Title of Thesis

Super Edge Magic Deficiency of Forests

Author(s)

Abdul Qudair Baig

Abstract
A labeling of a graph is a mapping that carries some set of graph elements(vertices, edges or both) into numbers (usually positive integers). An edge-magic total labeling of a graph with p vertices and q edges is a one-to-one mapping that takes the vertices and edges onto the integers 1; 2; : : : ; p + q, so that the sums of the label on the edges and the labels of their end vertices are always the same, thus they are independent any particular edge. Such a labeling is called super if the p smallest possible labels appear at the vertices. In 1970 Kotzig and Rosa [33] introduced the concept of edge-magic de ciency of a graph G, denoted by (G), which is the minimum nonnegative integer n such that G[nK1 is edge-magic total. Motivated by Kotzig and Rosa's concept of edge-magic de ciency, Figueroa-Centeno, Ichishima and Muntaner-Batle [17] de ned a similar concept for super edge-magic total labelings. The super edge-magic de ciency of a graph G, which is denoted by s(G), is the minimum nonnegative integer n such that G [ nK1 has a super edge-magic total labeling or it is equal to 1 if there exists no such n.
The thesis is devoted to study of super edge-magic de ciency of forests.We present new results on the super edge-magic de ciencies of forests including union of paths, stars, comb, banana trees and subdivisions of K1;3. In the thesis we also deal with the super edge-magic de ciencies of forests formed by a disjoint union of stars.

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2.1 Magic labelings
2.2 Antimagic labelings

3.1 Edge-magic total labeling
3.2 Super edge-magic total labeling
3.3 Edge-magic de ciency
3.4 Super edge-magic de ciency

4.1 Two paths and one star
4.2 Union of a comb and a star
4.3 Forest of banana trees
4.4 A banana tree and a star
4.5 Forest of subdivisions of a star

5.1 Upper bound
5.2 Galaxy of three stars
5.3 Galaxy of four stars
5.4 Galaxy of ve stars
5.5 Galaxy of k-isomorphic stars

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