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Abstract
In this thesis we introduce new results in four sorts of graph labelings. The thesis consists of five chapters as follows:
Chapter one: Is an introduction to present some basic ideas and concepts, as a trial to cover what the reader needs for the next chapters.
Chapter two: We deal with “The mean labeling of graphs”. We first define the labeling and present the most known results. Then, we introduce new results: We label all the graphs of order less than or equal to 6, which accept this labeling and we show why the rest of these graphs do not accept the labeling.
We give an upper bound of the number of edges of a mean graph with a known order, and we give the maximum possible vertex degree of a mean graph of a known size. Finally, we label some families of graphs using a new theorem.
These results are accepted to be published in the journal (Ars Combinatoria).
Chapter three: Here we deal with the so called “Super mean labeling of graphs”. We first define the labeling, we mention the known results and our new results: We label all graphs of order less than or equal to 5, which accept this labeling and show that the rest do not accept this labeling. We also determine the number of isolated vertices which may be added to a complete graph, so that it accepts the labeling.
We also label some families of graphs, depending upon a general theorem, such as: P_n⨀K_2,P_n⨀K ̅_2,P_n⨀P_3,P_n⨀P_4,P_n⨀C_4,
P_n⨀2K_2,C_n⨀K_1,C_n⨀K_2,C_n⨀P_3.
Afterwards we labeled all cycles, prisms, grids which contain a random edge in every square, odd prisms and C_n⨀K ̅_2, when n is odd.
These results are submitted to be refereed for publication.
Chapter four: Here we introduce the known labeling :” The prime cordial labeling”, and give an upper bound for the number of edges of a graph of order n to be prime cordial, consequently we prove that many graphs are not prime cordial. We improve this upper bound in case the graph is bipartite, so we show that many bipartite graphs are not prime cordial. Finally, we present all graphs of order less than or equal to 6, which are not prime cordial.
The results are submitted to be refereed for publication.
Chapter five: we define the so called “Permutation labeling”. What is known about this labeling is still limited. We determine all permutation graphs of order less than or equal to 9. We prove that every bipartite graph of order less than or equal to 50 is a permutation graph. We also conjecture that every bipartite graph is a permutation graph.
The results are submitted to be refereed for publication.