On The Bound For The Spectral Radius And Energy Of A Digraph

Assume that D is a simple digraph with n vertices. The adjacency matrix A= (aij) of D is the n×n matrix. The eigenvalues z1, z2,..., zn of A are called the eigenvalues of D and form the spectrum of D. The eigenvalues of D are in general complex numbers as the adjacency matrix A of D is not necessarily a symmetric matrix. The spectral radius of D, denoted by p(D), equals to the largest absolute value of an eigenvalue of D. The energy of a digraph is defined as where z* is the eigenvalue of the digraph D and Re(zi) denotes the real part of zi, i＝1,2,….n.In this thesis, we first give an improved lower bound for the spectral radius of a digraph D. Applying this result, we obtain some sharp upper bounds on the energy of D and characterize some extreme digraphs which attain these sharp upper bounds. Theoretical analysis shows that these results improve and generalize the known results. In addition, we also present a lower bound on the spectral radius of a digraph D with parametric a. These results improve some known results.