Some characterizations of dualizing complexes in terms of $G_{C}$-dimension

Let $(R,\fm)$ be a local ring and $C$ be a homologically bounded and finitely generated $R$-complex. Then, we prove that $C$ is a dualizing complex of $R$ if and only if $C$ is a Cohen-Macaulay semidualizing complex of type one or $\mu_R^{\inf C+\dim_R(C) }(\fm,R)=\beta_{\inf C}^R(C)$. Also, we show that a semidualizing complex $C$ is dualizing if and only if there exists a type one Cohen-Macaulay $R$-module of finite $G_{C}$-dimension or there exists a type one Cohen-Macaulay $R$-complex of finite $G_{C}$-dimension such that $\dim_R(X)=\dim_R(C)-\gr_C(X)$. Furthermore, for a semidualizing $R$-complex $C$, we prove that $C\sim R$ if and only if there exists a type one Cohen-Macaulay $R$-module $M$ which belongs to the Auslander class $\mathcal{A}_C(R)$.