Studies on color energy and its variations in graphs

Abstract

This thesis consists of studies on color energy and its variations in graphs. Apart from the exploration of color energy corresponding to various coloring schemes, the notion of P-energy as a generalization of color energy has been introduced. The computation of color energy and P-energy of graphs has been carried out using Python programs, while the general results are derived using research methods and proof techniques in linear algebra. The bounds of color energy for a graph G have been established in terms of several graph parameters such as chromatic number and#967;(G), domination number and#947;(G), maximum degree and#8710;(G) etc. It has been found out that the color energy of a graph G is greater than or equal to 1 n and#947;(G) q 2(m + mand#8242;c). Further, the bounds of color energy of a graph G in terms of extreme eigenvalues of color matrix of G have been obtained. The study on color energy with respect to the minimum number of colors and L(h, k)-coloring has been examined in detail for some families of graphs such as star graph, double star, crown graph and their color complements. We have also examined the variation of color energy in the specific cases of T-coloring and radio coloring for some families of graphs. The examination of color energy corresponding to some improper colorings such as Hamiltonian coloring, open neighborhood coloring and improper C-coloring has also been done. Moreover, the color equi-energetic families of graphs with respect to various coloring schemes have been investigated. The concept of P-energy has been introduced as a generalization of the concept of color energy. This stems from the fact that coloring problems in essence are vertex partition problems. For any vertex partition P having k elements, we define the P-matrix AP(G) having and#8722;1, 0, 1, 2 as off diagonal entries and diagonal entries represent the cardinality of the elements in partition P. Then, the P-energy EP(G) is defined as the sum of the absolute values of eigenvalues of P-matrix of G.