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 $m_G[0,1) \leq \gamma(G)$, and show that isolate-free graphs also satisfy $\gamma(G) \leq m_G[2,n]$. 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, $\gamma(G) \leq m_G[2, n] \leq (c + 1) \gamma(G)$ for $c$-cyclic graphs, $c \geq 1$. For trees $T$, $\gamma(T) \leq m_T[2, n] \leq 2 \gamma(G)$

A Luddite Archaeology of Hobbyist Computation

The practice element of this research includes the How-To documented here, as well as the collected recordings and artist events I have documented. It also includes the forms I have chosen and experimented with for the publication of this. This text was written primarily as a website, and the styling of different sections here references the web version. To read the website version visit: http://bruise.in/research/luddite-archaeology.html. Some additional notes on the construction of the page are included in the code which can be viewed in the inspector. Excerpts of the code are included in the Code Practice section