Improved upper and lower bounds for the spectral radius of digraphs A. Dilek Güngör a , Kinkar Ch. Das b, * a Selçuk Üniversity, Science Faculty, Department of Mathematics, 42031 Konya, Turkey b Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea article info Keywords: Digraph Spectral radius Bound abstract Let G be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius qðGÞ of G is the largest eigenvalue of its adjacency matrix. In this paper, sharp upper and lower bounds on qðGÞ are given. We show that some known bounds can be obtained from our bounds. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction Let G ¼ðV ; EÞ be a digraph with n vertices and m arcs without loops and multiarcs, V ¼fv 1 ; v 2 ; ... ; v n g. Let ðu; v Þ be an arc of G. Then u is called the initial vertex and v is called the terminal vertex of this arc. The outdegree d þ i of a vertex v i in the digraph G is defined to be the number of arcs in G with initial vertex v i . Let T þ i be the sum of the outdegrees of all the vertices in N þ ðv i Þ¼fv j : v i v j 2 Eg, and call it 2-outdegree. Furthermore, we call t þ i ¼ T þ i d þ i average 2-outdegree, 1 6 i 6 n. If the average 2-outdegrees of vertices in V are the same, we call G average 2-outdegree regular digraph. If V ¼ U [ W and the average 2-outdegrees of the vertices in U and W are t þ 1 and t þ 2 , respectively, we call G average 2-outdegree semiregular digraph. The spectral radius qðGÞ of G is defined to be the largest eigenvalue of its adjacency matrix AðGÞ. For applications it is cru- cial to be able to compute or at least estimate qðGÞ for a given digraph G. This is a classical problem with numerous results pertaining to it (see [1,2]). Here we list some known upper and lower bounds on the spectral radius of a digraph [4]: (1) min d þ i : v i 2 V   6 qðGÞ 6 max d þ i : v i 2 V   : ð1Þ Equality holds if and only if G is outdegree regular. (2) min t þ i : v i 2 V   6 qðGÞ 6 max t þ i : v i 2 V   : ð2Þ Equality holds if and only if G is average 2-outdegree regular. In [6], Zhang gave the following bounds for qðGÞ: (3) min ffiffiffiffiffiffi T þ i q : v i 2 V   6 qðGÞ 6 max ffiffiffiffiffiffi T þ i q : v i 2 V   : ð3Þ 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.080 * Corresponding author. E-mail addresses: drdilekgungor@gmail.com (A.D. Güngör), kinkar@lycos.com, kinkar@mailcity.com (K.Ch. Das). Applied Mathematics and Computation 216 (2010) 791–799 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc