U(N) Weingarten function, known in computing the U(N) link integral, is an essential ingredient in physics. Although fewer people pay attention to SU(N), the SU(N) Weingarten function is important in the lattice gauge theory and it differs from U(N) . In this talk, I will present the derivation of the SU(N) Weingarten function using character theory and emphasize some details about how it differs from the perspective of polynomial representation of $GL_N$. We will also explore the nice combinatorial interpretation of the 1/N expansion of the Weingarten function using Hurwitz-Cayley graph which serves as the Feynman diagram