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

TL;DR

This paper derives exact formulas for average hitting times in wheel graphs using Fibonacci and Lucas numbers, and relates these to the number of spanning trees with two identified vertices.

Contribution

It introduces a combinatorial approach to compute hitting times and links these to spanning tree counts in wheel graphs with identified vertices.

Findings

Average hitting times expressed via Fibonacci and Lucas numbers

Exact formula for spanning trees with two identified vertices

Connection between hitting times and effective resistance

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.