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arXiv · 2008.03144

Quartic Graphs with Minimum Spectral Gap

Maryam Abdi·Ebrahim Ghorbani
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Abstract

Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) \frac{3n^2}{2\pi^2}$. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected $k$-regular graph on $n$ vertices is at least $(1+o(1))\frac{2k\pi^2}{3n^2}$, and the bound is attained for at least one value of $k$. We determine the structure of connected quartic graphs on $n$ vertices with minimum spectral gap which enable us to show that the minimum spectral gap of connected quartic graphs on $n$ vertices is $(1+o(1))\frac{4\pi^2}{n^2}$. From this result, the Aldous--Fill conjecture follows for $k=4$.

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Maryam Abdi, Ebrahim Ghorbani. 2020-08-06. Quartic Graphs with Minimum Spectral Gap. https://arxiv.org/abs/2008.03144

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