[Paper Review] On the center of the Hurwitz graph

Yuval Hovannes Khachatryan-Raziel|arXiv (Cornell University)|Aug 11, 2015

Advanced Combinatorial Mathematics8 references3 citations

TL;DR

This paper investigates the center of the Hurwitz graph, a Cayley graph on maximal chains in the non-crossing partition lattice, by establishing a connection to geometric tree graphs. It proves that all stars (caterpillar trees with one central vertex) are central vertices in the extended Hurwitz graph, with radius exactly $n-2$, and provides bounds on the diameter of the extended graph as $[\frac{3}{2}n - 5, 2n - 4]$.

ABSTRACT

For any natural number n, the Hurwitz graph $\mathcal{H}(S_n )$ has vertices corresponding to reduced words of the cycle $(1 . . . n)$ in terms of transpositions in $S_n$, and has edges corresponding to local shifts. This graph is known to be connected, and some of its graph theoretic properties, such as its radius and upper bounds on its diameter, are known. We explore further the structure of the Hurwitz Graph, in particular its center and its connection to the geometric tree graph $G_n$. Finally, the extended Hurwitz Graph is defined and its properties such as the radius, characterization of some central elements and bounds on the diameter are obtained. Part of fulfillment of the requirements for the Master's Degree.

Motivation & Objective

• To identify central vertices in the Hurwitz graph, particularly in the extended version $\mathcal{E}(S_n)$, by linking it to geometric tree graphs.
• To determine the radius and diameter of the extended Hurwitz graph $\mathcal{E}(S_n)$, providing tight bounds.
• To establish a correspondence between factorizations of the $n$-cycle and non-crossing geometric trees via the geometric tree map $\Gamma$.
• To prove that all star configurations (e.g., $\mathcal{S}_1 = (1\,n)\dots(1\,2)$) are central vertices in $\mathcal{E}(S_n)$ using inductive arguments on word distance.
• To leverage known results on geometric tree graph diameter to derive lower bounds for the extended Hurwitz graph.

Proposed method

• Define the Hurwitz graph $\mathcal{H}(S_n)$ as the Cayley graph on maximal chains in the non-crossing partition lattice $NC(n)$, with edges induced by right and left Hurwitz moves $R_i$ and $L_i$.
• Introduce the geometric tree map $\Gamma$ that maps words in transpositions to non-crossing geometric trees in the plane, preserving edge structure.
• Use the partial product construction $\sigma_j = t_j \dots t_{n-1}$ to define a map $\Phi: F_n \to S_{n-1}$, which assigns a permutation to each word based on position of value decreases.
• Define a linear order $<_{i}$ on edges of a geometric tree centered at vertex $i$, used to compare words and prove centrality of stars.
• Apply induction on the number of differing transpositions between a word $w$ and a star $\mathcal{S}_i$, using Lemma 5.2.3 to perform 'pull' operations that reduce distance.
• Leverage the surjectivity of $\Gamma$ and the fact that adjacent vertices in $\mathcal{E}(S_n)$ map to adjacent or equal trees in $\mathcal{G}_n$ to relate distances in $\mathcal{E}(S_n)$ to those in $\mathcal{G}_n$.

Experimental results

Research questions

• RQ1 Which vertices in the extended Hurwitz graph $\mathcal{E}(S_n)$ are central, i.e., minimize eccentricity?
• RQ2 What is the exact radius of the extended Hurwitz graph $\mathcal{E}(S_n)$?
• RQ3 How does the diameter of $\mathcal{E}(S_n)$ relate to the diameter of the geometric tree graph $\mathcal{G}_n$?
• RQ4 Can the centrality of star configurations in $\mathcal{E}(S_n)$ be proven via inductive distance reduction using Hurwitz moves?
• RQ5 To what extent do the properties of geometric tree graphs (e.g., diameter bounds) transfer to the extended Hurwitz graph?

Key findings

• All star configurations in $\mathcal{E}(S_n)$, such as $\mathcal{S}_1 = (1\,n)\dots(1\,2)$, are central vertices, meaning they achieve the minimal eccentricity.
• The radius of the extended Hurwitz graph $\mathcal{E}(S_n)$ is exactly $n-2$, as shown by proving that every word is within distance $n-2$ of any star and that this bound is tight.
• The diameter of $\mathcal{E}(S_n)$ is bounded below by $\frac{3}{2}n - 5$ and above by $2n - 4$, with the lower bound derived from the known diameter of $\mathcal{G}_n$.
• The geometric tree map $\Gamma: F_n \to \mathcal{G}_n$ is surjective, meaning every non-crossing geometric tree arises as $\Gamma(w)$ for some $w \in F_n$.
• Adjacent vertices in $\mathcal{E}(S_n)$ map to either adjacent or equal trees in $\mathcal{G}_n$, preserving the graph structure under $\Gamma$.
• The proof of centrality relies on inductive reduction of word distance using 'pull' operations from Lemma 5.2.3, which replace a transposition not in the star with one that is, maintaining proximity to the star.