Energy of digraphs and signed digraphs

Abstract

The eigenvalues of a graph are the roots of characteristic polynomial of its adjacency matrix. For an n-vertex graph G, the energy of graph is defined as E(G) = n P i=1|λi|, where |λi| is the absolute value of the eigenvalue λi. Later the idea of energy of graph was extended to the energy of digraphs. Since adjacency matrix of a digraph is not always symmetric, its eigenvalues lie in the complex plane. Let D be an n-vertex digraph and zi, i = 1,...,n be its eigenvalues. Then energy of digraph D is E(D) = n P i=1|Re(zi)|, where |Re(zi)| is the absolute value of the real part of the eigenvalue zi. In this dissertation, our focus is the energy of digraphs. We find the extremal energies of bicyclic digraphs, which have vertex-disjoint directed cycles. We also introduce a new notion of energy of digraphs which we call the iota energy of digraphs. We find unicyclic digraphs with extremal iota energies among the class of unicyclic digraphs with fixed order. We calculate the iota energy formulae for the directed cycles. We define the complex adjacency matrix for the digraphs. We study the integral representation of iota energy of digraphs. We show that the Coulson’s integral formula remain valid for iota energy. We also study the iota equienergetic digraphs. We find two families of digraphs which has same iota energy. Furthermore, we find digraphs with extremal iota energy in the set of vertex-disjoint bicyclic digraphs. In the end, we find the extremal energy of vertex-disjoint bicyclic signed digraphs.