Fractional Medium Domination Number of Graphs

This work develops the framework of the fractional medium domination number(\( MD_f(G) \)), focusing on connected, undirected graphs without loops. The (\( MD_f(G) \)) is defined as the ratio of the fractional total domination value (\( T_{DV_f}(G) \)) to the total number of unordered pairs of distinct vertices in a graph. This new parameter expands on traditional domination concepts by incorporating fractional values, providing a more refined measure of domination in graphs. The fractional domination value between vertices is computed as the sum of fractional contributions from their common neighbors, where each contribution is inversely proportional to the degree of the respective vertex. The paper explores bounds for the fractional medium domination number across various graph families and presents computational methods for determining \( MD_f(G) \) using Python programming. Practical applications, such as network optimization and disaster relief, are also discussed to illustrate the significance of this parameter.