A Study on certain chromatic parameters and polynomials of graphs

Abstract

In graph theory, graph colouring pertains to the assignment of colours to the elements of a graph such as vertices, edges and faces. Because of the theoretical and practical implications of graph colourings in real-life situations, it is an adequate mathematical model for a wide range of applications such as network analysis, genomic, routing, newlineoptimisation techniques, digital networks and so forth.Motivated by various problems in chemical graph theory and information networks, chromatic topological indices were introduced in recent literature [81], opening ample and vibrant research area in graph theory.In the research reported in this thesis,the vertices of a graph are assigned with colours subject to certain conditions and manipulating their colour codes, a rich research area on chromatic topological indices and different chromatic polynomials are established. newlineAfter mentioning some fundamental terminologies, the study handles the notions of chromatic topological indices and chromatic irregularity indices. A detailed discussion of their upper and lower bounds concerning certain colouring conditions is carried out in this thesis. Chromatic topological indices of a wide variety of graph classes such as wheels, double wheels, helm graphs, closed helm graphs, flower graphs, sunflower graphs and blossom graphs are considered and investigated. The chromatic topological indices of certain derived graphs such as Mycielskian of paths and cycles are also included. Equitable chromatic topological and irregularity indices and injective chromatic topological and irregularity indices are defined and their values are determined for a handful of graph classes. As an indirect analogue to chromatic polynomials in the literature, the notion of chromatic Zagreb polynomials and chromatic irregularity polynomials are being introduced and the same is determined and discussed for paths, cycles and certain cycle related graph classes.