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Keywords:
unicyclic graph; Laplacian eigenvalue; multiplicity; bound
unicyclic graph; Laplacian eigenvalue; multiplicity; bound
Summary:
Let $G$ be a connected graph of order $n$ and $U$ a unicyclic graph with the same order. We firstly give a sharp bound for $m_{G}(\mu )$, the multiplicity of a Laplacian eigenvalue $\mu $ of $G$. As a straightforward result, $m_{U}(1)\le n-2$. We then provide two graph operations (i.e., grafting and shifting) on graph $G$ for which the value of $m_{G}(1)$ is nondecreasing. As applications, we get the distribution of $m_{U}(1)$ for unicyclic graphs on $n$ vertices. Moreover, for the two largest possible values of $m_{U}(1)\in \{n-5,n-3\}$, the corresponding graphs $U$ are completely determined.
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