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 $$\frac{\gamma (G)}{m_G[0,1)} \not \in O(\log n)$$ . However, $$\gamma (G) \le m_G[2, n] \le (c + 1) \gamma (G)$$ for c-cyclic graphs, $$c \ge 1$$ . For trees T, $$\gamma (T) \le m_T[2, n] < 2 \gamma (G)$$ .