This thesis studies graph matrices. Graph matrices are a type of random matrices that were invented as a powerful tool for analyzing large and complicated moment matrices which often arise in the analysis of the Sum of Square Hierarchy. They are also useful for other methods involving higher moments. Formally, a graph matrix is defined as a function mapping an n×n random input matrix to an output matrix which is generally larger and more complicated. This function is described by a small underlying shape. Previous studies on graph matrices mainly focused on their norm bounds. In this thesis, we further investigate their spectrum. We start with determining the spectrum of singular values of Z-shaped and multi-Z- shaped graph matrices when the entries of the random input matrix have distribution ±1 and the dimension parameter n → ∞. This result can be seen as an analog of Wigner’s Semicircle Law in the special case of ±1 distribution, instead of any arbitrary distribution with mean 0 and variance 1. We then generalize our result to multi-Z-shaped graph matrices with arbitrary input distributions with variance 1 and 0 odd moments. We achieve this using the ◦R operation, which mixes two distributions Ω and Ω′ via a random orthogonal matrix. This ◦R operation is closely connected to free probability theory, more precisely the free multiplicative convolution in the random matrix setting. As a part of our analysis, we prove a new direct formula for the moments of the product of two freely independent variables. We also prove some new results on non-crossing partitions which is an essential part of free probability theory.