Laplacian Minimum Domination Energy of Some Derived Graphs
Graph energy is an important concept in spectral graph theory with applications in mathematics and chemistry. In this paper, we study the Laplacian minimum domination energy of derived graphs of some standard graphs. The main aim is to obtain formulas, properties, and bounds for this energy measure. The study considers derived graphs of star graphs, complete bipartite graphs, friendship graphs, and healthy spider graphs. Using minimum dominating sets, minimum domination adjacency matrices, and Laplacian minimum domination matrices, the eigenvalues of these derived graphs are determined. Based on these eigenvalues, explicit formulas for the Laplacian minimum domination energy are obtained. Further, some basic properties related to eigenvalues are established. Upper and lower bounds for the Laplacian minimum domination energy are also derived using matrix methods and classical inequalities such as the Cauchy-Schwarz inequality. The results extend existing work on graph energy by combining domination concepts, Laplacian matrices, and derived graphs. The formulas, properties, and bounds obtained in this paper provide a better understanding of the spectral behavior of derived graphs and may be useful for further research in graph theory and its applications.