Number of spanning trees in a wheel graph with two identified vertices via hitting times

TL;DR

This paper addresses the exact average hitting-time problem for simple random walks on the wheel graph $W_{N+1}$, revealing parity-driven expressions in terms of Fibonacci numbers $F_n$ (when $N$ is odd) or Lucas numbers $L_n$ (when $N$ is even). A symmetry-reduced linear-system approach, avoiding spectral methods, yields closed-form formulas for $h(W_{N+1};0, ext{l})$ and $h(W_{N+1};N,0)$. By pairing these hitting-time results with Nash–Williams’ relation between effective resistance and hitting times and applying Kirchhoff’s matrix-tree theorem, the paper also provides exact counts of spanning trees after identifying two vertices, $ au(W_{N+1};0, ext{l})$ and $ au(W_{N+1};N,0)$, in terms of $F_n$ or $L_n$. Collectively, the results illuminate deep connections between random-walk metrics on symmetric graphs and classic integer sequences, with implications for network robustness and design, and point to natural extensions to more complex inner-vertex configurations.

Abstract

In this paper, we provide an exact formula for the average hitting times in a wheel graph $W_{N+1}$ using a combinatorial approach. For this wheel graph, the average hitting times can be expressed using Fibonacci numbers when the number of surrounding vertices is odd and Lucas numbers when it is even. Furthermore, combining the exact formula for the average hitting times with the general formula for the effective resistance of the graph allows determination of the number of spanning trees of the graph with two identified vertices.

Theorems & Definitions (14)

• Theorem 1
• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 2: Nash-Williams Nash
• Lemma 1
• proof
• Lemma 2
• proof