Perfect Coloring of Graphs Related to Irreducible Fullerenes in Carbon Structures

Abstract

Fullerenes are polyhedral molecules composed solely of carbon atoms, available in various sizes and shapes. These structures can also be depicted as graphs, with the vertices symbolizing the atoms and the edges representing the bonds between them. A fullerene graph is defined as a 3-connected, 3-regular planar graph that consists only of pentagonal and hexagonal faces. This paper examines the perfect 2- and 3-coloring of fullerene graphs, with a particular focus on irreducible fullerenes. The proposed approach begins by obtaining the adjacency matrix of the graphs and then comparing its eigenvalues with those of the parameter matrices. If the eigenvalues of a parameter matrix are a subset of the graph's eigenvalues, we retain these matrices for further analysis to determine their suitability for perfect coloring.