Iranian Journal of Mathematical Chemistry

Iranian Journal of Mathematical Chemistry

Extremal $\alpha$-Sum of the Randi\'c Index and its Reciprocal in Small k-Uniform Hypertrees

Document Type : Research Paper

Author
Nelson Berrocal Huamani
Department of Mathematics and Physics‎, ‎Universidad Nacional de San Crist\'obal de Huamanga‎, ‎Ayacucho‎, ‎Per\'u
10.22052/ijmc.2026.258426.2121
Abstract
‎For $\alpha>0$ we study a Randi'c-type functional on $k$-uniform hypergraphs‎,‎ $\Phi_\alpha(H)=\sum_{e\in E(H)}\Big(P_H(e)^{\alpha/2}+P_H(e)^{\alpha/2}\Big), ‎P_H(e)=\prod_{v\in e} d_H(v)‎,$‎‎which extends Gao’s $(\mathrm R+\mathrm{RR})$ at $\alpha=1$‎. We prove that splitting a (redundant) intersection strictly decreases $\Phi_\alpha$ for every $\alpha>0$‎, and deduce a sharp maximum with fixed order $n$‎: ‎among simple connected $k$-uniform hypergraphs on $n$ vertices‎, $\Phi_\alpha$ is uniquely maximized by the complete hypergraph $K_n^{(k)}$‎. We then study minimizers on small $k$-uniform hypertrees‎, where $m=|E(H)|$ denotes the number of hyperedges‎. For $m=3$ hyperedges‎, ‎there is a unique threshold $\alpha_0\approx 6.1412175821$ where the minimizer switches from the linear hyperpath $P_{3,k}$‎ to the sunflower $S(3,1,k)$ (with both extremal at $\alpha=\alpha_0$)‎. For m=4 no transition occurs‎: ‎$P_{4,k}$ is the unique minimizer for all $\alpha>0$‎. Finally‎, ‎we show a barrier at $\alpha=4$‎: ‎for $\alpha>4$ branching can decrease $\Phi_\alpha$‎, ‎invalidating direct pendent-path merging‎.
Keywords
Topological index
Degree product
Intersection splitting
Threshold phenomenon
Minimizer characterization
Subjects
[1] N. Trinajstic, Chemical Graph Theory, CRC Press, Boca Raton, 2018, https://doi.org/10.1201/9781315139111.
[2] M. Randic, On Characterization of Molecular Branching, J. Am. Chem. Soc. 97 (1975) 6609–6615, https://doi.org/10.1021/ja00856a001.
[3] C. Delorme, O. Favaron and D. Rautenbach, On the Randic Index, Discrete Math. 257 (2002) 29–38, https://doi.org/10.1016/S0012-365X(02)00256-X.
[4] G. Caporossi, I. Gutman, P. Hansen and L. Pavlovic, Graphs with maximum connectivity index, Comput. Biol. Chem. 27 (2003) 85–90, https://doi.org/10.1016/S00978485(02)00016-5.
[5] X. Li and Y. Shi, A Survey on the Randic Index, MATCH Commun. Math. Comput. Chem. 59 (2008) 127–156.
[6] I. Gutman, B. Furtula and C. Elphick, Three new/old vertex–degree–based topological indices, MATCH Commun. Math. Comput. Chem. 72 (2014) 617–632.
[7] W. Gao, The sum of a topological index and its reciprocal index, MATCH Commun. Math. Comput. Chem. 93 (2025) 535–547.
[8] E. V. Konstantinova and V. A. Skorobogatov, Application of hypergraph theory in chemistry, Discrete Math. 235 (2001) 365–383, https://doi.org/10.1016/S0012365X(00)00290-9.
[9] C. Berge, Hypergraphs: Combinatorics of Finite Sets, North-Holland Mathematical Library, 45, 1989.
[10] A. Bretto, Hypergraph Theory: An Introduction, Springer, Cham, 2013.
[11] G. H. Shirdel, A. Mortezaee and L. Alameri, General Randic index of uniform hypergraphs, Iranian J. Math. Chem. 14 (2023) 121–133, https://doi.org/10.22052/ijmc.2023.252836.1712.
[12] G. H. Shirdel, A. Mortezaee and L. Alaameri, Randic matrix and Randic energy of uniform hypergraphs, Math. Interdisc. Res. 9 (2024) 269–288, https://doi.org/10.22052/mir.2024.254116.1456.
[13] T. Vetrík, Degree-based indices of hypergraphs: definitions and first results, Asian-Eur. J. Math. 17 (2024) #2450023, https://doi.org/10.1142/S1793557124500232.
[14] A. Banerjee, On the spectrum of hypergraphs, Linear Algebra Appl. 614 (2021) 82–110, https://doi.org/10.1016/j.laa.2020.01.012.
[15] W. Gao, On the sum of the Randic index and the reciprocal Randic index of uniform hypergraphs, Discrete Math. Lett. 17 (2026) 7–14, https://doi.org/10.47443/dml.2025.164.
[16] J. Nieminen and M. Peltola, Hypertrees, Appl. Math. Lett. 12 (1999) 35–38, https://doi.org/10.1016/S0893-9659(98)00145-1.
Iranian Journal of Mathematical Chemistry
Volume 17, Issue 4
In Progress
Autumn 2026
Pages 363-380