Jeremy L. Martin, Victor Reiner
We give factorizations for weighted spanning tree enumerators of Cartesian products of complete graphs, keeping track of fine weights related to degree sequences and edge directions. Our methods combine Kirchhoff's Matrix-Tree Theorem with the technique of identification of factors.
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On stability of spanning tree degree enumerators
研究了连通图生成树度数枚举多项式的实稳定性，证明其充要条件为该图是距离遗传图。
On stability of weighted spanning tree degree enumerators
本文将生成树度枚举器的实稳定性结果从连通图推广到加权图，并据此定义了加权距离遗传图类。
Trees, Parking Functions and Factorizations of Full Cycles
本文扩展了停车函数、标号树与全循环分解之间的对应关系，引入了树逆序计数器的二元细化并证明其与分解计数器匹配，同时刻画了使Stanley双射成立的全部全循环，并建立了与Haglund弹跳统计量的联系。
The weighted spanning tree enumerator of a graph $G$ with weighted edges is the sum of the products of edge weights over all the spanning trees in $G$. In the special case that all of the edge weights equal $1$, the weighted spanning tree enumerator counts the number of spanning trees in $G$. The Weighted Matrix-Tree Theorem asserts that the weighted spanning tree enumerator can be calculated from the determinant of a reduced weighted Laplacian matrix of $G$. That determinant, however, is not always easy to compute. In this paper, we show how two well-known results from linear algebra, the Mat…
We give an interpretation of the coefficients of the two variable refinement $D_{\Sh_n}(q,t)$ of the distance enumerator of the Shi hyperplane arrangement $\Sh_n$ in $n$ dimensions. This two variable refinement was defined by Stanley \cite{stan-rota} for the general $r$-extended Shi hyperplane arrangements. We give an interpretation when $r=1$. We define three natural three-dimensional partitions of the number $(n+1)^{n-1}$. The first arises from parking functions of length $n$, the second from special posets on $n$ vertices defined by Athanasiadis and the third from spanning trees on $n+1$ ve…
Kreweras proved that the reversed sum enumerator for parking functions of length $n$ is equal to the inversion enumerator for labeled trees on $n+1$ vertices. Recently, Perkinson, Yang, and Yu gave a bijective proof of this equality that moreover generalizes to graphical parking functions. Using a depth-first search variant of Dhar's burning algorithm they proved that the codegree enumerator for $G$-parking functions equals the $κ$-number enumerator for spanning trees of $G$. The $κ$-number is a kind of generalized tree inversion number originally defined by Gessel. We extend the work of Perki…
We prove the following sharp estimate for the number of spanning trees of a graph in terms of its vertex-degrees: a simple graph $G$ on $n$ vertices has at most $(1/n^{2}) \prod_{v \in V(G)} (d(v)+1)$ spanning trees. This result is tight (for complete graphs), and improves earlier estimates of Alon from 1990 and Kostochka from 1995 by a factor of about $1/n$ (for dense graphs). We additionally show that an analogous bound holds for the weighted spanning tree enumerator of a (nonnegatively) weighted graph as well.
We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifte…
Networks with a high degree of symmetry are useful models for parallel processor networks. In earlier papers, we defined several global communication tasks (universal exchange, universal broadcast, universal summation) that can be critical tasks when complex algorithms are mapped to parallel machines. We showed that utilizing the symmetry can make network optimization a tractable problem. In particular, we showed that Cayley graphs have the desirable property that certain routing schemes starting from a single node can be transferred to all nodes in a way that does not introduce conflicts. In …
Spanning Factorizations in Vertex-Transitive Digraphs of Degree 2
针对出度为d的顶点传递有向图中生成1-因子分解的存在性问题，本文聚焦d=2的情况，利用交替环和块系结构提出块/阶段框架，得到同时包含两条出边的充分条件，证明块障碍的影响及强传递集的存在性，并给出可行块大小约束等结果。
A $1-$factorization of a complete graph on $2n$ vertices is said to be $G-$regular if it posseses an automorphism group $G$ acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (1985) on cyclic groups and, when $n$ is even, the problem is still open. An attempt to obtain a fairly precise description of groups and $1-$factorizations satisfying this symmetry constrain can be done by imposing further conditions. It was recently proved, see Rinaldi (2021) and Mazzuoccolo et al. (2019), that a …
The number of trees in distance-hereditary graphs and their friends
本文统一证明并推广了2020年代关于距离遗传图生成树计数的若干结果，通过研究顶点复制操作下度枚举子的行为，并证明了Ehrenborg猜想与其多项式形式的等价性。
Simplicial effective resistance and enumeration of spanning trees
本研究将图与电网的类比拓展至高维单纯复形，利用单纯有效电阻与加权树计数式的比值，证明色移复形新计数公式并证实相关猜想，同时重新推导移位复形的已知计数公式，为高维组合计数提供了电学类比的新方法。
All-to-all Routing on Digraph Networks
本文研究顶点传递有向图是否总存在生成1-因子分解这一开放问题，并揭示正则有向图中顶点传递性、消去性质与生成因子分解类型之间的内在联系。
Minimal transitive factorizations supported on quasi-threshold graphs
本文研究拟阈值图上单位置换的最小传递分解数，证明其可被星形图分解数整除，并给出基于带权生成树的组合公式。
For any connected multigraph $G=(V,E)$ and any $M\subseteq E$, if $M$ induces an acyclic subgraph of $G$ and removing all edges in $M$ yields a subgraph of $G$ whose components are complete graphs, a formula for $τ_G(M)$ is obtained, where $τ_G(M)$ is the number of spanning trees in $G$ which contain all edges in $M$. Applying this result, we can easily obtain a formula for the number of spanning trees in the line graph or the middle graph of an arbitrary graph. Applying this result, we also show that for any connected graph $G$ with a clique $U$ which is a cut-set of $G$, the number of spanni…
If $G$ is a strongly connected finite directed graph, the set $\mathcal{T}G$ of rooted directed spanning trees of $G$ is naturally equipped with a structure of directed graph: there is a directed edge from any spanning tree to any other obtained by adding an outgoing edge at its root vertex and deleting the outgoing edge of the endpoint. Any Schrödinger operator on $G$, for example the Laplacian, can be lifted canonically to $\mathcal{T}G$. We show that the determinant of such a lifted Schrödinger operator admits a remarkable factorization into a product of determinants of the restrictions of …
A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G(1), \ldots , G(k)$. It is called an $r$-factorization if every $G(i)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a decomposition of $G$ is said to be enclosed in a decomposition of $H$ if, for every $1 \leq i \leq k$, $G(i)$ is a subgraph of $H(i)$. Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of $λK_n$ to be enclosed in some $2$-edge-connected $r$-factorization of $μK_{m}$ for some range of values for the parameters $n$, $m$, $λ$, $μ$, $r$: $r=2$, $μ>λ$ an…
We introduce a new method for showing that the roots of the characteristic polynomial of certain finite lattices are all nonnegative integers. This method is based on the notion of a quotient of a poset which will be developed to explain this factorization. Our main theorem will give two simple conditions under which the characteristic polynomial factors with nonnegative integer roots. We will see that Stanley's Supersolvability Theorem is a corollary of this result. Additionally, we will prove a theorem which gives three conditions equivalent to factorization. To our knowledge, all other theo…
Eigenvalues and factors: a survey
本文从特征值角度综述图的因子相关研究，涵盖历史综述、刻画方法及最新进展。
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