Regarding the vector metric space introduced by Cevik and Altun, we first introduce the concept of a graphical vector metric space and then define several topological notions therein such as X-Dedekind complete Riesz spaces, vectorially G-continuous mappings, vectorially G-proximal mappings, vector P-property. Then some best proximity point theorems are proven in such spaces using these new concepts. Several consequences and an illustrative example are also provided to support our definitions and results. One important thing that can be distinguished between this work and similar researches is that it can also encompass all of those previously introduced by considering comparable and "-close elements.
