The chromatic number of a graph G, \(\chi (G)\) represents the minimum number of colors required to color the vertices of a graph such that no adjacent vertices share the same color. This paper provides a focused exploration of the chromatic number, emphasizing its mathematical significance and practical applications. In this work, the behavior of the chromatic number across various graph structures is investigated. Leveraging the computational tools, we develop algorithms to analyze the chromatic number across diverse graph structures and explore its implications for graph coloring problems. Additionally, we introduce visualization techniques to graphically represent graph coloring and thereby provide intuitive insights into the dynamics of the chromatic number. Experimental results include the visual representation of the graph, chromatic number of the given graph, and the execution time of the algorithm for various types of graphs. The algorithm can handle any number of nodes and the results are produced for the appropriate graph type for varying number nodes. This paper discusses the engineering applications of chromatic number in several fields such as education, networking, circuit design, integrated circuits, and so on.