The hyper-Zagreb index of cacti with perfect matchings. (English) Zbl 1473.05065
Summary: Let \(G\) be a simple connected graph. The hyper-Zagreb index is defined as \(HM(G) = \sum_{uv \in E(G)} (d_G (u)+d_G (v))^2\). A connected graph \(G\) is a cacti if all blocks of \(G\) are either edges or cycles. Let \(\zeta (2n,r)\) be the set of cacti of order \(2n\) with a perfect matching and \(r\) cycles. In this paper, we determine sharp upper bounds of the hyper-Zagreb index of cacti among \(\zeta (2n,r)\) and characterize the corresponding extremal cacti.
MSC:
| 05C09 | Graphical indices (Wiener index, Zagreb index, Randić index, etc.) |
| 05C07 | Vertex degrees |
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
| 05C40 | Connectivity |
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