(a;d)-distance antimagic labeling for some regular graphs

Abstract

Let $G=(V, E)$ be a graph of order $n$. Let $f: V(G) \longrightarrow\{1,2,3, \ldots, n\}$ be a bijective function. For every vertex $v$ in $G$, we define its weight $w(v)$ as the sum $\sum_{u \in N(v)} f(u)$, where $N(v)$ is the open neighborhood of $v$. If the set of all vertex weights forms an arithmetic progression $\{a, a+d, a+2 d, \ldots, a+(n-1) d\}$, then $f$ is called (a,d)-distance antimagic labeling and the graph $G$ is called (a,d)-distance antimagic graph. In this paper we prove circulant graph Circ $(2 n,\{1, n\})$ for odd $n$ and $n K_{2 n+1}$ for odd $n$ are (a,d)-distance antimagic graphs. We also give some necessary conditions for $m K_n$ to be (a,d)-distance antimagic graph for $d=2 k$, where $k$ is some positive integer.