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On the average hitting times of the directed wheel

Shunya Tamura

TL;DR

This work determines explicit average hitting times for a directed wheel $W_N^D$ with an absorbing center, deriving formulas that connect hitting times to Fibonacci numbers for odd $N$ and to Lucas numbers for even $N$. The authors formulate the problem via the Laplacian and a symmetry-reduced linear system, and solve it with elementary combinatorial methods, avoiding spectral theory. They also compute the number of spanning trees using Kirchhoff's Matrix-Tree Theorem, finding inward-rooted counts $L_{2N}-2$ and outward-rooted counts $N^2$ (except at the absorbing vertex). The results highlight a deep link between random-walk metrics on symmetric graphs and classical integer sequences, with potential connections to electrical-network analogies.

Abstract

In this paper, following the paper ``On the average hitting times of the squares of cycles,'' we provide an explicit formula for the average hitting times of a simple random walk on a directed graph with $N$ vertices, where the graph consists of a cycle with a single absorbing vertex at its center, using elementary methods. Also, we show that the average hitting times can be expressed in terms of the Fibonacci and Lucas numbers in general.

On the average hitting times of the directed wheel

TL;DR

This work determines explicit average hitting times for a directed wheel with an absorbing center, deriving formulas that connect hitting times to Fibonacci numbers for odd and to Lucas numbers for even . The authors formulate the problem via the Laplacian and a symmetry-reduced linear system, and solve it with elementary combinatorial methods, avoiding spectral theory. They also compute the number of spanning trees using Kirchhoff's Matrix-Tree Theorem, finding inward-rooted counts and outward-rooted counts (except at the absorbing vertex). The results highlight a deep link between random-walk metrics on symmetric graphs and classical integer sequences, with potential connections to electrical-network analogies.

Abstract

In this paper, following the paper ``On the average hitting times of the squares of cycles,'' we provide an explicit formula for the average hitting times of a simple random walk on a directed graph with vertices, where the graph consists of a cycle with a single absorbing vertex at its center, using elementary methods. Also, we show that the average hitting times can be expressed in terms of the Fibonacci and Lucas numbers in general.
Paper Structure (8 sections, 6 theorems, 18 equations)

This paper contains 8 sections, 6 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Fibonacci and Lucas numbers
  3. Our model
  4. Proof of Theorem 1
  5. Odd number of vertices
  6. Even number of vertices
  7. Spanning tree of $W_N^D$
  8. Conclusion

Key Result

Theorem 1

The exact formula for the average hitting times of a simple random walk on the graph $W_{N}^{D}$ (where $N \geq 3$) is as follows: where $F_{i}$ denotes the $i$-th Fibonacci number and $L_{i}$ denotes the $i$-th Lucas number.

Theorems & Definitions (6)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Theorem 2: Kirchhoff Kir, Tatte Tatte
  • Proposition 3
  • Proposition 4