On the extremal connective eccentricity index among trees with maximum degree

Document Type : Research Paper

Author

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, PR China

Abstract

The connective eccentricity index (CEI) of a graph $G$ is defined as $\xi^{ce}(G)=\sum_{v \in V(G)}\frac{d_G(v)}{\varepsilon_G(v)}$, where $d_G(v)$ is the degree of $v$ and $\varepsilon_G(v)$ is the eccentricity of $v$. In this paper, we characterize the unique trees with the maximum and minimum CEI among all $n$-vertex trees and $n$-vertex conjugated trees with fixed maximum degree, respectively.

Keywords

Main Subjects


[1] S. Gupta, M. Singh and A. K. Madan, Connective eccentricity index: a novel topological descriptor for predicting
biological activity, J. Mol. Graph. Model., 18 (2000) 18–25.
[2] F. Hayat, On the maximum connective eccentricity index among k-connected graphs, Discrete Math. Algorithms Appl., 13 (2021).
[3] A. Ilic and I. Gutman, Eccentric connectivity index of chemical trees, MATCH Commun. Math. Comput. Chem., 65
(2011) 731–744.
[4] H. Li, S. Li and H. Zhang, On the maximal connective eccentricity index of bipartite graphs with some given param-
eters, J. Math. Anal. Appl., 454 (2017) 453–467.
[5] S. Li and L. Zhao, On the extremal total reciprocal edge-eccentricity of trees, J. Math. Anal. Appl., 433 (2016)
587–602.
[6] M. J. Morgan, S. Mukwembi and H. C. Swart, On the eccentric connectivity index of a graph, Discrete Math., 311
(2011) 1229–1234.
[7] V. Sharma, R. Goswami and A. K. Madan, Eccentric connectivity index: a novel highly discriminating topological
descriptor for structure-property and structure-activity studies, J. Chem. Inf. Comput. Sci., 37 (1997) 273–282.
[8] L. Tang, X. Wang, W. Liu and L. Feng, The extremal values of connective eccentricity index for trees and unicyclic
graphs, Int. J. Comput. Math., 94 (2017) 437–453.
[9] N. Trinajstić, Chemical Graph Theory, 2nd revised ed., CRC Press, Boca Raton, 1992.
[10] K. Xu, K. C Das and H. Liu, Some extremal results on the connective eccentricity index of graphs, J. Math. Anal.
Appl. , 433 (2016) 803–817.
[11] G. Yu and L. Feng, On connective eccentricity index of graphs, MATCH Commun. Math. Comput. Chem., 69 (2013)
611–628.
[12] G. Yu, H. Qu, L. Tang and L. H. Feng, On the connective eccentricity index of trees and unicyclic graphs with given
diameter, J. Math. Anal. Appl. , 420 (2014) 1776–1786.
[13] J. Zhang, B. Zhou and Z. Liu, On the minimal eccentric connectivity indices of graphs, Discrete Math., 312 (2012)
819–829.
[14] B. Zhou and Z. Du, On eccentric connectivity index, MATCH Commun. Math. Comput. Chem., 63 (2010) 181–198.
[15] L. Zhao, H. Li and Y. Gao, On the extremal graphs with respect to the total reciprocal edge-eccentricity, J. Comb.
Optim. , https://doi.org/10.1007/s10878-019-00458-2.