Informatica

In this work labeling of planar graphs is taken up which involves labeling the p vertices, the q edges and the f internal faces such that the weights of the faces form an arithmetic progression with common difference d. If d=0, then the planar graph is said to have an Inner Magic labeling; and if d≠0, then it is Inner Antimagic labeling. Some new kinds of graphs have been developed which have been derived from Wheels by adding vertices in a certain way and it is proposed to give new names to these graphs namely Flower-1 and Flower-2. This paper presents the algorithms to obtain the Inner Magic and Inner Antimagic labeling for Wheels and the Inner Antimagic labeling for Flower-1 and Flower-2. The results thus found show much regularity in the labelings obtained.

Tree is one of the most studied and practically useful classes of graphs and is the attention of a great number of studies. There is absence of generalized results for tree as a class and even for one kind of labeling as whole. Only specialized results exist limited to specific types of trees only. A number of conjectures stand being unsolved. Graham and Sloane (1980) conjectured trees to be Harmonious and Ringel‐Kotzig conjectured trees to be Graceful about three decades ago. Kotzig and Rosa (1970) ask the question whether all trees are Magic or not. No generalized result for Antimagic labeling is given for trees so far. This paper presents the methodologies to obtain the major labeling schemes for trees viz., Harmonious, Sequential, Felicitous, Graceful, Antimagic and found the trees to be not Magic except T(2,1), thus solving the said conjectures. These findings could also be useful for those working in fields where graphs serve as models.