  This paper studies the eigenvalue distribution of the Watts-Strogatz random
graph, which is known as the "small-world" random graph. The construction of
the small-world random graph starts with a regular ring lattice of n vertices;
each has exactly k neighbors with equally k/2 edges on each side. With
probability p, each downside neighbor of a particular vertex will rewire
independently to a random vertex on the graph without allowing for self-loops
or duplication. The rewiring process starts at the first adjacent neighbor of
vertex 1 and continues in an orderly fashion to the farthest downside neighbor
of vertex n. Each edge must be considered once. This paper focuses on the
eigenvalues of the adjacency matrix A_n, used to represent the small-world
random graph. We compute the first moment, second moment, and prove the
limiting third moment as n goes to infinity of the eigenvalue distribution.
