associated to the finite Weyl groups, the other arising in the study of the infinite log-concavity of the Boros-Moll polynomials. For the former, Brändén has already proved that these $h$-polynomials have only real zeros. We establish certain recurrence relations for the two classes of Narayana polynomials, from which we derive the real-rootedness. To prove the real-rootedness, we use a sufficient condition, due to Liu and Wang, to …
It is well-known that the coordinator polynomials of the classical root lattice of type $A_n$ and those of type $C_n$ are real-rooted. They can be obtained, either by the Aissen-Schoenberg-Whitney theorem, or from their recurrence relations. In this paper, we develop a trigonometric substitution approach which can be used to establish the real-rootedness of coordinator polynomials of type $D_n$. We also find the coordinator polynomials of type $B_n$ are not real-rooted in general. As a conclusion, we obtain that all coordinator polynomials of Weyl group lattices are log-concave.
The binomial Eulerian polynomials, introduced by Postnikov, Reiner, and Williams, are $γ$-positive polynomials and can be interpreted as $h$-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations. In this paper, we generalize them to $\mathbf{s}$-inversion sequences and prove that these new polynomials have only real roots by the method of interlacing polynomials. Three applications of this result are presented. The first one is to prove the real-rootedness of binomial Eulerian polynomials, which confirms a conject…
In this paper we give a generalization of Chebyshev polynomials and using this we describe the Möbius function of the generalized subword order from a poset {a1,...as,c |ai<c}, which contains an affirmative answer for the conjecture by Björner, Sagan, Vatter.[5,10]
In algebraic, topological, and geometric combinatorics inequalities among the coefficients of combinatorial polynomials are frequently studied. Recently a notion called the alternatingly increasing property, which is stronger than unimodality, was introduced. In this paper, we relate the alternatingly increasing property to real-rootedness of the symmetric decomposition of a polynomial to develop a systematic approach for proving the alternatingly increasing property for several classes of polynomials. We apply our results to strengthen and generalize real-rootedness, unimodality, and alternat…
Bóna conjectured that the descent polynomials on $(n-2)$-stack sortable permutations have only real zeros. Brändén proved this conjecture by establishing a more general result. In this paper, we give another proof of Brändén's result by using the theory of $s$-Eulerian polynomials recently developed by Savage and Visontai.
As an instance of the B-polynomial, the circuit, or cycle, polynomial P(G(Gamma); w) of the generalized rooted product G(Gamma) of graphs was studied by Farrell and Rosenfeld ({\em Jour. Math. Sci. (India)}, 2000, \textbf{11}(1), 35--47) and Rosenfeld and Diudea ({\em Internet Electron. J. Mol. Des.}, 2002, \textbf{1}(3), 142--156). In both cases, the rooted product G(Gamma) was considered without any restrictions on graphs G and Gamma. Herein, we present a new general result and its corollaries concerning the case when the core graph G is restricted to be bipartite. The last instance of G(Gam…
We investigate a trivariate polynomial associated with rooted trees. It generalises a bivariate polynomial for rooted trees that was recently introduced by Liu. We show that this polynomial satisfies a deletion-contraction recursion and can be expressed as a sum over maximal antichains. Several combinatorial quantities can be obtained as special values, in particular the number of antichains, maximal antichains and cutsets. We prove that two of the three possible bivariate specialisations characterise trees uniquely up to isomorphism. One of these has already been established by Liu, the other…
Real-rootedness of the $Z$-polynomials of sparse paving matroids and strict interlacing for uniform matroids
School of Mathematical Sciences, Tianjin University of Technology（天津理工大学数学科学学院）；College of Mathematical Sciences & Institute of Mathematics and Interdisciplinary Sciences, Tianjin Normal University（天津师范大学数学科学学院与数学与交叉科学研究院）
We define a bivariate polynomial for unlabeled rooted trees and show that the polynomial of an unlabeled rooted tree $T$ is the generating function of a class of subtrees of $T$. We prove that the polynomial is a complete isomorphism invariant for unlabeled rooted trees. Then, we generalize the polynomial to unlabeled unrooted trees and we show that the generalized polynomial is a complete isomorphism invariant for unlabeled unrooted trees.
We prove that the genus polynomials of the graphs called iterated claws are real-rooted. This continues our work directed toward the 25-year-old conjecture that the genus distribution of every graph is log-concave. We have previously established log-concavity for sequences of graphs constructed by iterative vertex-amalgamation or iterative edge-amalgamation of graphs that satisfy a commonly observable condition on their partitioned genus distributions, even though it had been proved previously that iterative amalgamation does not always preserve real-rootedness of the genus polynomial of the i…
Gordon and McMahon defined a two-variable greedoid polynomial $ f(G;t,z) $ for any greedoid $ G $. They studied greedoid polynomials for greedoids associated with rooted graphs and rooted digraphs. They proved that greedoid polynomials of rooted digraphs have the multiplicative direct sum property. In addition, these polynomials are divisible by $ 1 + z $ under certain conditions. We compute the greedoid polynomials for all rooted digraphs up to order six. A greedoid polynomial $ f(D) $ of a rooted digraph $ D $ of order $ n $ GM-factorises if $ f(D) = f(G) \cdot f(H) $ such that $ G $ and $ H…
It is well known that the coefficients of the matching polynomial are unimodal. Unimodality of the coefficients (or their absolute values) of other graph polynomials have been studied as well. One way to prove unimodality is to prove real-rootedness.` Recently I. Beaton and J. Brown (2020) proved the for almost all graphs the coefficients of the domination polynomial form a unimodal sequence, and C. Barton, J. Brown and D. Pike (2020) proved that the forest polynomial (aka acyclic polynomial) is real-rooted iff $G$ is a forest. Let $\mathcal{A}$ be a graph property, and let $a_i(G)$ be the num…
We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomials and whether or not different rooted trees can have the same plucking polynomial.
Polynomials which afford nonnegative, real-rooted symmetric decompositions have been investigated recently in algebraic, enumerative and geometric combinatorics. Brändén and Solus have given sufficient conditions under which the image of a polynomial under a certain operator associated to barycentric subdivision has such a decomposition. This paper gives a new proof of their result which generalizes to subdivision operators in the setting of uniform triangulations of simplicial complexes, introduced by the first named author. Sufficient conditions under which these decompositions are also inte…
In this paper we give a sufficient and necessary condition for two rooted trees with the same plucking polynomial. Furthermore, we give a criteria for a sequence of non-negative integers to be realized as a rooted tree.
Based on the Hermite--Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $\mathbf{s}$-Eulerian polynomials. We also confirm Hyatt's conjectures on the interlacing property of half Eulerian polynomials. Borcea and Brändén's work on the characterization of linear operators preserving Hurwitz stability is critical to this approach.
In this paper, we study the root distribution of some univariate polynomials $W_n(z)$ satisfying a recurrence of order two with linear polynomial coefficients over positive numbers. We discover a sufficient and necessary condition for the overall real-rootedness of all the polynomials, in terms of the polynomial coefficients of the recurrence. Moreover, in the real-rooted case, we find the set of limits of zeros, which turns out to be the union of a closed interval and one or two isolated points; when non-real-rooted polynomial exists, we present a sufficient condition under which every polyno…
In 2009, Brown gave a set of conditions which when satisfied imply that a Feynman integral evaluates to a multiple zeta value. One of these conditions is called reducibility, which loosely says there is an order of integration for the Feynman integral for which Brown's techniques will succeed. Reducibility can be abstracted away from the Feynman integral to just being a condition on two polynomials, the first and second Symanzik polynomials. These polynomials can be defined from graphs, and thus reducibility is a property of graphs. We prove that for a fixed number of external momenta and no m…
We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Toeplitz matrices. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for $E$-polynomials of series-parallel posets and column-strict labelled Ferrers posets.
Athanasiadis conjectured that, for every positive integer $r$, the local $h$-polynomial of the $r$th edgewise subdivision of any simplex has only real zeros. In this paper, based on the theory of interlacing polynomials, we prove that a family of polynomials related to the desired local $h$-polynomial is interlacing and hence confirm Athanasiadis' conjecture.
Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial invariants for graphs such as the well-known Tutte polynomial have been studied for several years, and recently there has been interest to also define such invariants for phylogenetic networks, a special type of graph that arises in the area of evolutionary biology. Recently Liu gave a complete invariant for (phylogenetic) trees. However, the polynomial invariant…
This is an extended abstract of the talk given at the Oberwolfach Workshop "Algebraic Structures in Low-Dimensional Topology", 25 May -- 31 May 2014. My goal was to describe progress in distributive homology from the previous Oberwolfach Workshop June 3 - June 9, 2012, in particular my work on Yang-Baxter homology; however I concentrated my talk on my recent discovery of q-polynomial of a rooted tree; the appropriate topic as my talk was on May 30, 2014, the 30 anniversary of the Jones polynomial, and the polynomial has its roots in the Kauffman bracket approach to the Jones polynomial.
We introduce the concept of piecewise interlacing zeros for studying the relation of root distribution of two polynomials. The concept is pregnant with an idea of confirming the real-rootedness of polynomials in a sequence. Roughly speaking, one constructs a collection of disjoint intervals such that one may show by induction that consecutive polynomials have interlacing zeros over each of the intervals. We confirm the real-rootedness of some polynomials satisfying a recurrence with linear polynomial coefficients. This extends Gross et al.'s work where one of the polynomial coefficients is a c…
We prove a multivariate strengthening of Brenti's result that every root of the Eulerian polynomial of type $B$ is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability-a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and St…
Given a rooted tree $T$ with leaves $v_1,v_2,\ldots,v_n$, we define the ancestral matrix $C(T)$ of $T$ to be the $n \times n$ matrix for which the entry in the $i$-th row, $j$-th column is the level (distance from the root) of the first common ancestor of $v_i$ and $v_j$. We study properties of this matrix, in particular regarding its spectrum: we obtain several upper and lower bounds for the eigenvalues in terms of other tree parameters. We also find a combinatorial interpretation for the coefficients of the characteristic polynomial of $C(T)$, and show that for $d$-ary trees, a specific valu…
We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to readers is that we deal here with an elementary, interesting, new mathematics, and after reading this essay readers can take part in developing the topic, inventing new results and connections to other disciplines of mathematics, and likely, statistical mechanics, and combinatorial biology. We also provide a (free) translation of the paper in Polish.
