Minimum Domination Energy of Some Derived Graphs

In this study, we introduce and systematically explore the concept of minimum domination energy of derived graphs, representing a novel integration of two fundamental areas in graph theory: graph energy and domination. While both graph energy and domination theory have been extensively researched as separate topics, their combined study opens new avenues for understanding structural properties of graphs through the lens of spectral and domination parameters. The graph energy of a graph is a spectral invariant derived from the eigenvalues of its adjacency matrix and has applications in chemistry, physics, and network analysis. On the other hand, domination concerns selecting a minimal set of vertices such that every vertex in the graph is either in this set or adjacent to a vertex in the set, reflecting control or influence within the graph structure. We concentrate on the minimum domination energy, which is a measure of how these two ideas interact. For a number of derived graph classes, including the following, we calculate the precise values of the minimum domination energy: Star graphs, double star graphs, friendship graphs, healthy spider graphs, crown graphs, complete bipartite graphs, and cocktail party graphs. We also improve the theoretical framework by establishing strict lower and upper bounds for the minimum dominating energy. These findings pave the way for further study in graph theory and its applications and advance our knowledge of how domination affects the spectral characteristics of derived graphs.