Laplacian Distribution and Domination

Let $m_G(I)$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$, and let $\gamma (G)$ denote its domination number. We extend the recent result $m_G[0,1)\le \gamma (G)$, and show that isolate-free graphs also satisfy $\gamma (G)\le m_G[2,n]$. In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of $\gamma (G)$, showing that $\frac{\gamma (G)}{m_G[0,1)}\in ̸O(\log n)$. However, $\gamma(G) \leq m_G[2, n] \leq (c + 1) \gamma(G)$for$c$-cyclicgraphs,$c \geq 1$\mathrm{.}Fortrees$T$,$\gamma(T) \leq m_T[2, n] \leq 2 \gamma(G)$.