The largest Laplacian and adjacency indices of complete caterpillars of fixed diameter

Nair Abreu *

Universidade Federal do Ro de Janeiro

Brasil

Eber Lenes †

Universidad del Sinu

Colombia

Oscar Rojo ‡

Universidad Católica del Norte

Chile

---

ABSTRACT

A complete caterpillar is a caterpillar in which each internal vertex is a quasi-pendent vertex. In this paper, in the class of all complete caterpillars on n vertices and diameter d, the caterpillar attaining the largest Laplacian index is determined. In addition, it is proved that this caterpillar also attains the largest adjacency index.

Keywords : Caterpillar, Laplacian matrix, Laplacian index, adjacency matrix, index, spectral radius.

2010 AMS classification : 05C50.

---

REFERENCES

[1]    D. Cvetkovic, P. Rowlinson, S. K. Simic, An Introduction to the Theory of Graph Spectra, London Mathematical Society, Student Texts 75, Cambridge University Press, (2010).

[2]    J.M. Guo, J-Y. Shao, On the spectral radius of trees with fixed diameter, Linear Algebra and its Applications 413, pp. 131-147, (2006).

[3]    J.- M. Guo, On the Laplacian spectral radius of trees with fixed diameter, Linear Algebra Appl. 419, pp. 618-629, (2006).

[4]    J.- M. Guo, The effect on the Laplacian spectral radius of a graph by adding or grafting edges, Linear Algebra Appl. 413, pp. 59-71, (2006).

[5]    O. Rojo, L. Medina, N. Abreu and C. Justel, On the algebraic connectivity of some caterpillars: A sharp upper bound and a total ordering. Linear Algebra and its Applications 432, pp. 586-605, (2010).

[6]    O. Rojo, L. Medina, N. Abreu and C. Justel, Extremal algebraic connectivities of certain caterpillar classes and symmetric caterpillars, Electronic Journal of Linear Algebra, Vol. 20, pp. 136-157, (2010).

[7]    O. Rojo, L. Medina, Spectra of generalized Bethe trees attached to a path, Linear Algebra and its Applications 430, pp. 483-503, (2009).

[8]    O. Rojo, L. Medina, Spectra of weighted compound graphs of generalized Bethe trees, Electronic Journal of Linear Algebra 18, pp. 30-57, (2009).

[9]    S. K. Simic, E. M. L. Marzi, F. Belardo, On the index of caterpillars, Discrete Mathematics 308, pp. 324-330, (2008).

[10] R. S. Varga, Matrix Iterative Analysis, Springer Series in Computational Mathematics, Volume 27, (2000).

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

and

O. Rojo

Department of Mathematics Universidad Católica del Norte Antofagasta,

Chile

e-mail : orojo@ucn.cl

Received : January 2015. Accepted : May 2015

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

†thanks the support of Project Mecesup UCN 0711 and Project Fondecyt Regular 1130135, Chile.

‡thanks the 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.