  Let $q(H)$ be the signless Laplacian spectral radius of a graph $H$. In this
paper, we prove that \\1. Let $H$ be a proper subgraph of a $\Delta$-regular
graph $G$ with $n$ vertices and diameter $D$. Then $$2\Delta -
q(H)>\frac{1}{n(D-\frac{1}{4})}.$$ \\2. Let $H$ be a proper subgraph of a
$k$-connected $\Delta$-regular graph $G$ with $n$ vertices, where $k\geq 2$.
Then
$$2\Delta-q(H)>\frac{2(k-1)^{2}}{2(n-\Delta)(n-\Delta+2k-4)+(n+1)(k-1)^{2}}.$$
Finally, we compare the two bounds. We obtain that when
$k>2\sqrt{\frac{(n-\Delta)(n+\Delta-4)}{n(4D-3)-2}}+1$, the second bound is
always better than the first. On the other hand, when
$k<\frac{2(n-\Delta)}{\sqrt{n(4D-3)-2}}+1$, the first bound is always better
than the second.
