## Combinatorics and Graph Theory   ## The Eigen-Cover Ratio of Graphs: Asymptotes, Domination and Areas

Authors: Paul August Winter, Carol Lynne Jessop

The separate study of the two concepts of energy and vertex coverings of graphs has opened many avenues of research. In this paper we combine these two concepts in a ratio, called the eigen-cover ratio, to investigate the domination effect of the subgraph induced by a vertex covering of a graph (called the cover graph of ), on the original energy of , where large number of vertices are involved. This is referred to as the eigen-cover domination and has relevance, in terms of conservation of energy, when a molecule’s atoms and bonds are mapped onto a graph with vertices and edges, respectively. If this energy-cover ratio is a function of , the order of graphs belonging to a class of graph, then we discuss its horizontal asymptotic behavior and attach the graphs average degree to the Riemann integral of this ratio, thus associating eigen-cover area with classes of graphs. We found that the eigen-cover domination had a strongest effect on the complete graph, while this chromatic-cover domination had zero effect on star graphs. We show that the eigen-cover asymptote of discussed classes of graphs belong to the interval [0,1], and conjecture that the class of complete graphs has the largest eigen-cover area of all classes of graphs.