Minimum Total Dominating Energy in Some Chemical Graphs

Abstract

A graph  = ((), ()) is defined by an edge set () and a vertex set (). When every vertex in a subset  ⊆() is adjacent to at least one vertex in , the subset is referred to as a total dominating set (TDS). This means that every vertex must share an edge with at least one member of the set; no vertex is isolated or unrelated to the set . The smallest size of such a total dominating set is the total domination number (TDN), represented as. It stands for the minimum number of vertices required to "cover" the graph in such a way that every other vertex is next to at least one of them. Additionally, the total dominating energy (TDE) of a graph, denoted is calculated as the sum of the absolute values of all the Eigenvalues of the adjacency matrix of G. This energy concept provides insight into the structural properties and stability of a graph, and when applied to molecular graphs, it can help characterize chemical compounds in terms of graph-based descriptors. This study calculates the minimal total dominant energy for the molecular graphs of a number of compounds, including sodium chloride, acetaminophen, chloroquine, sesame, and washing soda. In order to better understand the role of TDE in examining molecular stability and efficiency based on graph theoretical models; these compounds were chosen to investigate how their structural configurations affect their graph energies.