\((n, m)\)-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:
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