A note on polyomino chains with extremum general sum-connectivity index

Document Type : Original paper

Authors

1 University of Ha'il

2 University of Management and Technology, Sialkot, Pakistan

Abstract

The general sum-connectivity index of a graph $G$ is defined as $\chi_{\alpha}(G)= \sum_{uv\in E(G)} (d_u + d_{v})^{\alpha}$ where $d_{u}$ is degree of the vertex $u\in V(G)$, $\alpha$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $\chi_{\alpha}$ values from a certain collection of polyomino chain graphs is solved for $\alpha<0$. The obtained results together with already known results (concerning extremum $\chi_{\alpha}$ values of polyomino chain graphs) give the complete solution of the aforementioned problem.

Keywords

Main Subjects

References

[1] M. Ahmad, M. Saeed, M. Javaid, and M. Hussain, Exact formula and improved bounds for general sum-connectivity index of graph-operations, IEEE Access 7 (2019), 167290–167299.
[2] S. Akhter, M. Imran, and Z. Raza, Bounds for the general sum-connectivity index of composite graphs, J. Inequal. Appl. 2017 (2017), no. 1, 1–12.
[3] A. Ali, An alternative but short proof of a result of Zhu and Lu concerning general sum-connectivity index, Asian-European J. Math. 11 (2018), no. 2, 1850030.
[4] A. Ali, S. Ahmed, Z. Du, W. Gao, and M.A. Malik, On the minimal general sumconnectivity index of connected graphs without pendant vertices, IEEE Access 7 (2019), 136743–136751.
[5] A. Ali, A.A. Bhatti, and Z. Raza, A note on the zeroth-order general Randić index of cacti and polyomino chains, Iranian J. Math. Chem. 5 (2014), no. 2, 143–152.
[6] A. Ali, A.A. Bhatti, and Z. Raza, Some vertex-degree-based topological indices of polyomino chains, J. Comput. Theor. Nanosci. 12 (2015), no. 9, 2101–2107.
[7] A. Ali and D. Dimitrov, On the extremal graphs with respect to bond incident degree indices, Discrete Appl. Math. 238 (2018), 32–40.
[8] A. Ali, D. Dimitrov, Z. Du, and F. Ishfaq, On the extremal graphs for general sum-connectivity index (χα) with given cyclomatic number when $α>1$, Discrete Appl. Math. 257 (2019), 19–30.
[9] A. Ali and Z. Du, On the difference between atom-bond connectivity index and Randić index of binary and chemical trees, Int. J. Quantum Chem. 117 (2017), no. 23, e25446.
[10] A. Ali, Z. Du, and K. Shehzadi, Estimating some general molecular descriptors of saturated hydrocarbons, Mol. Inform. 38 (2019), no. 11-12, 1900007.
[11] 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), no. 1, 5–84. 
[12] A. Ali, M. Javaid, M. Matejić, I.Z. Milovanović, and E.I. Milovanović, Some new bounds on the general sum–connectivity index, Commun. Comb. Optim. 5 (2020), no. 2, 97–109.
[13] A. Ali, Z. Raza, and A.A. Bhatti, Bond incident degree (BID) indices of polyomino chains: a unified approach, Appl. Math. Comput. 287 (2016), 28–37.
[14] A. Ali, L. Zhong, and I. Gutman, Harmonic index and its generalizations: extremal results and bounds, MATCH Commun. Math. Comput. Chem. 81 (2019), no. 2, 249–311.
[15] M. An and L. Xiong, Extremal polyomino chains with respect to general Randić index, J. Comb. Optim. 31 (2016), no. 2, 635–647.
[16] M. An and L. Xiong, Extremal polyomino chains with respect to general sum-connectivity index., Ars Combin. 131 (2017), 255–271.
[17] M. Arshad and I. Tomescu, Maximum general sum-connectivity index with $−1  ≤ α < 0$ for bicyclic graphs, Math. Reports 19 (2017), 5–84.
[18] B. Basavanagoud and A.P. Barangi, F-index and hyper-Zagreb index of four new tensor products of graphs and their complements, Discrete Math. Algorithms Appl. 11 (2019), no. 03, 1950039.
[19] J. A. Bondy and U. S. R. Murty, Graph theory, Springer, London, 2008.
[20] B. Borovicanin, K.C. Das, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017), no. 1, 17–100.
[21] Q. Cui and L. Zhong, On the general sum-connectivity index of trees with given number of pendent vertices, Discrete Appl. Math. 222 (2017), 213–221.
[22] K.C. Das, S. Balachandran, and I. Gutman, Inverse degree, Randić index and harmonic index of graphs, Appl. Anal. Discrete Math. 11 (2017), no. 2, 304–313.
[23] H. Deng, S. Balachandran, S.K. Ayyaswamy, and Y.B. Venkatakrishnan, The harmonic indices of polyomino chains, Natl. Acad. Sci. Lett. 37 (2014), no. 5, 451–455.
[24] S. Fajtlowicz, On conjectures of Graffiti-II, Congr. Numer. 60 (1987), 187–197.
[25] I. Gutman, B. Furtula, and V. Katanić, Randić index and information, AKCE Int. J. Graphs Comb. 15 (2018), no. 3, 307–312.
[26] I. Gutman, S. Klavžar, and A. Rajapakse, Average distances in square-cell configurations, Int. J. Quant. Chem. 76 (2000), no. 5, 611–617.
[27] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total πelectron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.
[28] F. Harary, Graph theory, Addison–Wesley, Reading, 1969.
[29] F. Harary and P.G. Mezey, Cell-shedding transformations, equivalence relations, and similarity measures for square-cell configurations, Int. J. Quant. Chem. 62 (1997), no. 4, 353–361.
[30] N.H. Husin, R. Hasni, Z. Du, and A. Ali, More results on extremum Randi´c indices of (molecular) trees, Filomat 32 (2018), no. 10, 3581–3590.
[31] M.K. Jamil and I. Tomescu, Minimum general sum-connectivity index of trees and unicyclic graphs having a given matching number, Discrete Appl. Math. 222 (2017), 143–150.
[32] F. Li and Q. Ye, The general connectivity indices of fluoranthene-type benzenoid systems, Appl. Math. Comput. 273 (2016), 897–911.
[33] R. Li, The hyper-Zagreb index and some Hamiltonian properties of graphs, Discrete Math. Lett. 1 (2019), 54–58.
[34] X. Li and Y. Shi, A survey on the Randi´c index, MATCH Commun. Math. Comput. Chem. 59 (2008), no. 1, 127–156.
[35] H. Liu and Z. Tang, The hyper-Zagreb index of cacti with perfect matchings, AKCE Int. J. Graphs Comb. 17 (2020), no. 1, 422–428.
[36] H. Liu and Z. Tang, Maximal hyper-Zagreb index of trees, unicyclic and bicyclic graphs with a given order and matching number, Discrete Math. Lett. 4 (2020), 11–18.
[37] S. Liu and J. Ou, On maximal resonance of polyomino graphs, J. Math. Chem. 51 (2013), no. 2, 603–619.
[38] T. Mansour, M.A. Rostami, S. Elumalai, and B.A. Xavier, Correcting a paper on the Randi´c and geometric-arithmetic indices, Turk. J. Math. 41 (2017), no. 1, 27–32.
[39] M. Matejić, I. Milovanović, and E. Milovanović, Some inequalities for general sum-connectivity index, J. Appl. Math. Inform. 38 (2020), no. 1 2, 189–200.
[40] E. Milovanović, M. Matejić, and I. Milovanović, Some new upper bounds for the hyper-Zagreb index, Discrete Math. Lett. 1 (2019), 30–35.
[41] I. Milovanović, E. Milovanović, and M. Matejić, Some inequalities for general sum–connectivity index, MATCH Commun. Math. Comput. Chem. 79 (2018), no. 2, 477–489.
[42] Q. Qin and Y. Shao, Minimum general sum-connectivity index of tricyclic graphs, Oper. Res. Trans. 22 (2018), no. 1, 142–150.
[43] H.S. Ramane, V.V. Manjalapur, and I. Gutman, General sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graph of subdivision graphs, AKCE Int. J. Graphs Comb. 14 (2017), no. 1, 92–
100.
[44] M. Randić, Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975), no. 23, 6609–6615.
[45] I. Tomescu, Proof of a conjecture concerning maximum general sum-connectivity index χα of graphs with given cyclomatic number when 1 < α < 2, Discrete Appl. Math. 267 (2019), 219–223.
[46] H. Wang, J.-B. Liu, S. Wang, W. Gao, S. Akhter, M. Imran, and M.R. Farahani, Sharp bounds for the general sum-connectivity indices of transformation graphs, Discrete Dyn. Nat. Soc. 2017, 2941615.
[47] Z. Yarahmadi, A.R. Ashrafi, and S. Moradi, Extremal polyomino chains with respect to Zagreb indices, Appl. Math. Lett. 25 (2012), no. 2, 166–171.
[48] L. Zhang, S. Wei, and F. Lu, The number of Kekulé structures of polyominos on the torus, J. Math. Chem. 51 (2013), no. 1, 354–368.
[49] L. Zhong and Q. Qian, The minimum general sum-connectivity index of trees with given matching number, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 2, 1527–1544.
[50] B. Zhou and N. Trinajstić, On a novel connectivity index, J. Math. Chem. 46 (2009), no. 4, 1252–1270.
[51] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010), no. 1, 210–218.
 
Communications in Combinatorics and Optimization
Volume 6, Issue 1
June 2021
Pages 81-91

Files

Share

How to cite

Statistics