Document Type : Original paper

**Authors**

Department of Mathematics, Karnatak University, Dharwad

**Abstract**

In this article the terminal status of a vertex and terminal status connectivity indices of a connected graph have introduced. Explicit formulae for the terminal status of vertices and for terminal status connectivity indices of certain graphs are obtained. Also some bounds are given for these indices. Further these indices are used for predicting the physico-chemical properties of cycloalkanes and it is observed that the correlation of physico-chemical properties of cycloalkanes with newly introduced indices is better than the correlation with other indices.

**Keywords**

- Terminal status of a vertex
- terminal status connectivity indices
- pendent vertex
- diameter of a graph
- molecular graph

**Main Subjects**

[1] A.R. Ashrafi and M. Ghorbani, Eccentric connectivity index of fullerenes, Novel Molecular Structure Descriptors-Theory and Applications II (I. Gutman and B. Furtula, eds.), Uni. Kragujevac, Kragujevac, 2010, pp. 183–192.

[2] X. Deng and J. Zhang, Equiseparability on terminal Wiener index, Algorithmic Aspects in Information and Management (A.V. Goldberg and Y. Zhou, eds.), Springer, Berlin, 2009, pp. 166–174.

[3] A.A. Dobrynin, R. Entringer, and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001), no. 3, 211–249.

[4] A.A. Dobrynin, I. Gutman, S. Klavžar, and P. Žigert, Wiener index of hexagonal systems, Acta Appl. Math. 72 (2002), no. 3, 247–294.

[5] A.A. Dobrynin and A.A. Kochetova, Degree distance of a graph: A degree analog of the Wiener index, J. Chem. Inf. Comput. Sci. 34 (1994), no. 5, 1082–1086.

[6] Z. Du, A. Jahanbai, and S.M. Sheikholeslami, Relationships between Randic index and other topological indices, Commun. Comb. Optim. 6 (2021), no. 1, 137–154.

7] I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34 (1994), no. 5, 1087–1089.

[8] I. Gutman and B. Furtula, A survey on terminal Wiener index, Novel Molecular Structure Descriptors-Theory and Applications I (I. Gutman and B. Furtula, eds.), Uni. Kragujevac, Kragujevac, 2010, pp. 173–190.

[9] , Distance in Molecular Graphs Theory, Univ. Kragujevac, Kragujevac, 2012.

[10] , Novel Molecular Structure Descriptors-Theory and Application I, Univ. Kragujevac, Kragujevac, 2012.

[11] I. Gutman, B. Furtula, K.C. Das, E. Milovanović, and I. Milovanović, Bounds in chemical graph theory-basics, Univ. Kragujevac, Kragujevac, 2017.

[12] I. Gutman, B. Furtula, and M. Petrović, Terminal Wiener index, J. Math. Chem. 46 (2009), no. 2, 522–531.

[13] I. Gutman, B. Ruˇsˇci´c, N. Trinajstić, and C.F. Wilcox Jr, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62 (1975), no. 9, 3399–3405.

[14] I. Gutman, Y.-N. Yeh, S.-L. Lee, and Y.-L. Luo, Some recent results in the theory of the Wiener number, Indian J. Chem. 32A (1993), 651–661.

[15] Ivan Gutman and Nenad Trinajstić, Graph theory and molecular orbitals. Total ϕ-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.

[16] A. Heydari and I. Gutman, On the terminal Wiener index of thorn graphs, Kragujevac J. Sci. 32 (2010), 57–64.

[17] B. Horvat, T. Pisanski, and M. Randić, Terminal polynomials and star-like graphs, MATCH Commun. Math. Comput. Chem. 60 (2008), no. 2, 493–512.

[18] I.r Milovanović, M. Matejić, E. Milovanović, and R. Khoeilar, A note on the first Zagreb index and coindex of graphs, Commun. Comb. Optim. 6 (2021), no. 1, 41–51.

[19] T.A. Naikoo, B.A. Rather, U. Samee, and S. Pirzada, On Zagreb index of tournaments, Kragujevac J. Math. 48 (2024), no. 2, 241–253.

[20] S. Nikolić and N. Trinajstić, The Wiener index: Development and applications, Croat. Chem. Acta 68 (1995), no. 1, 105–129.

[21] M. Pal, S. Samanta, and A. Pal, Handbook of Research on Advanced Applications of Graph Theory in Modern Society, IGI Global, Hershey PA, USA, 2020.

[22] S. Pirzada, B.A. Rather, T.A. Naikoo, and T.A. Chishti, On first general Zagreb index of tournaments, Creative Math. Inf. (to appear).

23] H.S. Ramane, K.P. Narayankar, S.S. Shirkol, and A.B. Ganagi, Terminal Wiener index of line graphs, MATCH Commun. Math. Comput. Chem. 69 (2013), no. 3, 775–782.

[24] H.S. Ramane and A.S. Yalnaik, Status connectivity indices of graphs and its applications to the boiling point of benzenoid hydrocarbons, J. Appl. Math. Comput. 55 (2017), no. 1, 609–627.

[25] H.S. Ramane, A.S. Yalnaik, and R. Sharafdini, Status connectivity indices and co-indices of graphs and its computation to some distance-balanced graphs, AKCE Int. J. Graphs Combin. 17 (2020), 98–108.

[26] M. Randic, Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975), no. 23, 6609–6615.

27] M. Randić and J. Zupan, Highly compact 2D graphical representation of DNA sequences, SAR and QSAR in Environ. Res. 15 (2004), no. 3, 191–205.

[28] M. Randić, J. Zupan, and D. Vikić-Topić, On representation of proteins by starlike graphs, J. Mol. Graph. Modell. 26 (2007), no. 1, 290–305.

[29] N.S. Schmuck, S.G. Wagner, and H. Wang, Greedy trees, caterpillars, and Wiener-type graph invariants, MATCH Commun. Math. Comput. Chem. 68 (2012), no. 1, 273.

[30] L.A. Székelys, H. Wang, and T. Wu, The sum of the distances between the leaves of a tree and the ‘semi-regular’ property, Discrete Math. 311 (2011), no. 13, 1197–1203.

[31] R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000.

[32] D. Vukiˇcević and A. Graovac, Note on the comparison of the first and second normalized Zagreb eccentricity indices., Acta Chim. Slov. 57 (2010), no. 3, 524–528.

[33] H.B. Walikar, H.S. Ramane, and V.S. Shigehalli, Wiener number of dendrimers, Proc. Nat. Conf. Math. Comput. Mod. (R. Nadarajan and G. Arulmozhi, eds.), Allied Publishers, New Delhi, 2003, pp. 361–368.

[34] H.B. Walikar, V.S. Shigehalli, and H.S. Ramane, Bounds on the Wiener number of a graph, MATCH Commun. Math. Comput. Chem. 50 (2004), no. 1, 117–132.

[35] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), no. 1, 17–20.

December 2022

Pages 275-300