On hyper-Kirchhoff index. (English) Zbl 1234.05074
Summary: The hyper-Kirchhoff index is introduced when the hyper-Wiener operator is applied to the resistance-distance matrix of a connected graph. We give lower and upper bounds for the hyper-Kirchhoff index, and determine the \(n\)-vertex unicyclic graphs with the smallest, the second and the third smallest as well as the largest, the second and the third largest hyper-Kirchhoff indices for \(n\geq 5\). We also determine the \(n\)-vertex unicyclic graphs of cycle length \(s\), \(3\leq s\leq n\), with the smallest and the largest hyper-Kirchhoff indices.
MSC:
| 05C12 | Distance in graphs |
| 05C90 | Applications of graph theory |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
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