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

‎Let G=(V‎, ‎E) be a graph‎. ‎By \emph{directional labeling (or‎ ‎d-labeling)} of an edge x=uv of G by an ordered n-tuple‎ ‎(a_۱,a_۲,\dots,a_n)‎, ‎we mean a labeling of the edge x such that‎ ‎we consider the label on uv as (a_۱,a_۲,\dots,a_n) in the‎ ‎direction from u to v‎, ‎and the label on x as‎ ‎(a_{n},a_{n-۱},\dots,a_۱) in the direction from v to u‎. ‎In‎ ‎this paper‎, ‎we study graphs‎, ‎called \emph{(n‎,d)-sigraphs}‎, ‎in‎ ‎which every edge is d-labeled by an n-tuple‎ ‎(a_۱,a_۲,\dots,a_n)‎, ‎where a_k \in \{+,-\}‎, ‎for ۱\leq k \leq‎ ‎n‎. ‎In this paper‎, ‎we give different notion of balance‎: ‎symmetric‎ ‎balance in a (n,d)-sigraph and obtain some characterizations‎.

Keywords:

Signed graphs , Directional labeling , Complementation , Balance