摘要:
对于任一自然数$b$,假设方程$b\mu(\mu-2)\!-\!(\mu-1)^2(\mu-3)\!=\!0$的第二大特征根分别为$l_G(b)$;假设方程$b\mu(\mu-2)\!-\!(\mu-1)^2(\mu-3)\!-\!(\mu-1)(\mu-2)\!=\!0$的第二大特征根分别为$l_T(b)$.\,本文首先证明了存在图序列$\{G_{n,b}\}$和$\{T_{n,b}\}$,其第三大拉普拉斯特征值的极限点分别为$l_G(b)$和$l_T(b)$,$(b\!=\!0,1,\cdots)$. 其次, 本文证明了$l_G(b)$,$l_T(b)$及$2$是第三大拉普拉斯特征值的所有小于等于$2$极限点
Abstract:
For a different parameter $b$, let $l_G(b)$ denote the second largest root of $b\mu(\mu-2)\!-\!(\mu-1)^2(\mu-3)\!=\!0$ $(b\!=\!0,1,\cdots)$ and $l_T(b)$ denote the second largest root of $b\mu(\mu-2)\!-\!(\mu-1)^2(\mu-3)\!-\!(\mu-1)(\mu-2)\!=\!0$$(b\!=\!0,1,\cdots)$. Firstly, we will prove that there exist sequences of graphs $\{G_{n,b}\}(b\!=\!0,1,\cdots)$ and $\{T_{n,b}\}(b\!=\!0,1,\cdots)$ such that their limit points of the third largest Laplacian eigenvalues are $l_G(b)$ and $l_T(b)$, respectively. Secondly, we will prove that $l_G(b)$, $l_T(b)$ and $2$ are all of the limit points of the third largest Laplacian eigenvalues which are no more than 2
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