Some new bounds on the general sum--connectivity index

Document Type : Original paper

Authors

1 Knowledge Unit of Science University of Management and Technology, Sialkot 51310, Pakistan

2 Faculty of Electronic Engineering, 18000 Nis, Serbia

3 Faculty of Electronic Engineering, Nis, Serbia

Abstract

Let $G=(V,E)$ be a simple connected graph with $n$ vertices, $m$ edges and sequence of vertex degrees $d_1 \ge d_2 \ge \cdots \ge d_n>0$, $d_i=d(v_i)$, where $v_i\in V$. With $i\sim j$ we denote adjacency of vertices $v_i$ and $v_j$. The general sum--connectivity index of graph is defined as $\chi_{\alpha}(G)=\sum_{i\sim j}(d_i+d_j)^{\alpha}$, where $\alpha$ is an arbitrary real number. In this paper we determine relations between $\chi_{\alpha+\beta}(G)$ and $\chi_{\alpha+\beta-1}(G)$, where $\alpha$ and $\beta$ are arbitrary real numbers, and obtain new bounds for $\chi_{\alpha}(G)$. Also, by the appropriate choice of parameters $\alpha$ and $\beta$, we obtain a number of old/new inequalities for different vertex--degree--based topological indices.

Keywords

Main Subjects

References

[1] A. Ali, I. Gutman, E. Milovanović, and I. Milovanović, Sum of powers of the degrees of graphs: extremal results and bounds, MATCH Commun. Math. Comput. Chem. 80 (2018), 5–84.
[2] A. Ali, L. Zhong, and I. Gutman, Harmonic index and its generalizations: extremal results and bounds, MATCH Commun. Math. Comput. Chem. 81 (2019), 249–311.
[3] B. Borovićanin, K.C. Das, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem 78 (2017), no. 1, 17–100.
[4] K.C. Das, S. Das, and B. Zhou, Sum-connectivity index of a graph, Front. Math. China 11 (2016), no. 1, 47–54.
[5] T. Došlic, B. Furtula, A. Graovac, I. Gutman, S. Moradi, and Z. Yarahmadi, On vertex-degree-based molecular structure descriptors, MATCH Commun. Math. Comput. Chem. 66 (2011), no. 2, 613–626.
[6] C. Elphick and P. Wocjan, Bounds and power means for the general Randić and sum-connectivity indices, In: Bounds in Chemical Graph Theory – Mainstreams, I. Gutman, B. Furtula, K.C. Das, E.Milovanović, I.Milovanović, Eds.), Mathematical Chemistry Monographs - MCM 19, Univ. Kragujevac, Kragujevac, 2017, pp. 121–133.
[7] S. Fajtlowicz, On conjectures of graffiti-II, Congr. Numer. 60 (1987), 187–197.
[8] F. Falahati-Nezhad and M. Azari, Bounds on the hyper-Zagreb index, J. Appl. Math. Inform. 34 (2016), no. 3-4, 319–330.
[9] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015), no. 4, 1184–1190.
[10] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), no. 1, 83–92.
[11] I. Gutman, E. Milovanović, and I. Milovanović, Beyond the Zagreb indices, AKCE Int. J. Graph Combin. (to appear).
[12] I. Gutman, B. Ruščić, N. Trinajstić, and C.F. Wilcox, Graph theory and molecular orbitals. XII. acyclic polyenes, J. Chem. Phys. 62 (1975), no. 9, 3399–3405.
[13] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total ϕ-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.
[14] M. Matejić, E. Milovanović, and I. Milovanović, On bounds for harmonic topological index, Filomat 32 (2018), no. 1, 311–317.
[15] M. Matejić, E. Milovanović, and I. Milovanović, On lower bounds for VDB topological indices of graphs, Appl. Math. Comput. Sci. 3 (2018), no. 1, 5–11.
[16] M. Matejić, E. Milovanović, and I. Milovanović, Some inequalities for general sum–connectivity index, MATCH Commun. Math. Comput. Chem. 79 (2018), 477–489.
[17] E. Milovanović, I. Milovanović, I. Gutman, and B. Furtula, Some inequalities for general sum–connectivity index, Int. J. Appl. Graph Theory 1, no. 1. 1--15
[18] D.S. Mitrinović, J.E. Pečarić, and A.M. Fink, Classical and new inequalities in analysis, Kluwer Academic Publishers, Dordrecht, 1993.
[19] D.S. Mitrinović and P.M. Vasić, Analytic inequalities, Springer Verlag, BerlinHeidelberg-New York, 1970.
[20] S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003), no. 2, 113–124.
[21] J. Radon, Über die absolut additiven mengenfunktionen, Wiener Sitzungsber 122 (1913), 1295–1438.
[22] T. Réti and I. Felde, Novel Zagreb indices-based inequalities with particular regard to semiregular and generalized semiregular graphs, MATCH Commun. Math. Comput. Chem. 76 (2016), 185–206.
[23] J.M. Rodríguez, J.L. Sánchez, and J.M. Sigarreta, CMMSE-on the first general Zagreb index, J. Math. Chem. 56 (2018), no. 7, 1849–1864.
[24] J.M. Rodríguez and J.M. Sigarreta, The harmonic index, In: Bounds in Chemical Graph Theory – Basics, (I. Gutman, B. Furtula, K. C. Das, E. Milovanović, I. Milovanović, Eds.), Mathematical Chemistry Monographs - MCM 19, Univ. Kragujevac, Kragujevac, 2017, pp. 229–281.
[25] J.M. Rodríguez and J.M. Sigarreta, New results on the harmonic index and its generalizations, MATCH Commun. Math. Comput. Chem. 78 (2017), 387–404.
[26] G.H. Shirdel, H. Rezapour, and A.M. Sayadi, The hyper-Zagreb index of graph operations, Iranian J. Math. Chem. 4 (2013), no. 2, 213–220.
[27] L. Zhong and K. Xu, Inequalities between vertex-degree-based topological indices, MATCH Commun. Math. Comput. Chem. 71 (2014), no. 3, 627–642.
[28] B. Zhou and N. Trinajstić, On a novel connectivity index, J. Math. Chem. 46 (2009), no. 4, 1252–1270.
[29] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010), no. 1, 210–218.
[30] B. Zhou and N. Trinajstić, Relations between the product-and sum-connectivity indices, Croat. Chem. Acta 85 (2012), no. 3, 363–365.
 
Communications in Combinatorics and Optimization
Volume 5, Issue 2
December 2020
Pages 97-109

Files

Share

How to cite

Statistics