Abstract
The resistance matrix \(R=R(G)\) of G is a matrix whose (i, j)th entry is equal to the resistance distance \(r_G(v_i, v_j)\). The resistance distance spectral radius, \(\partial _1^R(G)\), of G is the largest eigenvalues of R. In this paper, we obtain several edge-grafting transformations which are decreasing and/or increasing resistance distance spectral radius. As applications of these transformations, some lower and upper bounds on resistance distance spectral radius are determined and the corresponding extremal graphs are characterized.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bollobás, B.: Modern Graph Theory. Springer, Berlin (1998)
Babić, D., Klein, D.J., Lukovits, I., Nikolić, S., Trinajstić, N.: Resistance-distance matrix: a computational algorithm and its application. Int. J. Quantum Chem. 90, 166–176 (2002)
Chen, H., Zhang, F.: Resistance distance and the normalized Laplacian spectrum. Discrete Appl. Math. 155, 654–661 (2007)
Güngör Maden, A.D., Gutman, I., Cevik, A.S.: Bounds for resistance distance spectral radius. Hacet. J. Math. Stat. 42(1), 43–50 (2013)
Klein, D., Randić, M.: Resistance distance. J. Math. chem. 12, 81–95 (1993)
Minc, H.: Nonnegative Matrices. Wiley, New York (1988)
Sun, L., Wang, W., Zhou, J., Bu, C.: Some results on resistance distances and resistance matrices. Linear Multilinear Algebra 63(3), 523–533 (2015)
Xiao, W., Gutman, I.: On resistance matrices. MATCH Commun. Math. Comput. Chem. 49, 67–81 (2003)
Xiao, W., Gutman, I.: Resistance distance and Laplacian spectrum. Theor. Chem. Acc. 110, 284–289 (2003)
Xing, R., Zhou, B.: On the distance and distance signless Laplacian spectral radii of bicyclic graphs. Linear Algebra Appl. 439, 3955–3963 (2013)
Zhou, J., Wang, Z., Bu, C.: On the resistance matrix of a graph, Electron. J. Combin. 23, \(\sharp \) P1.41 (2016)
Acknowledgements
The authors would like to express their sincere gratitude to the referees for a very careful reading of the paper and for all their insightful comments and valuable suggestions, which led to a number of improvements in this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hossein Hajiabolhassan.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is partially supported by the Special Fund for Basic Scientific Research of Central Colleges, South Central University for Nationalities (CZY18032).
Rights and permissions
About this article
Cite this article
Zhu, Z., He, F. Some Properties on Resistance Distance Spectral Radius. Bull. Iran. Math. Soc. 46, 137–147 (2020). https://doi.org/10.1007/s41980-019-00246-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41980-019-00246-y