δ -Complement of a Graph

Amrithalakshmi Pai, Harshitha A. Rao, Sabitha D’Souza, Pradeep G. Bhat and Shankar Upadhyay
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Amrithalakshmi Pai: Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal 576104, Karnataka, India
Harshitha A. Rao: Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal 576104, Karnataka, India
Sabitha D’Souza: Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal 576104, Karnataka, India
Pradeep G. Bhat: Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal 576104, Karnataka, India
Shankar Upadhyay: Department of Mathematics, Maharaja Institute of Technology, Mysore 571438, Karnataka, India

Mathematics, 2022, vol. 10, issue 8, 1-11

Abstract: 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.

Keywords: ? -complement; ? ?-complement; self-complementary; degree sequence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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