Abstract
This article comprehensively explores matrices and their prerequisites for achieving positive semidefiniteness. The study delves into a series of theorems concerning pure quantum states in the context of weighted graphs. The main objective of this study is to establish a graph-theoretic framework for the study of quantum discord and to identify the necessary and sufficient conditions for zero quantum discord states using unitary operators. This research aims to advance the understanding of quantum discord and its implications for quantum information theory with a graph-theoretic framework.