Let $G=(V,E)$ be a graph of order n and with the Laplacian spectral radius $\lambda (G)$. For $v_i\in V$, denote the set of all neighbors of $v_i$ by $N_i$ and its number by $d_i$. The maximum degree of G is denoted by $\mathrm{\Delta }(G)$. It is shown that if G is connected and $\mathrm{\Delta }(G)<n-1$ then$\lambda (G)\ge \max \{m_i^{\prime }+(1+\frac{{(m_i^{\prime }-1)}^2}{d_{2,i}})\frac{d_i}{m_i^{\prime }}:v_i\in V\},$ where $m_i^{\prime }=\frac{{\sum }_{v_i{v}_j\in E}(d_j-|N_i\cap N_j|)}{d_i}$ and $d_{2,i}$ is the number of vertices at distance two from $v_i$.

Also it is shown that$\begin{array}{cc}\hfill \lambda (G)\ge \max \{& (p_{ij}+\frac{{(1-p_{ij})}^2}{p_{ij}\max \{1,d_i-1\}})\hfill \\ \hfill & \times |N_i\cup N_j|:v_i{v}_j\in E,d_i\ge d_j\},\hfill \end{array}$ where $p_{ij}=\frac{e(N_i,N_j-N_i)}{d_i(|N_i\cup N_j|-d_i)}$, $e(N_i,N_j-N_i)$ is the number of edges between $N_i$ and $N_j-N_i$.

让 $G=(伏,乙)$是一个n阶图，具有拉普拉斯谱半径$\lambda (G)$. 为了$v_{\mathrm{一世}}\in 伏$，表示所有邻居的集合 $v_{\mathrm{一世}}$ 经过 $N_{\mathrm{一世}}$ 和它的编号 $d_{\mathrm{一世}}$. G的最大程度表示为$\mathrm{\Delta }(G)$. 可以看出，如果G是连通的并且$\mathrm{\Delta }(G)<n-1$ 然后$\lambda (G)\ge \mathrm{最大限度}\{{米}_{\mathrm{一世}}^{\prime }+(1+\frac{{({米}_{\mathrm{一世}}^{\prime }-1)}^2}{d_{2,\mathrm{一世}}})\frac{d_{\mathrm{一世}}}{{米}_{\mathrm{一世}}^{\prime }}：v_{\mathrm{一世}}\in 伏\},$ 在哪里 ${米}_{\mathrm{一世}}^{\prime }=\frac{{\sum }_{v_{\mathrm{一世}}{v}_j\in 乙}(d_j-|N_{\mathrm{一世}}\cap N_j|)}{d_{\mathrm{一世}}}$ 和 $d_{2,\mathrm{一世}}$ 是距离为 2 处的顶点数 $v_{\mathrm{一世}}$.

还表明$\begin{array}{cc}\hfill \lambda (G)\ge \mathrm{最大限度}\{& ({磷}_{\mathrm{一世}j}+\frac{{(1-{磷}_{\mathrm{一世}j})}^2}{{磷}_{\mathrm{一世}j}\mathrm{最大限度}\{1,d_{\mathrm{一世}}-1\}})\hfill \\ \hfill & \times |N_{\mathrm{一世}}\cup N_j|：v_{\mathrm{一世}}{v}_j\in 乙,d_{\mathrm{一世}}\ge d_j\},\hfill \end{array}$ 在哪里 ${磷}_{\mathrm{一世}j}=\frac{\mathrm{电子}(N_{\mathrm{一世}},N_j-N_{\mathrm{一世}})}{d_{\mathrm{一世}}(|N_{\mathrm{一世}}\cup N_j|-d_{\mathrm{一世}})}$, $\mathrm{电子}(N_{\mathrm{一世}},N_j-N_{\mathrm{一世}})$ 是之间的边数 $N_{\mathrm{一世}}$ 和 $N_j-N_{\mathrm{一世}}$.