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Let <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> be a finite group and <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> be a non-empty subset of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> comprising of the non-conjugate elements. In this study, we introduced the non-conjugate graph associated with <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> with a coinciding set of vertices, such that two distinct vertices <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula> are adjacent only if <inline-formula> <tex-math notation="LaTeX">$x,y\in S$ </tex-math></inline-formula>. We then discussed some fundamental properties to ensure the algebraic and combinatorial structure of the graph. Afterward, we formulated the resolving set and resolving polynomial for a subclass of dicyclic groups.