Abstract
Let G be a simple connected graph. The eccentric complexity of graph G is introduced as the number of different eccentricities of its vertices. A graph with eccentric complexity equal to one is called self-centered. In this paper, we study the eccentric complexity of graph under several graph operations such as complement of graph, line graph, Cartesian product, sum, disjunction and symmetric difference of graphs. Also, we present an infinite family of non-vertex-transitive self-centered graphs and we prove that all such graphs are 2-connected. Further, for any D and k where \(k \le \frac{D-1}{2}\), we construct an infinite family of non-vertex-transitive graphs with eccentric complexity k and diameter D. Extremal graphs with minimum or maximum total eccentricity among all graphs with given eccentric complexity are determined. We also consider a family of nanotubes and show that it is extremal with respect to the eccentric complexity among all fullerene graphs. At the end we also indicate some possible directions of further research.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alizadeh, Y., Andova, V., Klavžar, S., Škrekovski, R.: Wiener dimension: fundamental properties and (5,0)-nanotubical fullerenes. MATCH Commun. Math. Comput. Chem. 72, 279–294 (2014)
Alizadeh, Y., Azari, M., Došlić, T.: Computing the eccentricity-related invariants of single-defect carbon nanocones. J. Comput. Theor. Nanosci. 10, 1297–1300 (2013)
Alizadeh, Y., Iranmanesh, A., Klavžar, S.: Interpolation method and topological indices: the case of fullerenes \(C_{12k+4}\). MATCH Commun. Math. Comput. Chem. 68, 303–310 (2012)
Alizadeh, Y., Klavžar, S.: Interpolation method and topological indices: 2-parametric families of graphs. MATCH Commun. Math. Comput. Chem. 69, 523–534 (2013)
Alizadeh, Y., Klavžar, S.: Complexity of topological indices: the case of connective eccentric index. MATCH Commun. Math. Comput. Chem. 76, 659–667 (2016)
Andova, V., Došlić, T., Krnc, M., Lužar, B., Škrekovski, R.: On the diameter and some related invariants of fullerene graphs. MATCH Commun. Math. Comput. Chem. 68, 109–130 (2012)
Azari, M., Iranmanesh, A.: Computing the eccentric-distance sum for graph operations. Discrete Appl. Math. 161, 2827–2840 (2013)
Behzad, M., Simpson, J.E.: Eccentric sequences and eccentric sets in graphs. Discrete Math. 16, 187–193 (1976)
Buckley, F.: Self-centered graphs. Ann. N. Y. Acad. Sci. 576, 71–78 (1989)
Došlić, T., Graovac, A., Ori, O.: Eccentric connectivity index of hexagonal belts and chains. MATCH Commun. Math. Comput. Chem. 65, 745–752 (2011)
Došlić, T., Saheli, M.: Augmented eccentric connectivity index. Miskolc Math. Notes 12, 149–157 (2011)
Došlić, T., Saheli, M.: Eccentric connectivity index of composite graphs. Util. Math. 95, 3–22 (2014)
Eskender, B., Vumar, E.: Eccentric connectivity index and eccentric distance sum of some graph operations. Trans. Combin. 2, 103–111 (2013)
Ghorbani, M., Hosseinzadeh, M.A.: New version of Zagreb indices. Filomat 26, 93–100 (2012)
Graovac, A., Ori, O., Faghani, M., Ashrafi, A.R.: Distance properties of fullerenes. Iran. J. Math. Chem. 2, 99–107 (2011)
Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs, 2nd edn. CRC Press, Boca Raton (2011)
Harary, F., Robinson, R.W.: The diameter of a graph and its complement. Am. Math. Mon. 92, 211–212 (1985)
Imrich, W., Klavžar, S.: Product Graphs: Structure and Recognition. Wiley, New York (2000)
Iranmanesh, A., Alizadeh, Y.: Eccentric connectivity index of \(HAC_5C_7[p, q]\) and \(HAC_5C_6C_7[p, q]\) nanotubes. MATCH Commun. Math. Comput. Chem. 69, 175–182 (2013)
Lesniak, L.: Eccentric sequences in graphs. Period. Math. Hung. 6, 287–293 (1975)
Liu, H., Pan, X.F.: On the Wiener index of trees with fixed diameter. MATCH Commun. Math. Comput. Chem. 60, 85–94 (2008)
Morgan, M.J., Mukwembi, S., Swart, H.C.: On the eccentric connectivity index of a graph. Discrete Math. 311, 1229–1234 (2011)
Sharma, V., Goswami, R., Madan, A.K.: Eccentric connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies. J. Chem. Inform. Model. 37, 273–282 (1997)
Wiener, H.: Structural determination of the paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)
Xu, K., Das, K.C., Liu, H.: Some extremal results on the connective eccentricity index of graphs. J. Math. Anal. Appl. 433, 803–817 (2016)
Yarahmadi, Z., Moradi, S.: The center and periphery of composite graphs. Iran. J. Math. Chem. 5, 35–44 (2014)
Zhou, B., Du, Z.: On eccentric connectivity index. MATCH Commun. Math. Comput. Chem. 63, 181–198 (2010)
Acknowledgements
We are grateful to three anonymous referees for their helpful comments, especially to one of them for an extremely careful reading of the manuscript and suggesting many improvements in it.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Sandi Klavžar.
Partial support of the Croatian Science Foundation via research projects BioAmpMode (Grant No. 8481) and LightMol (Grant No. IP-2016-06-1142) and of Croatian Ministry of Science, Education and Sports (Croatian-Chinese bilateral project “Graph-theoretical methods for nanostructures and nanomaterials”) is gratefully acknowledged by T. D. And K.X. is supported by NNSF of China (No. 11671202) and Chinese Excellent Overseas Researcher Funding in 2016 (No. 17005).
Rights and permissions
About this article
Cite this article
Alizadeh, Y., Došlić, T. & Xu, K. On the Eccentric Complexity of Graphs. Bull. Malays. Math. Sci. Soc. 42, 1607–1623 (2019). https://doi.org/10.1007/s40840-017-0564-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-017-0564-y