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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

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

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


Eber Lenes

Universidad del Sini


Oscar Rojo

Universidad Católica del Norte



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.


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Nair Abreu

Production Engineering Program, PEP/COPPE

Universidade Federal do Rio de Janeiro Rio de Janeiro,


e-mail : Eber Lenes

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



Oscar Rojo

Department of Mathematics Universidad Católica del Norte Antofagasta,



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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