On (a, d )-Distance Anti-Magic and 1-Vertex Bimagic Vertex Labelings of Certain Types of Graphs

Abstract

The results for the corona P n ∘ P1 are generalized, which make it possible to state that P n ∘ P1 is not an ( a, d)-distance antimagic graph for arbitrary values of a and d. A condition for the existence of an ( a, d)-distance antimagic labeling of a hypercube Q n is obtained. Functional dependencies are found that generate this type of labeling for Q n . It is proved by the method of mathematical induction that Q n is a (2 n + n − 1, n − 2) -distance antimagic graph. Three types of graphs are defined that do not allow a 1-vertex bimagic vertex labeling. A relation between a distance magic labeling of a regular graph G and a 1-vertex bimagic vertex labeling of G ∪ G is established.