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

Document Type: Research Paper

Authors

1 Dept. of Mathematics, Siddaganga Institute of Technology, B.H.Road,Tumkur-572103, India.

2 Berhampur University

Abstract

‎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 ‎$(a_1,a_2,\dots,a_n)$‎, ‎we mean a labeling of the edge $x$ such that‎ ‎we consider the label on $uv$ as $(a_1,a_2,\dots,a_n)$ in the‎ ‎direction from $u$ to $v$‎, ‎and the label on $x$ as‎ ‎$(a_{n},a_{n-1},\dots,a_1)$ 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 ‎$(a_1,a_2,\dots,a_n)$‎, ‎where $a_k \in \{+,-\}$‎, ‎for $1\leq k \leq‎ ‎n$‎. ‎In this paper‎, ‎we give different notion of balance‎: ‎symmetric‎ ‎balance in a $(n,d)$-sigraph and obtain some characterizations‎.

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