An Upper Bound on The Largest Signless Laplacian of an Odd Unicyclic Graph


  • Macarena Collao Universidad Católica del Norte.
  • Pamela Pizarro Universidad Católica del Norte.
  • Oscar Rojo Universidad Católica del Norte.



Laplacian matrix, signless Laplacian matrix, adjacency matrix, spectral radius, generalized Bethe tree, matriz laplaciana, matriz laplaciana sin signo, matriz de adyacencia, radio espectral, árbol de Bethe generalizado.


We derive an upper bound on the largest signless Laplacian eigen­value of an odd unicyclic graph. The bound is given in terms of the largest vertex degree and the largest height of the trees obtained re­moving the edges of the unique cycle in the graph.

Author Biographies

Macarena Collao, Universidad Católica del Norte.

Departamento de Matemáticas.

Pamela Pizarro, Universidad Católica del Norte.

Departamento de Matemáticas.

Oscar Rojo, Universidad Católica del Norte.

Departamento de Matemáticas.


[1] D. Cvetkovic, P. Rowlinson, S. K. Simic, Signless Laplacian of finite graphs, Linear Algebra Appl. 423, pp. 155-171, (2007).

[2] 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).

[3] R. Diestel, Graph Theory, Electronic Editions 2005, Springer-Verlag Hiedlberg, New York.

[4] G. H. Golub and C. F. van Loan, Matrix Computations, 2nd ed., John Hopkins University Press, (1989).

[5] C. S. Oliveira, L. S. de Lima, N. M. M. de Abreu, P. Hansen, Bounds on the index of the signless Laplacian of a graph, Discrete Applied Mathematics 158, pp. 355-360, (2010).

[6] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, (1991).

[7] S. Hu, The largest eigenvalue of unicyclic graphs, Discrete Math. 307, pp. 280-284, (2007).

[8] Y. Ikebe, T. Inagaki, S. Miyamoto, The Monotonicity Theorem, Cauchy’s Interlace Theorem and Courant-Fisher Theorem, American Mathematical Monthly Vol. 94, No. 4, April, pp. 352-354, (1987).

[9] S. Kouachi, Eigenvalues and eigenvectors of tridiagonal matrices, Electronic Journal of Linear Algebra 15, pp. 115-133, (2006).

[10] O. Rojo, New upper bounds on the spectral radius of unicyclic graphs, Linear Algebra Appl. 428, pp. 754-764, (2008).

[11] O. Rojo, Spectra of copies of a generalized Bethe tree atached to any graph, Linear Algebra Appl. 431, pp. 863-882, (2009).

[12] L. N. Trefethen and D. Bau, III Numerical Linear Algebra, Society for Industrial and Applied Mathematics, (1997).

[13] R. Varga, Matrix Iterative Analysis, Theory, Prentice-Hall, Inc., (1965).



How to Cite

M. Collao, P. Pizarro, and O. Rojo, “An Upper Bound on The Largest Signless Laplacian of an Odd Unicyclic Graph”, Proyecciones (Antofagasta, On line), vol. 31, no. 1, pp. 39-49, Jan. 2012.