Mathematics Interdisciplinary Research

Mathematics Interdisciplinary Research

Topological Indices and Structural Properties of a Graph Transformation

Articles in Press

Document Type : Original Scientific Paper

Author
Meysam Taheri
‎University of Applied Science and Technology, ‎Tehran‎, ‎I‎. ‎R‎. ‎Iran
10.22052/mir.2026.258147.1563
Abstract
‎Graph rewriting or graph transformation has existed for many years‎. ‎The $R_k$-Construction is obtained by replacing each vertex of a graph $G$ with $k$ independent copies‎. ‎Each edge $uv$ of $G$ is replaced by edge joining every copy of $u$ to every copy of $v$‎. ‎Hence‎, ‎every vertex in $R_k(G)$ has degree multiplied by $k$‎. ‎A topological index is a value associated a molecule's chemical structure‎. ‎Among various topological indices‎, ‎degree-based indices form an important and extensively studied class that is fundamental in chemical graph theory and structural graph analysis‎. ‎In this paper‎, ‎we study the structural invariants and metrics of the $R_k$-Construction‎. ‎Moreover‎, ‎we examine the behavior of certain topological indices on this transformation and demonstrate that an infinite family of graphs possesses an integer-valued topological index‎. ‎Furthermore‎, ‎we correct and generalize several previously reported results in the literature and calculate several topological indices for the graphs generated by this construction‎.
Keywords
Topological indices&lrm
Homogeneous function&lrm
, &lrm
Graph transformation
Subjects
[1] F. Harary, Graph Theory, Addison-Wesley Publishing Company, New York, 1969.
[2] P. Zhang and G. Chartrand, Introduction to Graph Theory, Tata McGraw-Hell Edition 2006.
[3] N. A. Abd Aziz, A note on graphs with integer Sombor index, Commun. Comb. Optim. 11 (2026) 205-209, https://doi.org/10.22049/cco.2024.29474.2013.
[4] M. Randic, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609-6615, https://doi.org/10.1021/ja00856a001.
[5] E. Estrada, L. Torres, L. Rodríguez and I. Gutman, An atom-bond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem. 37A (1998) 849-855.
[6] B. Furtula, A. Graovac and D. Vukicevic, Augmented Zagreb index, J. Math. Chem. 48 (2010) 370-380, https://doi.org/10.1007/s10910-010-9677-3.
[7] L. Zhong, The harmonic index for graphs, Appl. Math. Lett. 25 (2012) 561-566, https://doi.org/10.1016/j.aml.2011.09.059.
[8] I. Gutman, B. Furtula and C. Elphick, Three new/old vertex-degree-based topological indices, MATCH Commun. Math. Comput. Chem. 72 (2014) 617-632.
[9] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals: total $\phi$-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538, https://doi.org/10.1016/0009 2614(72)85099-1.
[10] D. Vukicevic and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46 (2009) 1369-1376, https://doi.org/10.1007/s10910-009-9520-x.
[11] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem. 86 (2021) 11-16.
[12] F. Movahedi, I. Gutman, I. Redžepovic and B. Furtula, Diminished Sombor index, MATCH Commun. Math. Comput. Chem. 95 (2026) 141-162.
[13] A. T. Balaban, P. V. Khadikar and S. Aziz, Comparison of topological indices based on iterated sum versus product operations, Iranian J. Math. Chem. 1 (2010) 43-67, https://doi.org/10.22052/ijmc.2010.5134.
[14] K. Elahi, A. Ahmad, M. A. Asim and R. Hasni, Computation of topological indices of binary and ternary trees using algorithmic approach, Iranian J. Math. Chem. 15 (2024) 107-115, https://doi.org/10.22052/ijmc.2024.252698.1701.
[15] G. H. Fath–Tabar, A. Azad and N. Elahinezhad, Some topological indices of tetrameric 1,3-adamantane, Iranian J. Math. Chem. 1 (2010) 111-118, https://doi.org/10.22052/ijmc.2010.5161.
[16] I. Gutman, Degree–based topological indices, Croat. Chem. Acta 86 (2013) 351-361, http://dx.doi.org/10.5562/cca2294.
[17] I. Gutman, Edge-decomposition of topological indices, Iranian J. Math. Chem. 6 (2015) 103-108, https://doi.org/10.22052/ijmc.2015.10107.
[18] S. Nikolic, G. Kovacevic, A. Milicevic and N. Trinajstic, The Zagreb indices 30 years after, Croat. Chem. Acta. 76 (2003) 113-124.
[19] M. Randic, The connectivity index 25 years after, J. Mol. Graph. Model. 20 (2001) 19-35, https://doi.org/10.1016/S1093-3263(01)00098-5.
[20] J. M. Rodríguez and J. M. Sigarreta, The harmonic index. In: I. Gutman, B. Furtula, K. C. Das, E. Milovanovic and I. Milovanovic (Eds.), Bounds in Chemical Graph Theory-Basics, Univ. Kragujevac, Kragujevac. (2017) 229-281.
[21] Y. Yuan, B. Zhou and N. Trinajstic, On geometric-arithmetic index, J. Math. Chem. 47 (2010) 833-841, https://doi.org/10.1007/s10910-009-9603-8.
Mathematics Interdisciplinary Research

Articles in Press, Accepted Manuscript
Available Online from 14 October 2026