An upper bound for the energy of radial digraphs

Let D be a digraph with n vertices, a arcs, c(2) closed walks of length 2 and spectral radius rho(D). Recently Ayyaswamy, Balachandran and Gutman (2011) [1] proved that when rho(D) >= a + c(2)/2n >= 1 it is possible to construct an upper bound for the energy of digraphs, which improves the McClelland inequality for the energy of strongly connected digraphs given in Rada (2009) [16], It is our interest in this paper to show that for general digraphs the inequality rho(D) >= a + c(2)/2n does not hold. However, we introduce the class of radial digraphs, that satisfy the spectral radius condition above, and for this class of digraphs we improve the bound for the energy given in Ayyaswamy et al. (2011) [1]. (C) 2013 Elsevier Inc. All rights reserved.