Research and Innovation Articles
Published 2020-01-22
Keywords
- Caterpillar,
- diameter,
- distance,
- index,
- tree
How to Cite
Lenes, E., García, H., Figueroa, A., & Mercado, F. (2020). Extremal graphs for alpha-index. Revista Integración, Temas De matemáticas, 38(1), 15–31. https://doi.org/10.18273/revint.v38n1-202002
Copyright (c) 2020 Revista Integración, temas de matemáticas
This work is licensed under a Creative Commons Attribution 4.0 International License.
Abstract
Let N(G) be the number of vertices of the graph G. Let Pl(Bi) be the tree obtained of the path Pl and the trees B1,B2, ...,Bl by identifying the root vertex of Bi with the i-th vertex of Pl. Let Vmn = {Pl(Bi) : N(Pl(Bi)) =n; N(Bi) ≥ 2; l ≥ m}. In this paper, we determine the tree that has the largest \alpha-index among all the trees in Vmn.
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References
[1] Abreu N., Lenes E. and Rojo O.,“The largest Laplacian and adjacency index of complete caterpillars of fixed diameter”, Proyecciones 34 (2015), No.2, 175–190.
[2] Cardoso D.M., Cvetković D., Rowlinson P. and Simić S.K., “A sharp lower bound for the least eigenvalue of the signless Laplacian of a non-bipartite graph”, Linear Algebra Appl. 429 (2008), No. 11-12, 2770–2780.
[3] Cvetković D., Rowlinson P. and Simić S. K., An Introduction to the Theory of Graph Spectra, London Mathematical Society Student Texts 75, Cambridge University Press, Cambridge, 2010.
[4] Cvetković D., Rowlinson P. and Simić S.K., “Signless Laplacian of finite graphs ”, Linear Algebra Appl. 423 (2007), No. 1, 155–171.
[5] Cvetković D. and Simić S.K., “Towards a spectral theory of graphs based on the signless
Laplacian I ”, Publ. Inst. Math. (Beograd) 85(99) (2009), 19–33.
[6] Cvetković D. and Simić S.K., “Towards a spectral theory of graphs based on the signless Laplacian II”, Linear Algebra Appl. 432 (2010), No. 9, 2257–2272.
[7] Grone R. and Merris R.,“The Laplacian spectrum of a graph II”, SIAM J. Discrete Math. 7 (1994), No. 2, 221–229.
[8] Grone R., Merris R. and Sunder V.S.,“The Laplacian spectrum of a graph”, SIAM J. Matrix Anal. Appl. 11 (1990), No. 2, 218–238.
[9] Guo J.M., “On the Laplacian spectral radius of trees with fixed diameter”, Linear Algebra Appl. 419 (2006), No. 2–3, 618–629.
[10] Guo J.M. and Shao J.Y., “On the spectral radius of trees with fixed diameter”, Linear Algebra Appl. 413 (2006), No. 1, 131–147.
[11] Merris R., “Laplacian matrices of graphs: a survey”, Linear Algebra Appl. 197–198 (1994), 143–176.
[12] Nikiforov V., “Merging the A-and Q-spectral theories”, Appl. Anal. Discrete Math. 11 (2017), No. 1, 81–107.
[13] Nikiforov V., Pastén G., Rojo O. and Soto R. L., “On the A-spectra of trees”, Linear Algebra Appl. 520 (2017), 286–305.
[14] Nikiforov V. and Rojo O.,“On the alpha-index of graphs with pendent paths”, Linear Algebra Appl. 550 (2018), 87–104.
[15] Rojo O. and Medina L., “Spectra of generalized Bethe trees attached to a path”, Linear Algebra Appl. 430 (2009), No. 1, 483–503.
[16] Rojo O. and Medina L., “Spectra of weighted compound graphs of generalized Bethe trees”,Electron. J. Linear Algebra18 (2009), 30–57.
[17] Rojo O., Medina L., Abreu N. and Justel C., “Extremal algebraic connectivities of certain caterpillar classes and symmetric caterpillars”,Electron. J. Linear Algebra 20 (2010), 136–157.
[18] Rojo O., Medina L., Abreu N. and Justel C., “On the algebraic connectivity of some caterpillars: A sharp upper bound and a total ordering”, Linear Algebra Appl. 432 (2010), No. 2–3, 586–605.
[19] Simić S. K., Li Marzi E. M. and Belardo F.,“On the index of caterpillars”,Discrete Math. 308 (2008), No. 2–3, 324–330.
[20] Varga R.S., Gershgorin and his Circles, Springer Series in computational Mathematics, Springer Verlag, Berlin, 2004.
[21] Varga R.S., Matrix Iterative Analysis, Prentice-Hall Inc., 1965.
[22] Varga R.S., Matrix Iterative Analysis, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2000.
[23] Xue J., Lin H., Liu S. and Shu J., “On the A-spectral radius of a graph”, Linear Algebra Appl. 550 (2018), 105–120.
[2] Cardoso D.M., Cvetković D., Rowlinson P. and Simić S.K., “A sharp lower bound for the least eigenvalue of the signless Laplacian of a non-bipartite graph”, Linear Algebra Appl. 429 (2008), No. 11-12, 2770–2780.
[3] Cvetković D., Rowlinson P. and Simić S. K., An Introduction to the Theory of Graph Spectra, London Mathematical Society Student Texts 75, Cambridge University Press, Cambridge, 2010.
[4] Cvetković D., Rowlinson P. and Simić S.K., “Signless Laplacian of finite graphs ”, Linear Algebra Appl. 423 (2007), No. 1, 155–171.
[5] Cvetković D. and Simić S.K., “Towards a spectral theory of graphs based on the signless
Laplacian I ”, Publ. Inst. Math. (Beograd) 85(99) (2009), 19–33.
[6] Cvetković D. and Simić S.K., “Towards a spectral theory of graphs based on the signless Laplacian II”, Linear Algebra Appl. 432 (2010), No. 9, 2257–2272.
[7] Grone R. and Merris R.,“The Laplacian spectrum of a graph II”, SIAM J. Discrete Math. 7 (1994), No. 2, 221–229.
[8] Grone R., Merris R. and Sunder V.S.,“The Laplacian spectrum of a graph”, SIAM J. Matrix Anal. Appl. 11 (1990), No. 2, 218–238.
[9] Guo J.M., “On the Laplacian spectral radius of trees with fixed diameter”, Linear Algebra Appl. 419 (2006), No. 2–3, 618–629.
[10] Guo J.M. and Shao J.Y., “On the spectral radius of trees with fixed diameter”, Linear Algebra Appl. 413 (2006), No. 1, 131–147.
[11] Merris R., “Laplacian matrices of graphs: a survey”, Linear Algebra Appl. 197–198 (1994), 143–176.
[12] Nikiforov V., “Merging the A-and Q-spectral theories”, Appl. Anal. Discrete Math. 11 (2017), No. 1, 81–107.
[13] Nikiforov V., Pastén G., Rojo O. and Soto R. L., “On the A-spectra of trees”, Linear Algebra Appl. 520 (2017), 286–305.
[14] Nikiforov V. and Rojo O.,“On the alpha-index of graphs with pendent paths”, Linear Algebra Appl. 550 (2018), 87–104.
[15] Rojo O. and Medina L., “Spectra of generalized Bethe trees attached to a path”, Linear Algebra Appl. 430 (2009), No. 1, 483–503.
[16] Rojo O. and Medina L., “Spectra of weighted compound graphs of generalized Bethe trees”,Electron. J. Linear Algebra18 (2009), 30–57.
[17] Rojo O., Medina L., Abreu N. and Justel C., “Extremal algebraic connectivities of certain caterpillar classes and symmetric caterpillars”,Electron. J. Linear Algebra 20 (2010), 136–157.
[18] Rojo O., Medina L., Abreu N. and Justel C., “On the algebraic connectivity of some caterpillars: A sharp upper bound and a total ordering”, Linear Algebra Appl. 432 (2010), No. 2–3, 586–605.
[19] Simić S. K., Li Marzi E. M. and Belardo F.,“On the index of caterpillars”,Discrete Math. 308 (2008), No. 2–3, 324–330.
[20] Varga R.S., Gershgorin and his Circles, Springer Series in computational Mathematics, Springer Verlag, Berlin, 2004.
[21] Varga R.S., Matrix Iterative Analysis, Prentice-Hall Inc., 1965.
[22] Varga R.S., Matrix Iterative Analysis, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2000.
[23] Xue J., Lin H., Liu S. and Shu J., “On the A-spectral radius of a graph”, Linear Algebra Appl. 550 (2018), 105–120.