Smarandachely t-path step signed graphs

A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S = (G;ae) (S = (G;")) where G = (V;E) is a graph called underlying graph of S and ae : E ! ( e1; e2;¢¢¢ ; ek) (" : V ! ( e1; e2;¢¢¢ ; ek)) is a function, where each ei 2 f+;ig. Particularly, a Smarandachely 2-signed graph or Smarandachely 2-marked graph is called abbreviated a signed graph or a marked graph. E. Prisner (9) in his book Graph Dynamics deflnes the t-path step operator on the class of flnite graphs. Given a graph G and a positive integer t, the t-path step graph (G)t of G is formed by taking a copy of the vertex set V (G) of G, joining two vertices u and v in the copy by a single edge e = uv whenever there exists a uiv path of length t in G. Analogously, one can deflne the Smarandachely t-path step signed graph (S)t = ((G)t;ae 0 ) of a signed graph S = (G;ae) is a signed graph whose underlying graph is (G)t called t-path step graph and sign of any edge e = uv in (S)t is "(u)"(v). It is shown that for any signed graph S, its (S)t is balanced. We then give structural characterization of Smarandachely t-path step signed graphs. Further, in this paper we characterize signed graphs which are switching equivalent to their Smarandachely 3-path step signed graphs.