EXTREMAL IOTA ENERGY OF A SUBCLASS OF TRICYCLIC DIGRAPHS AND SIDIGRAPHS

The iota energy of an n-vertex digraph D is defined by Ec () = ∑ 1 |Im( k)|, where z1, . . ., zn are eigenvalues of D and Im(zk) is the imaginary part of eigenvalue zk . The iota energy of an n-vertex sidigraph can be defined analogously. In this paper, we define a class Fn of n-vertex tricyclic digraphs containing five linear subdigraphs such that one of the directed cycles does not share any vertex with the other two directed cycles and the remaining two directed cycles are of same length sharing at least one vertex. We find the digraphs in Fn with minimal and maximal iota energy. We also consider a similar class of tricyclic sidigraphs and find extremal values of iota energy among the sidigraphs in this class

Energy of graphs and digraphs.

Spectral graph theory (Algebraic graph theory) which emerged in 1950s and 1960s is the study of properties of a graph in relationship to the characteristic polynomial, eigenvalues and eigenvectors of matrices associated to the graph. The major source of research in spectral graph theory has been the study of relationship between the structural and spectral properties of graphs. Another source has research in quantum chemistry. The 1980 monograph `spectra of graphs' by Cvetkovi,c, Doob and Sach summarised nearly all research to date in the area. In 1988 it was updated by the survey `Recent results in the theory of graph spectra'. The third edition of spectra of graphs (1995) contains a summary of the further contributions to the subject. Since then the theory has been developed to a greater extend and many research papers have been published. It is important to mention that spectral graph theory has a wide range of applications to other areas of mathematics and to other areas of sciences which include Computer Science, Physics, Chemistry, Biology, Statistics etc.Digital copy of ThesisUniversity of Kashmi