Abstract
In this paper, we first obtain the second lower bound on Wiener index for tricyclic graphs. As applications, those graphs with the first four minimum distance (resp. distance signless Laplacian) spectral radius among tricyclic graphs are characterized.
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References
Bollobás, B.: Morden Graph Theory. Springer, Berlin (1998)
Aouchiche, M., Hansen, P.: Two Laplacians for the distance matrix of a graph. Linear Algebra Appl. 439, 21–33 (2013)
Graham, R.L., Pollack, H.O.: On the addressing problem for loop switching. Bell Syst. Tech. J. 50, 2495–2519 (1971)
Graham, R.L., Lovász, L.: Distance matrix polynomials of trees. Adv. Math. 29, 60–88 (1978)
Stevanović, D., IIić, A.: Distance spectral radius of trees with fixed maximum degree. Electron. J. Linear Algebra 20, 168–179 (2010)
Yu, G., Wu, Y., Zhang, Y., Shu, J.: Some graft transformations and its application on a distance spectrum. Discret. Math. 311, 2117–2123 (2011)
Paul, S.: On the maximal distance spectral radius in a class of bicyclic graphs. Discret. Math. Algorithms Appl. 4, 1250061 (2012)
Bose, S.S., Nath, M., Paul, S.: On the maximal distance spectral radius of graphs without a pendent vertex. Linear Algebra Appl. 438, 4260–4278 (2013)
Nath, M., Paul, S.: On the distance spectral radius of bipartite graphs. Linear Algebra Appl. 436, 1285–1296 (2012)
Nath, M., Paul, S.: On the distance spectral radius of trees. Linear Multilinear Algebra 61, 847–855 (2013)
Zhou, B., Ilić, A.: On distance spectral radius and distance energy of graphs. MATCH Commun. Math. Comput. Chem. 64, 261–280 (2010)
Xing, R., Zhou, B., Li, J.: On the distance signless Laplacian spectral radius of graphs. Linear Multilinear Algebra 62, 1377–1387 (2014)
Xing, R., Zhou, B.: On the distance and distance signless Laplacian spectral radii of bicyclic graphs. Linear Algebra Appl. 439, 3955–3963 (2013)
Aouchiche, M., Hansen, P.: On the distance signless Laplacian of a graph. Linear Multilinear Algebra 64(6), 1113–1123 (2016)
Das, K.C.: Proof of conjectures on the distance signless Laplacian eigenvalues of graphs. Linear Algebra Appl. 467, 100–115 (2015)
Lin, H., Lu, X.: Bounds on the distance signless Laplacian spectral radius in terms of clique number. Linear Multilinear Algebra 63(9), 1750–1759 (2015)
Li, S., Li, X., Zhu, Z.: On tricyclic graphs with minimal energy. MATCH Commun. Math. Comput. Chem. 59, 397–419 (2008)
Wang, D., Tan, S., Zhu, L.: On the lower and upper bounds for different indices of tricyclic graphs. J. Appl. Math. Comput. 51, 1–11 (2016)
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The authors would like to express sincere gratitude to the editor and the reviewers for helpful comments in improving the quality of the original manuscript.
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Communicated by Sanming Zhou.
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This Research is supported by the Special Fund for Basic Scientific Research of Central Colleges, South-Central University for Nationalities (CZY18032).
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Zhu, Z., Zou, X. & Hong, Y. On the Distance and Distance Signless Laplacian Spectral Radii of Tricyclic Graphs. Bull. Malays. Math. Sci. Soc. 43, 2587–2604 (2020). https://doi.org/10.1007/s40840-019-00824-7
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DOI: https://doi.org/10.1007/s40840-019-00824-7