A complete characterization of spectra of the Randić matrix of level-wise regular trees

Articles in Press

Document Type : Special Issue for ICGTA23

Authors

1 Gujarat Technogical University, Ahmedabad - 382424, Gujarat (INDIA)

2 Lukhdhirji Engineering College, Morvi - 363642, Gujarat (INDIA)

Abstract

Let G be a simple finite connected graph with vertex set V(G)={v1,v2,…,vn} and di be the degree of the vertex vi. The Randić matrix \R(G)=[ri,j] of graph G is an n×n matrix whose (i,j)-entry ri,j is ri,j=1/didj if vi and vj are adjacent in G and 0 otherwise. A level-wise regular tree is a tree rooted at one vertex r or two (adjacent) vertices r and r′ in which all vertices with the minimum distance i from r or r′ have the same degree mi for 0≤i≤h, where h is the height of T. In this paper, we give a complete characterization of the eigenvalues with their multiplicity of the Randić matrix of level-wise regular trees. We prove that the eigenvalues of the Randić matrix of a level-wise regular tree are the eigenvalues of the particular tridiagonal matrices, which are formed using the degree sequence (m0,m1,…,mh−1) of level-wise regular trees.

Keywords

Main Subjects

References

[1] S. Alikhani and N. Ghanbari, Randić energy of specific graphs, Appl. Math. Comput. 269 (2015), 722–730.
https://doi.org/10.1016/j.amc.2015.07.112
[2] E. Andrade, H. Gomes, and M. Robbiano, Spectra and Randić spectra of caterpillar graphs and applications to the energy, MATCH Commun. Math. Comput. Chem. 77 (2017), 61–75.
[3] S¸.B. Bozkurt, A.D. Güngör, I. Gutman, and A.S. Çevik, Randić matrix and Randić energy, MATCH Commun. Math. Comput. Chem. 64 (2010), 239–250.
[4] D.M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs: Theory and Application, Academic Press, New York, 1980.
[5] M. Dehmer, M. Moosbrugger, and Y. Shi, Encoding structural information uniquely with polynomial-based descriptors by employing the Randi´c matrix, Appl. Math. Comput. 268 (2015), 164–168.
https://doi.org/10.1016/j.amc.2015.04.115
[6] R. Fernandes, H. Gomes, and E.A. Martins, On the spectra of some graphs like weighted rooted trees, Linear Algebra Appl. 428 (2008), no. 11–12, 2654–2674.
https://doi.org/10.1016/j.laa.2007.12.012
[7] B. Furtula and I. Gutman, Comparing energy and Randić energy, Maced. J. Chem. Chem. Eng. 32 (2013), no. 1, 117–123.
[8] Y. Gao, W. Gao, and Y. Shao, The minimal Randić energy of trees with given diameter, Appl. Math. Comput. 411 (2021), Article ID: 126489.
https://doi.org/10.1016/j.amc.2021.126489
[9] I. Gutman, B. Furtula, and S¸.B. Bozkurt, On Randić energy, Linear Algebra Appl. 442 (2014), 50–57.
https://doi.org/10.1016/j.laa.2013.06.010
[10] I. Gutman and B. Furtula (Eds.), Recent Results in the Theory of Randi´c Index, Univ. Kragujevac, Kragujevac, 2008.
[11] X. Li and I. Gutman, Mathematical Aspects of Randić-type Molecular Structure Descriptors, Univ. Kragujevac, Faculty of Science, Univ. Kragujevac, 2006.
[12] X. Li and Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem. 59 (2008), 127–156.
[13] B. Liu, Y. Huang, and J. Feng, A note on the Randić spectral radius, MATCH Commun. Math. Comput. Chem. 68 (2012), no. 3, 913–916.
[14] M. Randić, Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975), no. 23, 6609–6615.
https://doi.org/10.1021/ja00856a001
[15] M. Randić, On history of the Randić index and emerging hostility toward chemical graph theory, MATCH Commun. Math. Comput. Chem. 59 (2008), 5–124.
[16] B. Rather, S. Pirzada, I. Bhat, and T. Chishti, On Randić spectrum of zero divisor graphs of commutative ring Zn, Commun. Comb. Optim. 8 (2023), no. 1, 103–113.
https://doi.org/10.22049/cco.2021.27202.1212
[17] J.A. Rodríguez, A spectral approach to the Randić index, Linear Algebra Appl. 400 (2005), 339–344.
https://doi.org/10.1016/j.laa.2005.01.003
[18] J.A. Rodrıguez and J.M. Sigarreta, On the Randić index and conditional parameters of a graph, MATCH Commun. Math. Comput. Chem. 54 (2005), 403–416.
[19] O. Rojo, The spectrum of the laplacian matrix of a balanced binary tree, Linear Algebra Appl. 349 (2002), no. 1-3, 203–219.
https://doi.org/10.1016/S0024-3795(02)00256-2
[20] O. Rojo and R. Soto, The spectra of the adjacency matrix and laplacian matrix for some balanced trees, Linear Algebra Appl. 403 (2005), 97–117.
https://doi.org/10.1016/j.laa.2005.01.011
[21] L.N. Trefethen and D. Bau, Numerical Linear Algebra, Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), 1997.
[22] D. Yin, On the multiple eigenvalue of Randić matrix of trees, Linear Multilinear Algebra 69 (2021), no. 6, 1137–1150.
https://doi.org/10.1080/03081087.2019.1623856
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 05 September 2024

Files

Share

How to cite

Statistics