Unicyclic graphs with five Lapalcian eigenvalues dierent from 0 and 1
Received:September 01, 2020  Revised:November 17, 2020
Key Word: unicyclic graph   Laplacian eigenvalue   multiplicity  
Fund ProjectL:National Natural Science Foundation of China (No.11961041).
Author NameAffiliationAddress
Mengyue Yuan Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China
Fei Wen Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China
Muchun Li Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China
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Abstract:
      Let $U$ be a unicyclic graph of order $n$, and $m_{U}(1)$ the multiplicity of Laplacian eigenvalue $1$ of $U$. It is well-known that $0$ is a simple Lapalcian eigenvalue of connected graph. This means that if $U$ has five Laplacian eigenvalues different from $0$ and $1$, then $m_{U}(1)=n-6$. Recently, it is shown in [9] that $m_{U}(1)\leq n-3$ for any $U$, and the graphs with $m_{U}(1)\in \{n-3, n-4, n-5\}$ are characterized. In this paper, we completely characterize all the unicyclic graphs with $m_{U}(1)=n-6$.
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