$D$-Magic Strongly Regular Graphs

Abstract

For a set of distances $D$, a graph $G$ on $n$ vertices is said to be $D$-magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , n\}$ and a constant $k$ such that for any vertex $x$, $\sum_{y\in N_D(x)} f(y) = k$, where $N_D(x)=\{y|d(x,y)=i, i\in D\}$ is the $D$-neighbourhood set of $x$. In this paper we utilize spectra of graphs to characterize strongly regular graphs which are $D$-magic, for all possible distance sets $D$. In addition, we provide necessary conditions for distance regular graphs of diameter 3 to be $\{1\}$-magic.