Laplacian eigenvalue distribution and girth of graphs

2025/04/22 by Zhen, Wenhao, Wong, Dein, Xu, Songnian · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.15772

Abstract

Let G be a connected graph on n vertices with girth g. Let m_GI denote the number of Laplacian eigenvalues of graph G in an interval I. In this paper, we show that if G is not a cycle, then m_G(n-g+3,n]≤ n-g. Moreover, we prove that m_G(n-g+3,n]= n-g if and only if G≅ C₃ or G≅ K_3,2 or G≅ U₁, where U₁ is obtained from a cycle by joining a single vertex with a vertex of this cycle.

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