Alpha graphs with different pendent paths

Abstract

Graceful labelings are an effective tool to find cyclic decompositions of complete graphs and complete bipartite graphs. The strongest kind of graceful labeling, the α-labeling, is in the center of the research field of graph labelings, the existence of an α-labeling of a graph implies the existence of several, apparently non-related, other labelings for that graph. Furthermore, graphs with α-labelings can be combined to form new graphs that also admit this type of labeling. The standard way to combine these graphs is to identify every vertex of a base graph with a vertex of another graph. These methods have in common that all the graphs involved, except perhaps the base, have the same size. In this work, we do something different, we prove the existence of an α-labeling of a tree obtained by attaching paths of different lengths to the vertices of a base path, in such a way that the lengths of the pendent paths form an arithmetic sequence with difference one, where consecutive vertices of the base path are identified with paths which lengths are consecutive elements of the sequence. These α-trees are combined in several ways to generate new families of α-trees. We also prove that these trees can be used to create unicyclic graphs with an α-labeling. In addition, we show that the pendent paths can be substituted by equivalent α-trees to produce new α-trees, obtaining in this manner a quite robust category of α-trees.

A Study on C‐Exponential Mean Labeling of Graphs

“…The study of Kannan et al on the exponential mean labeling of a few different graphs studied through duplicate operations is examined for the present study [8]. Barrientos' study on alpha graphs has demonstrated the presence of α-labeling of a tree using various vertices and lengths of base path and proved that these trees can be utilized to demonstrate unicycle graphs with α-labeling [9]. Studies on cordial labeling between paths and cycles for a Cartesian product have dem-onstrated that these Cartesian products, under any conditions, are always cordial and even proved that two path Cartesian products are always cordial [10].…”

On graphs with &#945;- and <em>b</em>-edge consecutive edge magic labelings

“…Several papers have followed Rosa' seminal work and multiple classes of α-trees are known. For example, in [9], Kotzig proved that almost all trees can be α-labeled; other two families of α-trees are the path-like trees [1] and the triangular trees [4]. Gallian [7] devotes an entire section of his survey to this labeling.…”