Discussiones Mathematicae Graph Theory 32(2) (2012) 255-261
doi: 10.7151/dmgt.1612

[BIBTex] [PDF] [PS]

The Laplacian Spectrum of Some Digraphs Obtained from the Wheel

Li Su, Hong-Hai Li and Liu-Rong Zheng

College of Mathematics and Information Science
Jiangxi Normal University
Nanchang, 330022, P.R. China


The problem of distinguishing, in terms of graph topology, digraphs with real and partially non-real Laplacian spectra is important for applications. Motivated by the question posed in [R. Agaev, P. Chebotarev, Which digraphs with rings structure are essentially cyclic?, Adv. in Appl. Math. 45 (2010), 232-251], in this paper we completely list the Laplacian eigenvalues of some digraphs obtained from the wheel digraph by deleting some arcs.

Keywords: digraph, Laplacian matrix, eigenvalue, wheel

2010 Mathematics Subject Classification: 05C50, 15A18.


[1]R. Agaev and P. Chebotarev, Which digraphs with rings structure are essentially cyclic?, Adv. in Appl. Math. 45 (2010) 232--251, doi: 10.1016/j.aam.2010.01.005.
[2]R. Agaev and P. Chebotarev, On the spectra of nonsymmetric Laplacian matrices, Linear Algebra Appl. 399 (2005) 157--168, doi: 10.1016/j.laa.2004.09.003.
[3]W.N. Anderson and T.D. Morley, Eigenvalues of the Laplacian of a graph, Linear Multilinear Algebra 18 (1985) 141--145, doi: 10.1080/03081088508817681.
[4]J.S. Caughman and J.J.P. Veerman, Kernels of directed graph Laplacians, Electron. J. Combin. 13 (2006) R39.
[5]P. Chebotarev and R. Agaev, Forest matrices around the Laplacian matrix, Linear Algebra Appl. 356 (2002) 253--274, doi: 10.1016/S0024-3795(02)00388-9.
[6]P. Chebotarev and R. Agaev, Coordination in multiagent systems and Laplacian spectra of digraphs, Autom. Remote Control 70 (2009) 469--483, doi: 10.1134/S0005117909030126.
[7]C. Godsil and G. Royle, Algebraic Graph Theory (Springer Verlag, 2001).
[8]A.K. Kelmans, The number of trees in a graph I, Autom. Remote Control 26 (1965) 2118--2129.
[9]R. Merris, Laplacian matrices of graphs: A survey, Linear Algebra Appl. 197/198 (1994) 143--176, doi: 10.1016/0024-3795(94)90486-3.
[10]R. Olfati-Saber, J.A. Fax and R.M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE 95 (2007) 215--233, doi: 10.1109/JPROC.2006.887293.

Received 10 February 2011
Revised 10 May 2011
Accepted 10 May 2011