A note on antipodal signed graphs

A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S = (G, [sigma]) (S = (G, [micro])) where G = (V,E) is a graph called underlying graph of S and [sigma] : E [right arrow] ([[bar.e].sub.1], [[bar.e].sub.2], ..., [[bar.e].sub.k]) ([micro] : V [right arrow] ([[bar.e].sub.1], [[bar.e].sub.2], ..., [[bar.e].sub.k])) is a function, where each [[bar.e].sub.i] [member of] {+,-}. Particularly, a Smarandachely 1-signed graph or Smarandachely 1-marked graph is called abbreviated a signed graph or a marked graph. Singleton (1968) introduced the concept of the antipodal graph of a graph G, denoted by A(G), is the graph on the same vertices as of G, two vertices being adjacent if the distance between them is equal to the diameter of G. Analogously, one can define the antipodal signed graph A(S) of a signed graph S = (G, [sigma]) as a signed graph, A(S) = (A(G), [sigma]'), where A(G) is the underlying graph of A(S), and for any edge e = uv in A(S), [sigma]'(e) = [micro](u)[micro](v), where for any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is shown that for any signed graph S, its A(S) is balanced and we offer a structural characterization of antipodal signed graphs. Further, we characterize signed graphs [bar.S] for which S ~ A(S) and [bar.S] ~ A(S) where ~ denotes switching equivalence and A(S) and [bar.S] are denotes the antipodal signed graph and complementary signed graph of S respectively. Key Words: Smarandachely k-signed graphs, Smarandachely k-marked graphs, signed graphs, marked graphs, balance, switching, antipodal signed graphs, complement, negation. AMS(2010): 05C22