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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.35 no.4 Antofagasta Dec. 2016

http://dx.doi.org/10.4067/S0716-09172016000400003 

Energy of strongly connected digraphs whose underlying graph is a cycle

 

Juan Monsalve

Juan Rada 

Universidad de Antioquia

Colombia 


ABSTRACT

The energy ofa digraph is defined as E (D) =∑1n|Re (zk)|, where z1,..., zn are the eigenvalues of the adjacency matrix of D. This is a generalization of the concept of energy introduced by I. Gutman in 1978 [3]. When the characteristic polynomial ofa digraph D is ofthe form

where bo (D) = 1 and bk(D) ≥ 0 for all k, we show that

This expression for the energy has many applications in the study of extremal values of the energy in special classes of digraphs. In this paper we consider the set D* (Cn) of all strongly connected digraphs whose underlying graph is the cycle Cn, and characterize those whose characteristic polynomial is ofthe form (0.1). As a consequence, we find the extremal values ofthe energy based on (0.2).

Keywords : digraphs; energy; characteristic polynomial; strongly connected; directed cycles.

AMS Subject Classification: 05C35; 05C50; 05C90.


REFERENCES

[1]    R. Cruz, H. Giraldo, J. Rada, An upper bound for the energy of radial digraphs, Linear Alg. Appl. 442, pp. 75-81, (2014).         [ Links ]

[2]    D. Cvetkovic, M. Doob, H. Sachs, Spectra of graphs - Theory and Application, Academic, New York, (1980).         [ Links ]

[3]    I. Gutman, The energy of a graph. Ber. Math.-Statist. Sekt. Forschungsz. Graz 103, pp. 1-22, (1978).         [ Links ]

[4]    X. Li, Y. Shi, I. Gutman, Graph energy, Springer-Verlag, New York, (2012).         [ Links ]

[5]    M. Mateljevic, V. BoZin, I. Gutman, Energy of a polynomial and the Coulson integral formula. J. Math. Chem. 48, pp. 1062-1068, (2010).         [ Links ]

[6]    I. Pena, J. Rada, Energy of digraphs, Lin. Multilin. Alg. 56, pp. 565579, (2008).         [ Links ]

[7]    J. Rada, I. Gutman, R. Cruz, The energy of directed hexagonal systems, Linear Alg. Appl. 439, pp. 1825-1833, (2013).         [ Links ]

Juan Monsalve

Instituto de Matematicas, Universidad de Antioquia Medellln,

Colombia

e-mail : jdmonsal@gmail.com

Juan Rada

Instituto de Matematicas, Universidad de Antioquia Medellín,

Colombia

e-mail : pablo.rada@udea.edu.co

Received : June 2015. Accepted : July 2016

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