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On average hitting time and Kemeny's constant for weighted trees

Ji Zeng

Abstract

For a connected graph $G$, the average hitting time $α(G)$ and the Kemeny's constant $κ(G)$ are two similar quantities, both measuring the time for the random walk on $G$ to travel between two randomly chosen vertices. We prove that, among all weighted trees whose edge weights form a fixed multiset, $α$ is maximized by a special type of ``polarized'' paths and is minimized by a unique weighted star graph. We also obtain a similar characterization of the $κ$-maximizing and $κ$-minimizing elements among such a collection of weighted trees. Our proofs are based on the forest formulas for $α$ and $κ$.

On average hitting time and Kemeny's constant for weighted trees

Abstract

For a connected graph , the average hitting time and the Kemeny's constant are two similar quantities, both measuring the time for the random walk on to travel between two randomly chosen vertices. We prove that, among all weighted trees whose edge weights form a fixed multiset, is maximized by a special type of ``polarized'' paths and is minimized by a unique weighted star graph. We also obtain a similar characterization of the -maximizing and -minimizing elements among such a collection of weighted trees. Our proofs are based on the forest formulas for and .

Paper Structure

This paper contains 5 sections, 9 theorems, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['minmax_aht']}
  4. Proof of Theorem \ref{['minmax_kemeny']}
  5. Remarks

Key Result

Theorem 1.1

Among all unweighted trees of a fixed size, the path has the largest $\alpha$ and $\kappa$, and the star graph has the smallest $\alpha$ and $\kappa$. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: From left to right: a weighted tree in $\mathcal{T}_W$ where $W=\{7,5,4,2,2,1\}$ and the numbers indicate edge weights; the weighted star $S_W$; a polarized path in $\mathcal{T}_W$ where $c(e_1)=c(e_6)=1$, $c(e_2)=c(e_5)=2$, and $c(e_3)=c(e_4)=3$.
  • Figure 2: $T$ edge-transfers to $T'$ with respect to size.
  • Figure 3: $T\setminus e$ and $T'\setminus e$ give the same component partition of vertices.
  • Figure 4: Hasse diagram of the partial order $\succeq$ on unweighted trees of size $7$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['minmax_aht']}
  • ...and 5 more