On maximum degree (signless) Laplacian matrix of a graph
DOI:
https://doi.org/10.22199/issn.0717-6279-5434Keywords:
maximum degree matrix, maximum degree Laplacian matrix, maximum degree signless Laplacian matrixAbstract
Let G be a simple graph on n vertices and v1, v2, . . . , vn be the vertices of
G. We denote the degree of a vertex vi in G by dG(vi) = di. The maximum
degree matrix of G, denoted by M(G), is the real symmetric matrix with
its ijth entry equal to max{di, dj} if the vertices vi and vj are adjacent in
G, 0 otherwise. In analogous to the definitions of Laplacian matrix and
signless Laplacian matrix of a graph, we consider Laplacian and signless
Laplacian for the maximum degree matrix, called the maximum degree
Laplacian matrix and the maximum degree signless Laplacian matrix,
respectively. Also, we introduce maximum degree Laplacian energy and
maximum degree signless Laplacian energy of a graph. Then we determine
the maximum degree (signless) Laplacian energy of some graphs in terms
of ordinary energy, and (signless) Laplacian energy. We compute the
maximum degree (signless) Laplacian spectra of some graph compositions.
A lower and upper bound for the largest eigenvalue of the (signless) Laplacian
matrix is established and also we determine an upper bound for the second
smallest eigenvalue of maximum degree Laplacian matrix in terms of vertex
connectivity. We also determine bounds for the maximum degree (signless)
Laplacian energy in terms of first Zagreb index.
Downloads
References
N. Abreu, D. M. Cardoso, I. Gutman, E. A. Martins, and M. Robbiano, “Bounds for the signless Laplacian Energy”, Linear Algebra and its Applications, vol. 435, no. 10, pp. 2365–2374, 2011. https://doi.org/10.1016/j.laa.2010.10.021
C. Adiga and B. R. Rakshith, “Upper Bounds for the extended energy of graphs and some extended equienergetic graphs”, Opuscula Mathematica, vol. 38, no. 1, p. 5-13, 2018. https://doi.org/10.7494/opmath.2018.38.1.5
C. Adiga and M. Smitha, “On maximum degree energy of a graph”, International Journal of Contemporary Mathematical Sciences, vol. 4, pp. 385-396, 2009. [On line]. Available: https://bit.ly/3Nv3GqN
D. M. Cvetković, M. Doob and H. Sachs, Spectra of Graphs: Theory and Application. New York: Academic Press, 1980.
D. Cvetković and S. Simić, “Towards a spectral theory of graphs based on the signless Laplacian, III”, Applicable Analysis and Discrete Mathematics, vol. 4, no. 1, pp. 156–166, 2010. https://doi.org/10.2298/aadm1000001c
K. C. Das, I. Gutman, I. Milovanović, E. Milovanović, and B. Furtula, “Degree-based energies of graphs”, Linear Algebra and its Applications, vol. 554, pp. 185–204, 2018. https://doi.org/10.1016/j.laa.2018.05.027
K. Das and S. A. Mojalal, “On energy and Laplacian energy of graphs”, The Electronic Journal of Linear Algebra, vol. 31, pp. 167–186, 2016. https://doi.org/10.13001/1081-3810.3272
K. C. Das and S. A. Mojallal, “Relation between signless Laplacian Energy, energy of graph and its line graph”, Linear Algebra and its Applications, vol. 493, pp. 91–107, 2016. https://doi.org/10.1016/j.laa.2015.12.006
R. Frucht and F. Harary, “On the corona of two graphs”, Aequationes Mathematicae, vol. 4, no. 3, pp. 322–325, 1970. https://doi.org/10.1007/bf01844162
I. Gutman, “The energy of a graph”, Ber. Math.-Statist. Sekt. Forschungsz. Graz, vol. 103, pp. 1-22, 1978.
I. Gutman and B. Furtula, “Survey of graph energies”, Mathematics Interdisciplinary Research, vol. 2, pp. 85-129, 2017. [On line]. Available: https://bit.ly/3fxWKgb
I. Gutman, E. Milovanović, and I. Milovanović, “Beyond the Zagreb indices”, AKCE International Journal of Graphs and Combinatorics, vol. 17, no. 1, pp. 74–85, 2020. https://doi.org/10.1016/j.akcej.2018.05.002
I. Gutman and H. S. Ramane, “Research on graph energies in 2019”, MATCH Communications in Mathematical and in Computer Chemistry, vol. 84, pp. 277-292, 2020. [On line]. Available: https://bit.ly/3WCNzMe
I. Gutman and B. Zhou, “Laplacian energy of a graph”, Linear Algebra and its Applications, vol. 414, no. 1, pp. 29–37, 2006. https://doi.org/10.1016/j.laa.2005.09.008
H. Hatefi, H. A. Ahangar, R. Khoeilar, and S. M. Sheikholeslami, “On the inverse sum indeg energy of trees”, Asian-European Journal of Mathematics, vol. 15, no. 09, 2021. https://doi.org/10.1142/s1793557122501765
R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge University Press, 2012.
G. Indulal, A. Vijayakumar, “On a pair of equienergetic graphs”, MATCH Communications in Mathematical and in Computer Chemistry, vol. 55, pp. 83-90, 2006. [On line]. Available: https://bit.ly/3E5OQnT
A. Jahanbani, R. Khoeilar, and H. Shooshtari, “On the Zagreb matrix and Zagreb Energy,” Asian-European Journal of Mathematics, vol. 15, no. 01, 2022. https://doi.org/10.1142/s179355712250019x
X. Li, Y. Shi and I. Gutman, Graph Energy. New York: Springer, 2012.
A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications. New York: Academic Press, 1979.
R. Merris, “Laplacian matrices of graphs: A survey”, Linear Algebra and its Applications, vol. 197-198, pp. 143–176, 1994. https://doi.org/10.1016/0024-3795(94)90486-3
E. Sampathkumar, “On duplicate graphs”, Journal of the Indian Mathematical Society, vol. 37, pp. 285-293, 1973. [On line]. Available: https://bit.ly/3haZv7m
Downloads
Published
Issue
Section
License
Copyright (c) 2022 R. Rangarajan, V. D. Raghu, B. R. Rakshith
This work is licensed under a Creative Commons Attribution 4.0 International License.
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.