On the spectral radius of weighted digraphs

Abstract

We consider the weighted digraphs in which the arc weights are positive definite matrices. We obtain some upper bounds for the spectral radius of these digraphs and characterize the digraphs achieving the upper bounds. Some known upper bounds are then special cases of our results.

Spectral radius and signless Laplacian spectral radius of strongly connected digraphs

“…, by directly calculating, we see that the bound of (2.1) is better than the bounds of (1.1)-(1.4) in [5] (see Table 1) because…”

“…Then q(D 1 ) = 3 + √ 3 by Theorem 2.7. We can see from Table 1 that the bound of (2.1) is better than the bounds of (1.1)-(1.4) in [5] because q(…”

“…Then we give a new sharp upper bound of the signless Laplacian spectral radius among all simple digraphs and compare them with the upper bounds given in [5] as inequalities (1.1)-(1.4). The technique used in the result is motivated by [8,17] et al By the definition of Q(D), the i-th row sum of Q(D) is 2d + i .…”

“…Recently, R.A. Brualdi wrote a stimulating survey on the spectra of digraphs [4]. Furthermore, some upper or lower bounds on the spectral radius or the signless Laplacian spectral radius of digraphs were obtained by the outdegrees and the average 2outdegrees [5,8]. Some extremal digraphs which attain the maximum or minimum spectral radius and the distance spectral radius of digraphs with given parameters, such as given connectivity, given arc connectivity, given dichromatic number, given clique number, given girth and so on, were characterized, see e.g.…”

“…In [5], S. Burcu Bozkurt and Durmus Bozkurt gave some sharp upper and lower bounds for the signless Laplacian spectral radius as follows:…”