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Laplacian eigenvalue distribution and girth of graphs
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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 $m_G(n-g+3,n]\leq n-g$. Moreover, we prove that $m_G(n-g+3,n]= n-g$ if and only if $G\cong C_3$ or $G\cong K_{3,2}$ or $G\cong U_1$, where $U_1$ is obtained from a cycle by joining a single vertex with a vertex of this cycle.
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Girth and Laplacian eigenvalue distribution
For graphs of girth g≥4, at most n-g Laplacian eigenvalues lie in [n-g-k+4,n]; the extremal graphs for k=1,2 and the girth-3 analogues are classified.
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