Original Research Article

Article volume = 2024 and issue = 1

Pages: 29–36

Article publication Date: December 05, 2023

You can download PDF file of the article here: Download

Visited 177 times and downloaded 81 times

# Terminal eccentricity indices of graphs

#### Harishchandra S. Ramane(a), Deepa V. Kitturmath(b), and Kavita Bhajantri(c)

(a) Department of Mathematics, Karnatak University, Pavate Nagar, Dharwad - 580003, India.

(b) Department of Mathematics, KLE Technological University, Vidyanagar, Hubballi - 580031, India.

(c) Department of Mathematics, JSS Banashankari Arts, Commerce and S.K. Gubbi Science College, Dharwad - 580004, India.

##### Abstract:

Let $V_T(G)$ be the set of pendent vertices of $G$ and $d(u, v)$ be the distance between the vertices $u$ and $v$. The terminal eccentricity of a vertex $u$ is defined as $te_G(u)= \max \{d(u, v) \; | \; v \in V_T(G)\}$. In this paper we introduce new topological indices of graphs called as first and second terminal eccentricity indices and are defined as \[ TE_1(G) = \sum_{uv \in E(G)} [te_G(u)+te_G(v)] \;\;\; \text{and} \;\;\; TE_2(G) = \sum_{uv \in E(G)} te_G(u)te_G(v), \] where $E(G)$ is the edge set of $G$. We discuss properties of terminal eccentricity indices of graphs and carry regression analysis of these indices with the chemical properties of benzenoid hydrocarbons. The terminal eccentricity indices are useful in the graphs where pendent vertices exists.

##### Keywords:

Eccentricity of a vertex, Terminal eccenticity of a vertex, Terminal eccentricity indices.

##### References:

- [1] A. R. Ashrafi, M. Ghorbani, Eccentric connectivity index of fullerenes, in: I. Gutman, B. Furtula (Eds.), Novel Molecular Structure Descriptors-Theory and Applications II, Uni. Kragujevac, Kragujevac, (2010), 183–192. 1
- [2] N. De, On eccentric connectivity index and polynomial of thorn graph, Appl. Math., 3 (2012), 931–934. 1
- [3] M. V. Diudea, I. Gutman, Wiener-type topological indices, Croat. Chem. Acta, 71 (1998), 21–51. 1
- [4] I. Gutman, Degree-based topological indices, Croat. Chem. Acta, 86 (2013), 351–361. 1
- [5] I. Gutman, K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem., 50 (2004), 83–92. 1
- [6] I. Gutman, B. Furtula, M. Petrović, Terminal Wiener index, J. Math. Chem., 46 (2009), 522–531. 1
- [7] H. S. Ramane, K. Bhajantri, D. V. Kitturmath, Terminal status of vertices and terminal status connectivity indices of graphs with its applications to properties of cycloalkanes, Commun. Combin. Optimiz., 7 (2022), 275–300. 1, 1
- [8] H. S. Ramane, A. S. Yalnaik, Status connectivity indices of graphs and its applications to the boiling point of benzenoid hydrocarbons, J. Appl. Math. Comput., 55 (2017), 609–627. 1
- [9] V. Sharma, R. Goswami, A. K. Madan, Eccentric connectivity index: A novel highly discriminating topological descriptor for structure-property and structure-activity studies, J. Chem. Infor. Model., 37(2) (1997), 273–282. 1
- [10] D. Vukičević, A. Graovac, Note on the comparison of the first and second normalized Zagreb eccentricity indices, Acta Chem. Slovenica, 57 (2010), 524–528. 1
- [11] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69(1) (1947), 17–20. 1

##### Cite this article as:

- Harishchandra S. Ramane, Deepa V. Kitturmath, and Kavita Bhajantri, Terminal eccentricity indices of graphs, Communications in Combinatorics, Cryptography & Computer Science, 2024(1), PP.29–36, 2023
- Export citation to BibTeX