  The brain is one of the most studied and highly complex systems in the
biological world. It is the information center behind all vertebrate and most
invertebrate life, and thus has become a major focus in current research. While
many of these studies have concentrated on studying the brain directly, our
focus is the structure of the brain itself: at its core an interconnected
network of nodes (neurons). A better understanding of the structural aspects of
the brain should elucidate some of its functional properties. In this paper we
analyze the brain of the nematode Caenorhabditis elegans. Consisting of only
302 neurons, it is one of the better-understood neural networks. Using a
Laplacian matrix of the 279-neuron "giant component" of the network, we use an
eigenvalue counting function to look for fractal-like self similarity. This
matrix representation is also used to plot (in eigenfunction coordinates) both
2 and 3 dimensional visualizations of the neural network. Further analysis
examines the small-world properties of the system, including average path
length and clustering coefficient. We then test for localization of
eigenfunctions, using graph energy and spacial variance. To better understand
these results, all of these calculations are also performed on random networks,
branching trees, and known fractals, as well as fractals which have been
"rewired" to have small-world properties. This analysis is one of many
stepping-stones in the research of neural networks. While many of the
structures and functions within the brain are known, understanding how the two
interact is also important. A firmer grasp on the structural properties of the
neural network is a key step in this process
