Abstract: The signless Laplacian matrix of a graph is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are denoted by $q_1\geq q_2\geq,\cdots ,\geq q_n$. Let $\mathscr{C}(n,m)$ be a set of connected graphs in which every graph has $n$ vertices and $m$ edges, where $1\leq n-1\leq\ m \leq\bigl(\begin{smallmatrix}n\\2\end{smallmatrix}\bigr)$. A graph $G^\star \in \mathscr{C}(n,m)$ is called maximum if $\ q_1(G^\star )\geq q_1(G)$ for any $G\in \mathscr{C}(n,m)$. In this paper, we proved that for any given positive integer $a=m-n+1$, $n-\frac{1}{2}+a+\frac{1}{2}\sqrt{1+12a+12a^2}$, which leads to $q_1(G)-\frac{1}{2}+a+\frac{1}{2}\sqrt{1+12a+12a^2}$.

Key words: signless Laplacian, maximum graph, nested split graph, Q-spectral radius