Gutman index of polyomino chains

Document Type : Full Length Article

Authors

Department of Mathematics, Faculty of Basic Sciences, Shahed University, Tehran, I. R. Iran

Abstract

‎The Gutman index is a degree-distance-based topological descriptor of connected graphs‎. ‎In this paper‎, ‎we derive explicit analytic expressions for its expected value in polyomino chains built by sequentially attaching square tiles via one of two fixed local connection modes‎. ‎This expectation is expressed as a cubic polynomial in the number of tiles $n$‎. ‎We then identify which attachment patterns yield the extremal (maximum and minimum) values and compute the overall average of the Gutman index across all polyomino chains of length $n$‎. ‎These results enhance the topological analysis of square-tiled networks with applications in chemical graph theory‎, ‎polymer science‎, ‎and materials design‎.

Graphical Abstract

Gutman index of polyomino chains

Keywords

Main Subjects


[1] J. A. Bondy, U. S. R. Murty, Graph theory, Springer, New York, 2008.
https://www.zib.de/userpage/groetschel/teaching/WS1314/BondyMurtyGTWA.pdf
[2] F. Buckley, F. Harary, Distance in graphs, Addison-Wesley, Reading, 1989.
[3] A. L. Chen, F. J. Zhang, Wiener index and perfect matchings in random
phenylene chains, MATCH Commun. Math. Comput. Chem. 61 (2009) 623–630.
https://match.pmf.kg.ac.rs/electronic versions/Match61/n3/match61n3 623-630.pdf
[4] X. Cheng, X. Li, Extremal general Gutman index of trees, MATCH Commun. Math. Comput.
Chem. 89 (2023) 567–582. https://doi.org/10.46793/match.89-3.567C
[5] H. Deng, Wiener indices of spiro and polyphenyl hexagonal chains, Math. Comput. Model. 55
(2012) 634–644. https://doi.org/10.48550/arXiv.1006.5488
[6] H. L. Donald, M. A. Whitehead, Molecular geometry and bond energy. III. Cyclooctatetraene and related compounds, J. Am. Chem. Soc. 91 (1969) 238–242.
https://doi.org/10.3389/fchem.2022.1067874
[7] R. C. Entringer, D. E. Jackson, D. A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (1976)
283–296. http://eudml.org/doc/12941
[8] E. Estrada, D. Bonchev, P. Zhang, Chemical graph theory, Discrete Math. Appl. (2013) 1–24.
https://doi.org/10.1201/b16132-92
[9] D. R. Flower, On the properties of bit string-based measures of chemical similarity, J. Chem. Inf.
Comput. Sci. 38 (1998) 379–386. https://doi.org/10.1021/ci970437z
[10] S. W. Golomb, Polyominoes, Princeton Univ. Press, Princeton, 1994.
[11] A. Granados, A. Portilla, Y. Quintana, E. Touris, Bounds for the Gutman-Milovanovic index and
some applications, J. Math. Chem. 63 (2025) 406–418. https://doi.org/10.1007/s10910-024-01677-7
[12] F. Guo, H. Wang, M. Yang, S. Wang, The limiting behaviors for the Gutman index and
the Schultz index in a random (2k − 1)-polygonal chain, Filomat 37 (2023) 2487–2501.
https://doi.org/10.2298/FIL2407487G
[13] H. Hosoya, Topological index, a newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn. 44 (1971) 2332–2339.
https://doi.org/10.1246/bcsj.44.2332
[14] L. Ma, H. Bian, B. J. Liu, H. Z. Yu, The expected values of the Wiener indices in the random
phenylene and spiro chains, Ars Combin. 130 (2017) 267–274.
[15] Y. Mao, K. C. Das, Steiner Gutman index, MATCH Commun. Math. Comput. Chem. 79 (2018)
779–794. https://match.pmf.kg.ac.rs/electronic versions/Match79/n3/match79n3 779-794.pdf
[16] E. D. Molina, J. M. Rodriguez-Garcia, J. M. Sigarreta, S. J. Torralbas Fitz, On the GutmanMilovanovic index and chemical applications, AIMS Mathematics 10 (2025) 1998–2020.
https://doi.org/10.3934/math.2025094
[17] A. I. Pavlyuchko, E. V. Vasiliev, L. A. Gribov, Quantum chemical estimation of the overtone contribution to the IR spectra of hydrocarbon halogen derivatives, J. Struct. Chem. 51 (2010) 1045–1051.
https://doi.org/10.1007/s10947-010-0161-5
[18] J. F. Qi, M. L. Fang, X. Y. Geng, The expected value for the Wiener index in the random spiro chains,
Polycycl. Aromat. Comp. (2022). https://doi.org/10.1080/10406638.2022.2038218
[19] S. C. Sigarreta, S. M. Sigarreta, H. Cruz-Suarez, On degree-based topological indices of random
polyomino chains, Math. Biosci. Eng. 19 (2022) 8760–8773. https://doi.org/10.3934/mbe.2022406
[20] S. Sigarreta, H. Cruz-Suarez, Zagreb connection indices on polyomino chains and random polyomino chains, Mathematics 12 (2024) 57. https://doi.org/10.48550/arXiv.2405.12389
[21] C. Tao, S. Tang, X. Geng, The limiting behaviors of the Gutman and Schultz indices in random
2k-sided chains, Axioms 13 (2024) 518. https://doi.org/10.3390/axioms13080518
[22] L. L. Zhang, Q. S. Li, S. C. Li, M. J. Zhang, The expected values for the Schultz index, Gutman index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff
index of a random polyphenylene chain, J. Discrete Appl. Math. 282 (2020) 243–256.
https://doi.org/10.1016/j.dam.2019.11.007
Volume 10, Issue 4
December 2025
Pages 375-392
  • Receive Date: 23 September 2025
  • Revise Date: 07 October 2025
  • Accept Date: 23 November 2025
  • First Publish Date: 25 November 2025
  • Publish Date: 01 December 2025