Directionally $n$-Signed Graphs-III: The Notion of Symmetric Balance

Let G = (V, E) be a graph. By directional labeling (or d-labeling) of an edge x = uv of G by an ordered n-tuple (a1, a2, . . . , an), we mean a labeling of the edge x such that we consider the label on uv as (a1, a2, . . . , an) in the direction from u to v, and the label on x as (an, an−1, . . . , a1) in the direction from v to u. In this paper, we study graphs, called (n, d)-sigraphs, in which every edge is d-labeled by an n-tuple (a1, a2, . . . , an), where ak ∈ {+,−}, for 1 ≤ k ≤ n. In this paper, we give different notion of balance: symmetric balance in a (n, d)-sigraph and obtain some characterizations.