2012/05/17 by James R. Lee, Lee, James R., Teng Qin +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1205.3980
arxiv created 2012/05/17 · openalex publication_date 2012/05/17 · arxiv updated 2012/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present an infinite family of finite planar graphs \Xn\ with degree at most five and such that for some constant c > 0, λ1(Xn) ≥ c((log \diam(Xn))/(\diam(Xn)))2 , where λ1 denotes the smallest non-zero eigenvalue of the graph Laplacian. This significantly simplifies a construction of Louder and Souto. We also remark that such a lower bound cannot hold when the diameter is replaced by the average squared distance: There exists a constant c > 0 such that for any family \Xn\ of planar graphs we have λ1(Xn) ≤ c ((1)/(|Xn|2) ∑x,y ∈ Xn d(x,y)2)-1 , where d denotes the path metric on Xn.