## The eigen-chromatic ratio of classes of graphs : molecular stability, asymptotes and area.

##### Abstract

This dissertation involves combining the two concepts of energy and the chromatic number of
classes of graphs into a new ratio, the eigen-chromatic ratio of a graph G. Associated with this
ratio is the importance of its asymptotic convergence in applications, as well as the idea of area
involving the Rieman integral of this ratio, when it is a function of the order n of the graph G
belonging to a class of graphs.
The energy of a graph G, is the sum of the absolute values of the eigenvalues associated with
the adjacency matrix of G, and its importance has found its way into many areas of research
in graph theory. The chromatic number of a graph G, is the least number of colours required
to colour the vertices of the graph, so that no two adjacent vertices receive the same colour.
The importance of ratios in graph theory is evident by the vast amount of research articles:
Expanders, The central ratio of a graph, Eigen-pair ratio of classes of graphs , Independence
and Hall ratios, Tree-cover ratio of graphs, Eigen-energy formation ratio, The eigen-complete
difference ratio, The chromatic-cover ratio and "Graph theory and calculus: ratios of classes of
graphs". We combine the two concepts of energy and chromatic number (which involves the
order n of the graph G) in a ratio, called the eigen-chromatic ratio of a graph. The chromatic
number associated with the molecular graph (the atoms are vertices and edges are bonds between
the atoms) would involve the partitioning of the atoms into the smallest number of sets of like
atoms so that like atoms are not bonded. This ratio would allow for the investigation of the
effect of the energy on the atomic partition, when a large number of atoms are involved. The
complete graph is associated with the value 1
2 when the eigen-chromatic ratio is investigated
when a large number of atoms are involved; this has allowed for the investigation of molecular
stability associated with the idea of hypo/hyper energetic graphs. Attaching the average degree
to the Riemann integral of this ratio (as a function of n) would result in an area analogue for
investigation.
Once the ratio is defned the objective is to find the eigen-chromatic ratio of various well known
classes of graphs such as the complete graph, bipartite graphs, star graphs with rays of length
two, wheels, paths, cycles, dual star graphs, lollipop graphs and caterpillar graphs. Once the
ratio of each class of graph are determined the asymptote and area of this ratio are determined
and conclusions and conjectures inferred.