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The walk matrix of an n-vertex graph G with adjacency matrix A, denoted by $ W(G) $ W (G) , is $ [e,Ae,\ldots,A^{n-1}e] $ [ e , Ae , ... , A n − 1 e ] , where e is the all-ones vector. Let $ G\circ P_m $ G ∘ P m be the rooted product of G and a rooted path $ P_m $ P m (taking an endvertex as the root), i.e. $ G\circ P_m $ G ∘ P m is a graph obtained from G and n copies of $ P_m $ P m by identifying each vertex of G with an endvertex of a copy of $ P_m $ P m . Mao et al. [A new method for constructing graphs determined by their generalized spectrum. Linear Algebra Appl. 2015;477:112–127.] and Mao and Wang [Generalized spectral characterization of rooted product graphs. Linear Multilinear Algebra. 2022. DOI:10.1080/03081087.2022.2098226.] proved that, for m = 2 and $ m\in \{3,4\} $ m ∈ { 3 , 4 } , respectively \[ \det W(G\circ P_m)=\pm a_0^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m, \] det W (G ∘ P m) = ± a 0 ⌊ m 2 ⌋ (det W (G)) m , where $ a_0 $ a 0 is the constant term of the characteristic polynomial of G. Furthermore, in the same paper, Mao and Wang conjectured that the formula holds for any $ m\ge 2 $ m ≥ 2. In this paper, we verify this conjecture using the technique of Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

Proudfoot, Xu, and Young introduced the Z-polynomial for any matroid and conjectured the polynomial has only real roots. Recently, Ferroni, Nasr, and Vecchi introduced the γ-polynomial of a matroid. In this paper, we prove that both the Z-polynomials and γ-polynomials of uniform matroids are ultra log-concave, which due to Newton's inequality partially supports the real-rootedness conjecture. We also give an alternative formula for the γ-polynomials of uniform matroids. As an application, we use this formula to provide a new proof of the γ-positivity of sparse paving matroids. [ABSTRACT FROM AUTHOR]

This is the second paper in a series of two. The goal of the series is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time-algorithm that starts with a 4-precoloring of a graph with no induced six-vertex path and outputs a polynomial-sized collection of so-called excellent precolorings. Excellent precolorings are easier to handle than general ones, and, in addition, in order to determine whether the initial precoloring can be extended to the whole graph, it is enough to answer the same question for each of the excellent precolorings in the collection. The first paper in the series deals with excellent precolorings, thus providing a complete solution to the problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the images of polynomial derivations and their simplicity. We prove that the images of simple Shamsuddin derivations are not Mathieu–Zhao spaces. In addition, we show that the images of some simple derivations in dimension three are not Mathieu-Zhao spaces. Thus, we conjecture that the images of simple derivations in dimension greater than one are not Mathieu–Zhao spaces. We also prove that locally nilpotent derivations are not simple in dimension greater than one. [ABSTRACT FROM AUTHOR]

A long-standing conjecture due to R. Fox states that the coefficients of the Alexander polynomial of an alternating knot exhibit a trapezoidal pattern. In other words, these coefficients increase, stabilize, then decrease in a symmetric way. A stronger version of this conjecture states that these coefficients form a log-concave sequence. This conjecture has been recently highlighted by J. Huh as one of the most interesting problems on log-concavity of sequences. In this expository paper, we shall review the various versions of the conjecture, highlight settled cases and outline some future directions. [ABSTRACT FROM AUTHOR]

In this paper, we give a formula for the number of lattice points in the dilations of Schubert matroid polytopes. As applications, we obtain the Ehrhart polynomials of uniform and minimal matroids as special cases, and give a recursive formula for the Ehrhart polynomials of (a, b)-Catalan matroids. Ferroni showed that uniform and minimal matroids are Ehrhart positive. We show that all sparse paving Schubert matroids are Ehrhart positive and their Ehrhart polynomials are coefficient-wisely bounded by those of minimal and uniform matroids. This confirms a conjecture of Ferroni for the case of sparse paving Schubert matroids. Furthermore, we introduce notched rectangle matroids, which include minimal matroids, sparse paving Schubert matroids and panhandle matroids. We show that three subfamilies of notched rectangle matroids are Ehrhart positive, and conjecture that all notched rectangle matroids are Ehrhart positive. [ABSTRACT FROM AUTHOR]

Finding the sharp estimate of max | z | = 1 | p ′ (z) | in terms of max | z | = 1 | p (z) | for the class of polynomials p(z) satisfying p (z) ≡ z n p (1 / z) has been a well-known open problem for a long time and many papers in this direction have appeared. The earliest result is due to Govil, Jain and Labelle [9] who proved that for polynomials p(z) satisfying p (z) ≡ z n p (1 / z) and having all the zeros either in left half or right half-plane, the inequality max | z | = 1 | p ′ (z) | ≤ n 2 max | z | = 1 | p (z) | holds. A question was posed whether this inequality is sharp. In this paper, we answer this question in the negative by obtaining a bound sharper than n 2 . We also conjecture that for such polynomials max | z | = 1 | p ′ (z) | ≤ ( n 2 - 2 - 1 4 (n - 2)) max | z | = 1 | p (z) | and provide evidence in support of this conjecture. [ABSTRACT FROM AUTHOR]

Partial cubes are the graphs which can be embedded into hypercubes. The cube polynomial of a graph G is a counting polynomial of induced hypercubes of G, which is defined as C(G,x)≔∑i⩾0αi(G)xi, where αi(G) is the number of induced i‐cubes (hypercubes of dimension i) of G. The clique polynomial of G is defined as Cl(G,x)≔∑i⩾0ai(G)xi, where ai(G) (i⩾1) is the number of i‐cliques in G and a0(G)=1. Equivalently, Cl(G,x) is exactly the independence polynomial of the complement G¯ of G. The crossing graphG# of a partial cube G is the graph whose vertices are corresponding to the Θ‐classes of G, and two Θ‐classes are adjacent in G# if and only if they cross in G. In the present paper, we prove that for a partial cube G, C(G,x)⩽Cl(G#,x+1) and the equality holds if and only if G is a median graph. Since every graph can be represented as the crossing graph of a median graph, the above necessary‐and‐sufficient result shows that the study on the cube polynomials of median graphs can be transformed to the one on the clique polynomials of general graphs (equivalently, on the independence polynomials of their complements). In addition, we disprove the conjecture that the cube polynomials of median graphs are unimodal. [ABSTRACT FROM AUTHOR]

Denote by A(p, n, k) the number of commuting p-tuples of permutations on [n] that have exactly k distinct orbits. It was conjectured in Abdesselam (Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations, http://arxiv.org/abs/2309.07358, 2023) that A(p, n, k) is log-concave with respect to k for every p ≥ 2 , n ≥ 3 , and the log-concavity was proved in " p = ∞ " case. In this paper, we prove that for k = n - α , the log-concavity for A(p, n, k) holds for every p ≥ 2 for sufficiently large n. [ABSTRACT FROM AUTHOR]

Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order n is unimodal with the maximum value occurring at ⌊ n 2 ⌋. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter. [ABSTRACT FROM AUTHOR]

For a vertex v of a graph, the local complementation at v is an operation that replaces the neighborhood of v by its complement graph. Two graphs are locally equivalent if one is obtained from the other by a sequence of local complementations. A graph H is a vertex-minor of a graph G if H is an induced subgraph of a graph locally equivalent to G. Although this concept was introduced in the 1980s, it was not widely known and except for the survey paper of Bouchet published in 1990, there is no comprehensive survey listing all the new developments. We survey classic and recent theorems and conjectures on vertex-minors and related concepts such as circle graphs, cut-rank functions, rank-width, and interlace polynomials. [ABSTRACT FROM AUTHOR]

Let I(G; x) denote the independence polynomial of a graph G. Alavi, Malde, Schwenk and Erdős conjectured for any tree T that I(T; x) is unimodal. In this paper, we derive that rooted products of some trees preserve real-rootedness of independence polynomials. In consequence, we obtain more and more trees whose independence polynomials have only real zeros. This in particular implies that their independence polynomials are unimodal. [ABSTRACT FROM AUTHOR]

In this paper we construct new classes of mixed singularities that provide realizations of real algebraic links in the 3-sphere. Especially, we describe this construction in the case of semiholomorphic polynomials, which are mixed polynomials that are holomorphic in one variable. Classifications and characterizations of real algebraic links are still open. These new classes of mixed singularities may help to shed light on the Benedetti–Shiota conjecture, which states that any fibered link on the 3-sphere is a real algebraic link. [ABSTRACT FROM AUTHOR]

In the paper, we study the isotropy groups of K [ x ] if D is a simple derivation. We first study the derivation of the form: D = ∂ x 1 + ∑ i = 2 n (a i x i + b i) ∂ x i with a i , b i ∈ K [ x 1 , ... , x i − 1 ] for all 2 ≤ i ≤ n . We prove that Aut (K [ x ]) D = { i d } if D is simple with deg x i − 1 a i ≥ 1 for all 2 ≤ i ≤ n or a i ∈ K * for all 3 ≤ i ≤ n or 0 ≤ deg x j b i ≤ deg x j a i for all 2 ≤ j ≤ i − 1 , 3 ≤ i ≤ n . Thus, we conjecture that Aut (K [ x ]) D = { i d } if D is simple with ∏ i = 2 n a i ≠ 0 , a i , b i ∈ K [ x 1 , ... , x i − 1 ] for all 2 ≤ i ≤ n . Then we prove that Aut (K [ x ]) D = { (x 1 , ... , x n − 1 , x n + c) | c ∈ K } if D = (x 1 s x 2 + g) ∂ x 1 + x 1 s − 1 ∂ x 2 + g 3 ∂ x 3 + ⋯ + g n ∂ x n or D = (x 1 s x 2 t + c) ∂ x 1 + x 1 r ∂ x 2 + g 3 ∂ x 3 + ⋯ + g n ∂ x n with g ∈ K [ x 1 ] , deg (g) ≤ s , g (0) ≠ 0 and g i ∈ K [ x i − 1 ] \ K for all 3 ≤ i ≤ n . [ABSTRACT FROM AUTHOR]

Let (Zk)k⩾1$(Z_k)_{k\geqslant 1}$ be a sequence of independent and identically distributed complex random variables with common distribution μ$\mu$ and let Pn(X):=∏k=1n(X−Zk)$P_n(X):=\prod _{k=1}^n (X-Z_k)$ be the associated random polynomial in C[X]$\mathbb {C}[X]$. Kabluchko established the conjecture stated by Pemantle and Rivin that the empirical measure νn$\nu _n$ associated with the critical points of Pn$P_n$ converges weakly in probability to the base measure μ$\mu$. In this note, we establish that the convergence, in fact, holds in the almost sure sense. Our result positively answers a question raised by Kabluchko and formalized as a conjecture in the recent paper (Michelen and Vu [arXiv:2212.11867]). [ABSTRACT FROM AUTHOR]

Let F be any field of characteristic different from two and let E be the Grassmann algebra of an infinite dimensional F -vector space L. In this paper we will provide a condition for a Z 2 -grading on E to behave like the natural Z 2 -grading E c a n. More specifically, our aim is to prove the validity of a weak version of a conjecture presented in [10]. The conjecture poses that every Z 2 -grading on E has at least one non-zero homogeneous element of L. As a consequence, we obtain a characterization of E c a n by means of its Z 2 -graded polynomial identities. Furthermore we construct a Z 2 -grading on E that gives a negative answer to the conjecture. [ABSTRACT FROM AUTHOR]

We prove two conjectures involving permutation polynomials in a paper of Dmytrenko, Lazebnik, Williford, in a low degree regime, using the theory of algebraic curves over finite fields. More precisely, we prove that Conjecture A holds whenever q ≥ max ⁡ { (2 k − 1) 2 + 1 , 1.823 (4 k 2 − 14 k + 12) } , whereas Conjecture B holds if q ≥ 2.233 (9 k 2 − 21 k + 12). Although one of these conjectures was already proved by Hou without any restriction on the degree of the polynomials, we consider the proof contained in this paper is more direct and less computational. [ABSTRACT FROM AUTHOR]

We conjecture a formula for the rational q, t-Catalan polynomial C r / s that is symmetric in q and t by definition. The conjecture posits that C r / s can be written in terms of symmetric monomial strings indexed by maximal Dyck paths. We show that for any finite d ∗ , giving a combinatorial proof of our conjecture on the infinite set of functions { C r / s d : r ≡ 1 mod s , d ≤ d ∗ } is equivalent to a finite counting problem. [ABSTRACT FROM AUTHOR]

A hypertree is a connected hypergraph without cycles. Further a hypertree is called an r -tree if, additionally, it is r -uniform. Note that 2-trees are just ordinary trees. A classical result states that for any 2-tree T with characteristic polynomial ϕ T (λ) and matching polynomial φ T (λ) , then ϕ T (λ) = φ T (λ). More generally, suppose T is an r -tree of size m with r ≥ 2. In this paper, we extend the above classical relationship to r -trees and establish that ϕ T (λ) = ∏ H ⊑ T φ H (λ) a H , where the product is over all connected subgraphs H of T , and the exponent a H of the factor φ H (λ) can be written as a H = b m − e (H) − | ∂ (H) | c e (H) (b − c) | ∂ (H) | , where e (H) is the size of H , ∂ (H) is the boundary of H , and b = (r − 1) r − 1 , c = r r − 2. In particular, for r = 2 , the above correspondence reduces to the classical result for ordinary trees. In addition, we resolve a conjecture by Clark and Cooper (2018) [7] and show that for any subgraph H of an r -tree T with r ≥ 3 , φ H (λ) divides ϕ T (λ) , and additionally ϕ H (λ) divides ϕ T (λ) , if either r ≥ 4 or H is connected when r = 3. Moreover, a counterexample is given for the case when H is a disconnected subgraph of a 3-tree. [ABSTRACT FROM AUTHOR]

We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave. [ABSTRACT FROM AUTHOR]

From Carlitz's identity, we deduce two new q-supercongruences modulo the square of a cyclotomic polynomial, which were originally conjectured by Guo. These results establish new q-analogues of a supercongruence of Sun. [ABSTRACT FROM AUTHOR]

In this work we review and derive some elementary properties of the discrete renewal sequences based on a positive, finite and integer-valued random variable. Our results consider these sequences as dependent on the probability masses of the underlying random variable. In particular we study the minima and the maxima of these sequences and prove that they are attained for indices of the sequences smaller or equal than the support of the underlying random variable. Noting that the minimum itself is a minimum of multi-variate polynomials we conjecture that one universal polynomial envelopes the minimum from below and that it is maximal in some sense and largest in another. We prove this conjecture in a special case. [ABSTRACT FROM AUTHOR]

Avila's Almost Reducibility Conjecture (ARC) is a powerful statement linking purely analytic and dynamical properties of analytic one-frequency cocycles. It is also a fundamental tool in the study of spectral theory of analytic one-frequency Schrödinger operators, with many striking consequences, allowing to give a detailed characterization of the subcritical region. Here we give a proof, completely different from Avila's, for the important case of Schrödinger cocycles with trigonometric polynomial potentials and non-exponentially approximated frequencies, allowing, in particular, to obtain all the desired spectral consequences in this case. [ABSTRACT FROM AUTHOR]

Let f_t(z)=z^2+t. For any z\in \mathbb {Q}, let S_z be the collection of t\in \mathbb {Q} such that z is preperiodic for f_t. In this article, assuming a well-known conjecture of Flynn, Poonen, and Schaefer [Duke Math. J. 90 (1997), pp. 435–463], we prove a uniform result regarding the size of S_z over z\in \mathbb {Q}. In order to prove it, we need to determine the set of rational points on a specific non-hyperelliptic curve C of genus 4 defined over \mathbb {Q}. We use Chabauty's method, which requires us to determine the Mordell-Weil rank of the Jacobian J of C. We give two proofs that the rank is 1: an analytic proof, which is conditional on the BSD rank conjecture for J and some standard conjectures on L-series, and an algebraic proof, which is unconditional, but relies on the computation of the class groups of two number fields of degree 12 and degree 24, respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for J. [ABSTRACT FROM AUTHOR]

We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3^{[2]} deformation type. This proves in particular a conjecture of O'Grady stating that hyper-Kähler fourfolds of K3^{[2]} numerical type are of K3^{[2]} deformation type. Our topological assumption concerns the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions. There are two main ingredients in the proof. We first prove a topological version of the statement, by showing that our topological assumption forces the Betti numbers, the Fujiki constant, and the Huybrechts–Riemann–Roch polynomial of the hyper-Kähler fourfold to be the same as those of K3^{[2]} hyper-Kähler fourfolds. The key part of the article is then to prove the hyper-Kähler SYZ conjecture for hyper-Kähler fourfolds for divisor classes satisfying the numerical condition mentioned above. [ABSTRACT FROM AUTHOR]