Randi´c Matrix and Randi´c Energy of Uniform Hypergraphs

Document Type : Original Scientific Paper

Authors

Department of Mathematics, University of Qom, ‎Qom‎, ‎I‎. ‎R‎. ‎Iran

10.22052/mir.2024.254116.1456

Abstract

‎The Randi´c matrix $R=[r_{ij}]$ of a graph $ G=(V,E) $ was defined as $r_{ij}=\frac{1}{\sqrt{d_id_j}}$ if vertices $v_i$ and $v_j$ are adjacent and $r_{ij}=0$ otherwise‎, ‎where $d_i$ is the degree of the vertex $v_i\in V$‎. ‎In this paper‎, ‎we define the Randi´c matrix of a uniform hypergraph and study some its spectral properties‎. ‎We also define the Randi´c energy of a uniform hypergraph and determine some upper and lower bound for it‎.

Keywords

Main Subjects

References

[1] X. Li, Y. Shi and I. Gutman, Graph Energy, Springer, New York, 2012.
[2] I. Gutman, The energy of a graph, Ber. Math.— Statist. Sekt. Forschungsz. Graz. 103 (1978) 1- 22.
[3] E. Andrade, M. Robbiano and B. San Martín, A lower bound for the energy of symmetric matrices and graphs, Linear Algebra Appl. 513 (2017) 264-275, https://doi.org/10.1016/j.laa.2016.10.022.
[4] I. Gutman, Relating graph energy with vertex-degree-based energies, Mill. Tech. Cour. 68 (2020) 715 - 725.
[5] L. Wang and X. Ma, Bounds of graph energy in terms of vertex cover number, Linear Algebra Appl. 517 (2017) 207 - 216, https://doi.org/10.1016/j.laa.2016.12.015.
[6] I. Gutman and B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29 - 37, https://doi.org/10.1016/j.laa.2005.09.008.
[7] N. Abreu, D. M. Cardoso, I. Gutman, E. A. Martins and M. Robbiano, Bounds for the signless Laplacian energy, Linear Algebra Appl. 435 (2011) 2365-2374, https://doi.org/10.1016/j.laa.2010.10.021.
[8] M. Cavers, S. Fallat and S. Kirkland, On the normalized Laplacian energy and general Randic index R1 of graphs, Linear Algebra Appl. 433 (2010) 172-190, https://doi.org/10.1016/j.laa.2010.02.002.
[9] B. Bozkurt, A. D. Maden, I. Gutman and A. S Çevik, Randi'c matrix and Randi'c energy, MATCH Commun. Math. Comput. Chem. 64 (2010) 239-250.
[10] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl. 326 (2007) 1472 - 1475, https://doi.org/10.1016/j.jmaa.2006.03.072.
[11] M. R. Jooyandeh, D. Kiani and M. Mirzakhah, Incidence energy of a graph, MATCH Commun. Math. Comput. Chem. 62 (2009) 561 - 572.
[12] B. Cheng and B. Liu, The normalized incidence energy of a graph, Linear Algebra Appl. 438 (2013) 4510-4519, https://doi.org/10.1016/j.laa.2013.01.003.
[13] L. Shi and H.Wang, The Laplacian incidence energy of graphs, Linear Algebra Appl. 439 (2013) 4056 - 4062, https://doi.org/10.1016/j.laa.2013.10.028.
[14] R. Gu, F. Huang and X. Li, Randi'c incidence energy of graphs, Trans. Comb. 3 (2014) 1 - 9.
[15] R. Cruz, J. Monsalve and J. Rada, Randi'c energy of digraphs, Heliyon 8 (2022) e11874, https://doi.org/10.1016/j.heliyon.2022.e11874.
[16] K. Cardoso, R. Del-Vecchio, L. Portugal and V. Trevisan, Adjacency energy of hypergraphs, Linear Algebra Appl., 648 (2022) 181 - 204, https://doi.org/10.1016/j.laa.2022.04.018.
[17] K. Cardoso and V. Trevisan, Energies of hypergraphs, Electron. J. Linear Algebra 36 (2020) 293- 308, https://doi.org/10.13001/ela.2020.5025.
[18] N. F. Yalçın, On Laplacian energy of r-uniform hypergraphs, Symmetry 15 (2023) p. 382, https://doi.org/10.3390/sym15020382.
[19] K. Sharma and S. K. Panda, On the distance energy of k-uniform hypergraphs, Spec. Matrices 11 (2023) p. 20230188, https://doi.org/10.1515/spma-2023-0188.
[20] C. Berge, Hypergraphs: Combinatorics of Finite Sets, North-Holland, Amsterdam, The Netherlands, 1973.
[21] J. Cooper and A. Dutle, Spectra of uniform hypergraphs, Linear Algebra Appl. 436 (2012) 3268 - 3292, https://doi.org/10.1016/j.laa.2011.11.018.
[22] A. Banerjee, On the spectrum of hypergraphs, Linear Algebra Appl. 614 (2021) 82 - 110, https://doi.org/10.1016/j.laa.2020.01.012.
[23] F. R. K. Chung, Spectral Graph Theory, American Mathematical Society, Providence, 1997.
[24] F. R. Gantmacher, The Theory of Matrices, Chelsea, New York, 1959.
[25] K. Cardoso and V. Trevisan, The signless Laplacian matrix of hypergraphs, Spec. Matrices 10 (2022) 327-342, https://doi.org/10.1515/spma-2022-0166.
[26] M. Randi'c, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609 - 6615, https://doi.org/10.1021/ja00856a001.
[27] B. Bollobas and P. Erdös, Graphs of extremal weights, Ars Comb. 50 (1998) 225 - 233.
[28] G. H. Shirdel, A. Mortezaee and L. Alameri, General Randic index of uniform hypergraphs, Iranian J. Math. Chem. 14 (2023) 121 - 133, https://doi.org/10.22052/IJMC.2023.252836.1712.
[29] X. Li and Y. Yang, Sharp bounds for the general Randi'c index, MATCH Commun. Math. Comput. Chem. 51 (2004) 155 - 166.
[30] K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A. 37 (1951) 760-766, https://doi.org/10.1073/pnas.37.11.760.
[31] P. Cerone and S. S. Dragomir, A refinement of the Grüss inequality and applications, Tamkang J. Math. 38 (2007) 37 - 49,https://doi.org/10.5556/j.tkjm.38.2007.92.
Mathematics Interdisciplinary Research
Volume 9, Issue 3
In progress
September 2024
Pages 269-288

Files

Share

How to cite

Statistics