آمار
| تعداد نشریات | 6 |
| تعداد شماره‌ها | 131 |
| تعداد مقالات | 1,631 |
| تعداد مشاهده مقاله | 1,879,815 |
| تعداد دریافت فایل اصل مقاله | 1,741,139 |
On the graphs attaining the minimum for the multiplicative variable Euler--Sombor index | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 16 مهر 1405 اصل مقاله (428.97 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.31791.2973 | ||
| نویسنده | ||
| Mehdi Eliasi* | ||
| Department of Mathematics, Khansar Faculty, University of Isfahan Isfahan, Iran | ||
| چکیده | ||
| We introduce the multiplicative variable Euler–Sombor index, \[\Pi_{\mathrm{ES}_\lambda}(G) = \prod_{uv\in E(G)} \sqrt{d_u^2 + d_v^2 + \lambda d_u d_v}, \] where $\lambda$ is a real parameter. For $0\le \lambda < 2$, we characterize the graphs that attain the minimum of this quantity. A key intermediate result asserts that any connected graph attaining the minimum with fixed order and size must satisfy $\Delta(G)-\delta(G)\le 1$. This structural fact yields a complete description of the graphs that attain the minimum in several families: paths among trees, cycles among unicyclic graphs, and, for bicyclic graphs, the book and single-bridge dumbbell graphs. In the general setting of $k$-cyclic graphs (with $k\ge 3$ and $n\ge 5(k-1)$), the graphs that attain the minimum are precisely the bidegreed graphs with degree set $\{2,3\}$, and the value they attain is exactly \[ (18+9\lambda)^{\frac{3k-4}{2}}(13+6\lambda)(8+4\lambda)^{\frac{n-2k+1}{2}}. \] Our framework provides the first extremal characterization for the multiplicative Euler–Sombor index (the case $\lambda=1$), together with a complete description for all $0\le \lambda<2$. This simultaneously extends the known results for the classical multiplicative Sombor index ($\lambda=0$) and introduces a new parametric family that has not been previously investigated. | ||
| کلیدواژه‌ها | ||
| Sombor index؛ Euler--Sombor index؛ multiplicative topological index؛ graphs attaining the minimum؛ k-cyclic graphs | ||
| مراجع | ||
|
[1] M.O. Albertson, The irregularity of a graph, Ars Combin. 46 (1997), 219–226.
[2] A. Ali, I. Gutman, T. Réti, A.M. Albalahi, and A.E. Hamza, Survey on extremal results and bounds for elliptic Sombor and Euler–Sombor indices, Discrete Math. Lett. 17 (2026), 101–112. https://doi.org/10.47443/dml.2026.056
[3] M. Azari and F. Falahati-Nezhad, Inequalities involving multiplicative Sombor index, J. Math. Inequal. 20 (2026), 425–448. https://doi.org/10.7153/jmi-2026-20-26
[4] K.C. Das, Extremal graphs on the multiplicative Sombor index, Discrete Math. 349 (2026), no. 10, 115192. https://doi.org/10.1016/j.disc.2026.115192
[5] R. Diestel, Graph Theory, Grad. Texts in Math., vol. 173, Springer, Heidelberg, 2005.
[6] M. Eliasi, On the minimum variable Euler–Sombor index, MATCH Commun. Math. Comput. Chem. 97 (2027), no. 2, 785–815. https://doi.org/10.46793/match.97-2.23926
[7] L. Euler, Nova series infinita maxime convergens perimetrum ellipsis exprimens, Novi Comm. Acad. Sci. Petrop. 18 (1773), 71–84.
[8] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem. 86 (2021), 11–16.
[9] I. Gutman, Relating Sombor and Euler indices, Milit. Techn. Courier 72 (2024), no. 1, 1–12. https://doi.org/10.5937/vojtehg72-48818
[10] I. Gutman, B. Furtula, and M.S. Oz, Geometric approach to vertex-degree-based topological indices – elliptic Sombor index, theory and application, Int. J. Quantum Chem. 124 (2024), e27346. https://doi.org/10.1002/qua.27346
[11] I. Gutman, I. Redžepović, G.Ö. Kizilirmak, and V. R. Kulli, Euler–Sombor index and its congeners, Open J. Math. Sci. 9 (2025), 141–148. https://doi.org/10.30538/oms2025.0249
[12] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538. https://doi.org/10.1016/0009-2614(72)85099-1
[13] D. Haji Mohammed, The Sombor index and multiplicative Sombor index of some products of graphs, Commun. Comb. Optim. 11 (2026), no. 3, 787–826. https://doi.org/10.22049/cco.2024.30123.2321
[14] D.-Y. Huh, Extremal multiplicative Sombor index of graphs with a given chromatic number, Math. Rep. 27(77) (2025), no. 1-2, 103–120. https://doi.org/10.59277/mrar.2025.27.77.1.2.103
[15] V.R. Kulli, Multiplicative Sombor indices of certain nanotubes, Int. J. Math. Arch. 12 (2021), no. 3, 1–5.
[16] , Multiplicative elliptic Sombor and multiplicative modified elliptic Sombor indices, Ann. Pure Appl. Math. 29 (2024), 19–23. https://doi.org/10.22457/apam.v29n1a03927
[17] H. Liu, Multiplicative Sombor index of graphs, Discrete Math. Lett. 9 (2022), 80–85.
[18] H. Liu, Z. Du, Y. Huang, H. Chen, and S. Elumalai, Note on the minimum bond incident degree indices of $k$-cyclic graphs, MATCH Commun. Math. Comput. Chem. 91 (2024), 255–266. https://doi.org/10.46793/match.91-1.255L
[19] G.G. Nyauli and J. Buragohain, Geometric approach to degree-based topological indices: Cosine-rule generalized Sombor indices, MATCH Commun. Math. Comput. Chem. 96 (2026), no. 1, 295–324. https://doi.org/10.46793/match.96-1.24725
[20] Z. Tang, Y. Li, and H. Deng, The Euler Sombor index of a graph, Int. J. Quantum Chem. 124 (2024), no. 9, e27387. https://doi.org/10.1002/qua.27387 [21] R. Todeschini and V. Consonni, Molecular Descriptors for Chemoinformatics, Wiley-VCH, Weinheim, 2009.
[22] K. Varmuza, M. Dehmer, and D. Bonchev, Statistical Modelling of Molecular Descriptors in QSAR/QSPR, Wiley, 2012. | ||
|
آمار تعداد مشاهده مقاله: 22 تعداد دریافت فایل اصل مقاله: 34 |
||