Sigma Labeling and Circulant Graphs

This book is a copy of my Ph.D. thesis entitled “? -Labeled Graphs and Circulant Graphs” submitted in March 1994 under the guidance of Dr. M.I. Jinnah University of Kerala India. Defined ? -Labeling a new type of labeling of graphs and developed its theory. Got motivation to define sigma labeling by observing two similar situations in the number labeled to the faces of dice where the sum of the numbers assigned to each pair of opposite faces is 7 and of magic squares. Suppose we have an n×n magic square in which each row sum is r. If we form a complete multipartite graph with each row of the square representing a partite set and if we label each vertex with the corresponding integers in the magic squares we find that the sum of labels of all vertices in the neighborhood set for each vertex is the same equal to r(n-1). In the study we could notice that many regular ? -labeled graphs are circulant graphs and extended our study on circulant graphs.