In this paper, the effects on the signless Laplacian spectral radius of a graph are studied when some operations, such as edge moving, edge subdividing, are applied to the graph. Moreover, the largest signless Laplacian spectral radius among the all unicyclic graphs with $n$ vertices and $k$ pendant vertices is identified. Furthermore, we determine the graphs with the largest Laplacian spectral radii among the all unicyclic graphs and bicyclic graphs with $n$ vertices and $k$ pendant vertices,...

Let $G$ be a graph with $n$ vertices, $m$ edges and a vertex degree sequence $(d_1,d_2,\cdots ,d_n)$, where $d_1\ge d_2\ge \cdots \ge d_n$. The spectral radius and the largest Laplacian eigenvalue are denoted by $\rho \left(G\right)$ and $\mu \left(G\right)$, respectively. We determine the graphs with $$\rho \left(G\right)=\frac{d_n-1}{2}+\sqrt{2m-n{d}_n+\frac{{(d_n+1)}^2}{4}}$$ and the graphs with $d_n\ge 1$ and $$\mu \left(G\right)=d_n+\frac{1}{2}+\sqrt{{\sum }_{i=1}^n{d}_i(d_i-d_n)+{\left(d_n-\frac{1}{2}\right)}^2}.$$ We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.

The eccentricity $e\left(v\right)$ of a vertex $v$ is defined as the distance to a farthest vertex from $v$. The radius of a graph $G$ is defined as a $r\left(G\right)={min}_{u\in V\left(G\right)}\left\{e\left(u\right)\right\}$. A graph $G$ is radius-edge-invariant if $r(G-e)=r\left(G\right)$ for every $e\in E\left(G\right)$, radius-vertex-invariant if $r(G-v)=r\left(G\right)$ for every $v\in V\left(G\right)$ and radius-adding-invariant if $r(G+e)=r\left(G\right)$ for every $e\in E\left(\overline{G}\right)$. Such classes of graphs are studied in this paper.