δ -Complement of a Graph

Let G ( V , X ) be a finite and simple graph of order n and size m . The complement of G , denoted by G ¯ , is the graph obtained by removing the lines of G and adding the lines that are not in G . A graph is self-complementary if and only if it is isomorphic to its complement. In this paper, we define δ -complement and δ ′ -complement of a graph as follows. For any two points u and v of G with deg u = deg v remove the lines between u and v in G and add the lines between u and v which are not in G . The graph thus obtained is called δ -complement of G . For any two points u and v of G with deg u ≠ deg v remove the lines between u and v in G and add the lines between u and v that are not in G . The graph thus obtained is called δ ′ -complement of G . The graph G is δ ( δ ′ ) -self-complementary if G ≅ G δ ( G ≅ G δ ′ ) . The graph G is δ ( δ ′ ) -co-self-complementary if G δ ≅ G ¯ ( G δ ′ ≅ G ¯ ) . This paper presents different properties of δ and δ ′ -complement of a given graph.