AN INTRODUCTION TO DISTANCE D MAGIC GRAPHS
Abstract: For a graph G of
order jV (G)j = n and a real-valued mapping f : V (G) ! R, if S ⊂ V (G) then f(S) = Pw2S f(w) is called the weight of Sunder f.
When there exists a bijection f : V (G) ! [n] such that the weight of allopen
neighborhoods is the same, the graph is said to be 1-vertex magic, or Σ
labeled. In this paper we generalize the notion of 1-vertex magic by defining a
graph G of diameter d to be D-vertex magic when for D ⊂ f0; 1; : : : ; dg, we have that Pu2ND(v) f(u) is constant for
all v 2 V (G). We provide several existence criteriafor graphs to be D-vertex
magic and use them to provide solutions to several open problems presented at
the IWOGL 2010 Conference. In addition, we extend the notion of vertex magic
graphs by providing measures describing how close a non-vertex magic graph is
to being vertex magic. The general viewpoint is to consider how to assign a set
W of weights to the vertices so as to have an equitable distribution overthe
D-neighborhoods.
Author: Allen O’Neal, Peter J.
Slater
Journal Code: jpmatematikagg110009
