On equitable coloring of splitting of block circulate graph with spectral clustering

Abstract

In this thesis, the author investigated an efficient way of equitable newlineallotment of colors to vertices of the graphs through clustering based on its newlinespectrum pertaining to the adjacency matrices of splitting of graphs. This newlineapproach of spectrum aims in suitable partitioning of vertices and does the newlinecoloring work to be simpler. The equitable vertex coloring is achieved for newlinesplitting of n-Pan, n-Sun and n-Sunlet graphs. newlineThe author has also proposed an equitable vertex coloring for the newlinesplitting of block circulant graphs. The block circulant graphs comprises newlineblock circulant matrices, where each block is itself a matrix. These blocks in newlineeach row are cyclically shifted one place to the right from those of the newlineprevious row. We approached such block circulant graphs in matrix newlinerepresentation and derived their independent sets using the neighborhoods of newlineeach vertex. This classification makes the vertex coloring process to be newlinesimpler and equitable in most cases. In this framework, the equitable newlinechromatic numbers are obtained for splitting on block circulant graphs, newlinenamely Prism, Antiprism, Crossed Prism and Closed Sun graphs. newlineThe author has also investigated the coloring of all the edges of a newlinegraph G with the minimum number of colors, such that the adjacent edges newlineare allotted a different color is known as the proper edge coloring. It is said to newlinebe equitable, if the number of edges in any two color classes differ by atmost newlineone and we obtain the equitable edge chromatic number of splitting graphs of newlinesome wheel graph families such as Wheel graph, Double Wheel graph and newlineGear graph. Also investigated the same with some prism graph families such newlineas Prism graph, Antiprism graph and Crossed Prism graph and finally with newlinePan graph newline