A labeling of a
graph is a map that carries graph elements to the numbers (usually
positive or non-negative integers). The most common choices of
domain are the set of all vertices (vertex labelings), the edge set
alone (edge labelings), or the set of all vertices and edges (total
In many cases, it is interesting to consider the sum of all labels
associated with a graph element. It is called the weight of the
element: the weight of a vertex or the weight of an edge.
In this thesis, we consider a total k-labeling as a labeling of the
vertices and edges of graph G with labels from the set f1; 2; : : :
; kg. A total k-labeling is de ned to be an edge irregular total
k-labeling of the graph G if edge-weights are di erent for all pairs
of distinct edges and to be a vertex irregular total k-labeling of G
if vertex-weights are di erent for all pairs of distinct vertices.
The minimum value of k for which the graph G has an edge irregular
total k- labeling is called the total edge irregularity strength of
the graph G, tes(G). Analogously, the total vertex irregularity
strength of G, tvs(G), is de ned as the minimum k for which there
exists a vertex irregular total k-labeling of G. In this thesis, we
present new results on the total edge irregularity strength and the
total vertex irregularity strength.