Chemical graph theory, a subset of analytical chemistry, focuses on studying chemical graphs and systems using graph theory as a mathematical model for chemical phenomena. Topological indices, like connectivity indices, served as molecular descriptors derived from the molecular graphs of chemical compounds. These numeric parameters, acting as graph invariants, find applications in quantitative structure-activity relationships for correlating biological activity with chemical structure. In theoretical chemistry, drugs are represented as molecular graphs. Shifting focus to cancer, a diverse group of diseases, cancer results from abnormal cell proliferation, invading neighboring parts, and spreading through metastasis. Melanoma, a dangerous skin cancer, arises from uncontrolled melanocyte growth, demanding early detection due to its high metastatic potential. Immunotherapy, particularly against melanoma, has emerged as an effective cancer treatment. Vemurafenib, a B-Raf enzyme inhibitor, is used in advanced melanoma therapy. The current study explores polynomial approaches to examine topological indices for the chemical graph of vemurafenib.