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 2 f+; ??g, for 1  k  n. In this paper, we give dierent notion of balance: symmetric balance in a (n; d)-sigraph and obtain some characterizations.