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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.34 no.4 Antofagasta dic. 2015

http://dx.doi.org/10.4067/S0716-09172015000400006 

Proyecciones Journal of Mathematics Vol. 34, No 4, pp. 379-390, December 2015. Universidad Católica del Norte Antofagasta - Chile

Computing the maximal signless Laplacian index among graphs of prescribed order and diameter

Nair Abreu *

Universidad Federal de Rio de Janeiro

Brasil

Eber Lenes

Universidad del Sini

Colombia 

Oscar Rojo

Universidad Católica del Norte

Chile.


ABSTRACT

A bug Bugp,r1r2 is a graph obtained from a complete graph Kp by deleting an edge uv and attaching the paths Pri and Pr2 by one of their end vertices at u and v, respectively. Let Q(G) be the signless Laplacian matrix of a graph G and q1(G) be the spectral radius of Q(G). It is known that the bug maximizes q1(G) among all graphs G of order n and diameter d. For a bug B of order n and diameter d, n - d is an eigenvalue of Q(B) with multiplicity n - d - 1. In this paper, we prove that remainder d +1 eigenvalues of Q(B), among them q1(B), can be computed as the eigenvalues of a symmetric tridiagonal matrix of order d +1. Finally, we show that q1(B0) can be computed as the largest eigenvalue of a symmetric tridiagonal matrix of order whenever d is even.

Keyword : Signless Laplacian index, diameter, bug, H-join.

2000 AMS classification : 05C50, 05C35, 15A18.


REFERENCES

[1]    D. M. Cardoso, M. A. A. de Freitas, E. Martins., M. Robbiano, Spectra of graphs obtained by a generalization of the join graph operation, Discrete Mathematics 313, pp. 733-741, (2013).         [ Links ]

[2]    D. M. Cardoso, E. Martins., M. Robbiano, O. Rojo, Eigenvalues of a H-generalized operation constrained by vertex subsets, Linear Algebra Appl. 438, pp. 3278-3290, (2013).         [ Links ]

[3]    H. Liu, M. Lu, A conjecture on the diameter and signless Laplacian index of graphs, Linear Algebra Appl. 450, pp. 158-174, (2014).         [ Links ]

[4]    D. Cvetkovic, P. Rowlinson, S.K. Simic, Eigenvalue bounds for the signless Laplacian, Publications de L’Institute Mathematique, Nouvelle serie, tome 81 (95), pp. 11-27, (2007).

[5]    P. Hansen, C. Lucas, Bounds and conjectures for the signless Laplacian index of graphs, Linear Algebra Appl. 432, pp. 3319-3336, (2010).         [ Links ]

[6]    Miao-Lin Ye, Yi-Zheng Fan, Hai Feng Wang, Maximizing signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity, Linear Algebra Appl. 433, pp. 1180-1186, (2010).         [ Links ]

[7]    G. Yu, On the maximal signless Laplacian spectral radius of graphs with given matching number, Proc. Japan Acad. Ser. A 84, pp. 163166, (2008).         [ Links ]

Nair Abreu

Production Engineering Program, PEP/COPPE

Universidade Federal do Rio de Janeiro Rio de Janeiro,

Brazil

e-mail : nairabreunovoa@gmail.com Eber Lenes

Departamento de Investigaciones Universidad del Sinu. Elias Bechara Zainum Cartagena,

Colombia

e-mail: elenes@ucn.cl

Oscar Rojo

Department of Mathematics Universidad Católica del Norte Antofagasta,

Chile

e-mail: orojo@ucn.cl

Received : July 2015. Accepted : August 2015

*Thanks the support of Grant 305372/2009-2, CNPq, Brazil.

†Thanks the support of Projects Mecesup UCN 0711, Mecesup UCN 1102 and Fonde-cyt Regular 1130135

‡Thanks he support of Project Fondecyt Regular 1130135, Chile, and the hospitality of the Center For Mathematical Modeling, Universidad de Chile, Chile, in which this research was finished.

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