Smarandache directionally n-signed graphs--a survey

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 ([a.sub.1], [a.sub.2], ... , [a.sub.n]), we mean a labeling of the edge x such that we consider the label on uv as ([a.sub.1], [a.sub.2], ... , [a.sub.n]) in the direction from u to v, and the label on x as ([a.sub.n], [a.sub.n-1], ... , [a.sub.1]) in the direction from v to u. In this survey, we study graphs, called (n, d)-sigraphs, in which every edge is d-labeled by an n-tuple ([a.sub.1], [a.sub.2], ... , [a.sub.n]), where [a.sub.k] ∈ {+, -}, for 1 [less than or equal to] k ≤ n. Several variations and characterizations of directionally n-signed graphs have been proposed and studied. These include the various notions of balance and others. Key Words: Signed graphs, directional labeling, complementation, balance. AMS(2010): 05C 22