Cospectral and hyper-energetic self complementary comparability graphs

  • Merajuddin, Merajuddin (Department of Applied Mathematics, Faculty of Engg. & Tech. AMU) ;
  • Kirmani, S.A.K. (Department of Applied Mathematics, Faculty of Engg. & Tech. AMU) ;
  • Ali, Parvez (Department of Applied Mathematics, Faculty of Engg. & Tech. AMU) ;
  • Pirzada, S. (Department of Mathematics, University of Kashmir)
  • Published : 2007.09.30

Abstract

A graph G is self-complementary (sc) if it is isomorphic to its complement. G is perfect if for all induced subgraphs H of G, the chromatic number of H (denoted ${\chi}$(H)) equals the number of vertices in the largest clique in H (denoted ${\omega}$(H)). An sc graph which is also perfect is known as sc perfect graph. A comparability graph is an undirected graph if it can be oriented into transitive directed graph. An sc comparability (scc) is clearly a subclass of sc perfect graph. In this paper we show that no two non-isomorphic scc graphs with n vertices each, (n<13) have same spectrum, and that the smallest positive integer for which there exists hyper-energetic scc graph is 13.