\((n, m)\)-graphs with maximum exponential second Zagreb index. (English) Zbl 1529.05046
Summary: For a graph \(G = (V, E)\), the exponential second Zagreb index is defined as \(e^{M_2} (G) = \sum_{u v \in E} e^{d(u) d(v)}\), where \(d(v)\) is the degree of vertex \(v \in V\). Recently, M. Eliasi [ibid. 307, 172–179 (2022; Zbl 1479.05063)] posed a conjecture about \((n, m)\)-graphs with maximum \(e^{M_2}\) when \(n \leq m \leq 2 n - 3\). In this paper, we show that this conjecture is true. Furthermore, we determine the graphs with maximum \(e^{M_2}\) among all \((n, m)\)-graphs when \(2n - 3 < m\).
MSC:
| 05C09 | Graphical indices (Wiener index, Zagreb index, Randić index, etc.) |
| 05C07 | Vertex degrees |
| 05C92 | Chemical graph theory |
| 05C35 | Extremal problems in graph theory |
| 92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |
Citations:
Zbl 1479.05063References:
| [1] | Ahlswede, R.; Katona, G. O.H., Graphs with maximal number of adjacent pairs of edges. Acta Math. Hungar., 97-120 (1978) · Zbl 0386.05036 |
| [2] | Balachandran, S.; Vetrik, T., Exponential second Zagreb index of chemical trees. Trans. Comb., 97-106 (2021) |
| [3] | Bollobás, B.; Erdös, P.; Sarkar, A., Extremal graphs for weights. Discrete Math., 5-19 (1999) · Zbl 0933.05081 |
| [4] | Cruz, R.; Monsalve, J. D.; Rada, J., The balanced double star has maximum exponential second Zagreb index. J. Comb. Optim., 544-552 (2021) · Zbl 1464.05060 |
| [5] | Cruz, R.; Rada, J., The path and the star as extremal values of vertex-degree-based topological indices among trees. MATCH Commun. Math. Comput. Chem., 715-732 (2019) · Zbl 1472.92285 |
| [6] | Das, K. C.; Elumalai, S.; Balachandran, S., Open problems on the exponential vertex-degree-based topological indices of graphs. Discrete Appl. Math., 38-49 (2021) · Zbl 1460.05045 |
| [7] | Eliasi, M., Unicyclic and bicyclic graphs with maximum exponential second Zagreb index. Discrete Appl. Math., 172-179 (2022) · Zbl 1479.05063 |
| [8] | Gutman, I.; Trinajstić, N., Graph theory and molecular orbitals. Total \(\pi \)-electron energy of alternant hydrocarbons. Chem. Phys. Lett., 4, 535-538 (1972) |
| [9] | Horoldagva, B.; Das, K. C., On Zagreb indices of graphs. MATCH Commun. Math. Comput. Chem., 295-301 (2021) · Zbl 1473.92072 |
| [10] | Horoldagva, B.; Das, K. C.; Selenge, Ts., Complete characterization of graphs for direct comparing Zagreb indices. Discrete Appl. Math., 146-154 (2016) · Zbl 1346.05037 |
| [11] | Horoldagva, B.; Selenge, Ts.; Buyantogtokh, L.; Dorjsembe, Sh., Upper bounds for the reduced second Zagreb index of graphs. Trans. Comb., 137-148 (2021) · Zbl 1488.05099 |
| [12] | Rada, J., Exponential vertex-degree-based topological indices and discrimination. MATCH Commun. Math. Comput. Chem., 29-41 (2019) · Zbl 1472.92333 |
| [13] | Selenge, Ts.; Horoldagva, B., Extremal kragujevac trees with respect to Sombor indices. Commun. Comb. Optim. (2023) |
| [14] | Sigarreta, J. M., Extremal problems on exponential vertex-degree-based topological indices. Math. Biosci. Eng., 6985-6995 (2022) · Zbl 1516.92134 |
| [15] | Wang, F.; Wu, B., The reduced Sombor index and the exponential reduced Sombor index of a molecular tree. J. Math. Anal. Appl. (2022) · Zbl 1497.05045 |
| [16] | Xu, K.; Das, K. C.; Balachandran, S., Maximizing the Zagreb indices of \(( n , m )\)-graphs. MATCH Commun. Math. Comput. Chem., 641-654 (2014) · Zbl 1464.05118 |
| [17] | Xu, C.; Horoldagva, B.; Buyantogtokh, L., The exponential second Zagreb index of \(( n , m )\)-graphs. Mediterr. J. Math., 181 (2023) · Zbl 1511.05052 |
| [18] | Zeng, M.; Deng, H., An open problem on the exponential of the second Zagreb index. MATCH Commun. Math. Comput. Chem., 367-373 (2021) · Zbl 1473.92110 |
| [19] | Zhou, W.; Pang, S.; Liu, M.; Ali, A., On bond incident degree indices of connected graphs with fixed order and number of pendent vertices. MATCH Commun. Math. Comput. Chem., 625-642 (2022) · Zbl 1505.92305 |
| [20] | https://cocalc.com/share/public_paths/4295e1dffd1ad3afd96e40a10604cced930d959d |
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