Cheng Yeaw Ku, K. B. Wong
We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it.
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We show that a monic univariate polynomial over a field of characteristic zero, with $k$ distinct non-zero known roots, is determined by its $k$ proper leading coefficients by providing an explicit algorithm for computing the multiplicities of each root. We provide a version of the result and accompanying algorithm when the field is not algebraically closed by considering the minimal polynomials of the roots. Furthermore, we show how to perform the aforementioned algorithm in a numerically stable manner over $\mathbb{C}$, and then apply it to obtain new characteristic polynomials of hypergraph…
On the $q$-multiplicity of sums of distinct simple roots of $\mathfrak{sl}_{r+1}(\mathbb{C})$
本文研究了sl_{r+1}(C)不同简单根之和的q-乘法性，通过确定Weyl交替集，利用Fibonacci数列的乘积公式计算权重的q-乘法性。
On the multiplicity of matching polynomial roots and $θ$-critical graphs
本文研究匹配多项式非零根的最大重数及达到该界的图，将其推广至任意 $θ$ 并联系 $θ$-临界图，构造性回答 Godsil 问题，同时证明各阶 $1$-临界树与图的存在性。
Godsil observed the simple fact that the multiplicity of 0 as a root of the matching polynomial of a graph coincides with the classical notion of deficiency. From this fact he asked to what extent classical results in matching theory generalize, replacing "deficiency" with multiplicity of $θ$ as a root of the matching polynomial. We prove an analogue of the Stability Lemma for any given root, which describes how the matching structure of a graph changes upon deletion of a single vertex. An analogue of Gallai's Lemma follows. Together these two results imply an analogue of the Gallai-Edmonds St…
Given a bivariate system of polynomial equations with fixed support sets $A, B$ it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity $i$ for all $i$ in the range $\{0,1,...,|A|-|\mbox{conv}(A)\ominus B|-1\}$, where $A\ominus B$ is the set of all integral vectors that shift B to a subset of $A$. As an application of this result we classify all pairs $(A, B)$ such that the system supported at $(A, B)$ does not have a solution of multiplicity $3$.
Computing the $q$-Multiplicity of the Positive Roots of $\mathfrak{sl}_{r+1}(\mathbb{C})$ and Products of Fibonacci Numbers
本研究利用Kostant权重重数公式，证明了$\mathfrak{sl}_{r+1}(\mathbb{C})$伴随表示中正根重数的非零贡献项数为斐波那契数乘积，并推导出其$q$-重数表达式，验证了$q=1$时的已知结论。
In this paper we complete the proof of the X=K conjecture, that for every family of nonexceptional affine algebras, the graded multiplicities of tensor products of symmetric power Kirillov-Reshetikhin modules known as one-dimensional sums, have a large rank stable limit X that has a simple expression (called the K-polynomial) as nonnegative integer combination of Kostka-Foulkes polynomials. We consider a subfamily of Lusztig's q-analogues of weight multiplicity which we call stable KL polynomials and denote by KL. We give a type-independent proof that K=KL. This proves that X=KL: the family of…
If for any k the k-th coefficient of a polynomial I(G;x)is equal to the number of stable sets of cardinality k in graph G, then it is called the independence polynomial of G (Gutman and Harary, 1983). A graph G is very well-covered (Favaron, 1982) if it has no isolated vertices, its order equals 2*alpha(G), where alpha(G) is the size of a maximum stable set, and it is well-covered (i.e., all its maximal independent sets are of the same size, Plummer, 1970). For instance, appending a single pendant edge to each vertex of G yields a very well-covered graph, which we denote by G*. Under certain c…
Characterized are all simple undirected graphs $G$ such that any real symmetric matrix that has graph $G$ has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general fields), but only certain partial 2-trees guarantee maximum multiplicity 2. Among partial linear 2-trees, they are only those whose vertices can be covered by two "parallel" induced paths. The remaining graphs that guarantee maximum multiplicity 2 are comprised by certain identified families of "exceptional" partial 2-trees that are not linear.
On higher multiplicity hyperplane and polynomial covers for symmetry preserving subsets of the hypercube
本文研究超立方体保对称子集的高重数超平面覆盖问题，证明求解对应的多项式覆盖问题即可获得紧致界，并指出该方法在处理更广的块对称集时已达极限。
Covering Hypercube $mB^n$
本文研究了覆盖超立方体 $mB^n$ 中所有点至少 $k$ 次的超平面最小数量，推导了多项式消去条件的下界，并给出了 $k=1,2$ 时的精确值及 $k \ge 3$ 时的上下界。
Let G = (V;E) be a simple graph. We consider domination polynomial, matching polynomial and edge cover polynomial of G. Graphs which their polynomials have few roots can give sometimes a very surprising information about the structure of the graph. In this paper we study graphs which their domination polynomial, independence polynomial and edge cover polynomial have few roots.
We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed formula for certain root multiplicities. Using this connection we give a second interpretation, namely that the root multiplicity of a given root coincides with the number of acyclic orientations with a unique sink of a certain graph (depending on the root). Finally, using the combinatorics of Lyndon words we construct a basis for the root spaces correspondin…
Face covers and rooted minors in bounded genus graphs
本文研究有界亏格曲面上3-连通有根图的面覆盖问题，在无有根K_{2,t}子式且面宽足够大的条件下给出大小受g和t函数控制的面覆盖，并在平面情形得到无条件O(t^4)上界，改进已有结果。
We introduce two polynomials (in $q$) associated with a finite poset $P$ that encode some information on the covering relation in $P$. If $P$ is a distributive lattice, and hence $P$ is isomorphic to the poset of dual order ideals in a poset $L$, then these polynomials coincide and the coefficient of $q$ equals the number of $k$-element antichains in $L$. In general, these two covering polynomials are different, and we introduce a deviation polynomial of $P$, which measures the difference between these two. We then compute all these polynomials in the case, where $P$ is one of the posets assoc…
Construction of commuting difference operators for multiplicity free spaces
本文研究了满足特定公理的根系与主导权重基，构造了能同时对角化的差分算子，并推导出Pieri型规则，扩展了移位Jack多项式结果。
Combinatorics of higher-dimensional tropical covers
本研究构建组合框架研究高维热带覆盖类多面映射，证明平衡条件与带指标分支覆盖的等价性，实现连通性提升并解决重数扩展问题，给出强连通性假设缺失时的反例。
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