Directionally n-signed graphs-III: The notion of symmetric balance

Abstract

Let $G=(V, E)$ be a graph. By emph{directional labeling (ord-labeling)} of an edge $x=uv$ of $G$ by an ordered $n$-tuple$(a_1,a_2,...,a_n)$, we mean a labeling of the edge $x$ such thatwe consider the label on $uv$ as $(a_1,a_2,...,a_n)$ in thedirection from $u$ to $v$, and the label on $x$ as$(a_{n},a_{n-1},...,a_1)$ in the direction from $v$ to $u$. Inthis paper, we study graphs, called emph{(n, d)-sigraphs}, inwhich every edge is $d$-labeled by an $n$-tuple$(a_1,a_2,...,a_n)$, where $a_k in {+,-}$, for $1leq k leqn$. In this paper, we give different notion of balance: symmetricbalance in a $(n,d)$-sigraph and obtain some characterizations.

Keywords

• Signed graphs
• Directional labeling
• Complementation
• Balance