The harmonic index of a graph. (English) Zbl 1305.05095
Summary: The harmonic index of a graph \(G\) is defined as the sum of weights \(\frac{2}{d(v_i)+d(v_j)}\) of all edges \(v_iv_j\) of \(G\), where \(d(v_i)\) denotes the degree of the vertex \(v_i\) in \(G\). In this paper, we study how the harmonic index behaves when the graph is under perturbations. These results are used to provide a simpler method for determining the unicyclic graphs with maximum and minimum harmonic index among all unicyclic graphs, respectively. Moreover, a lower bound for harmonic index is also obtained.
MSC:
| 05C22 | Signed and weighted graphs |
| 05C07 | Vertex degrees |
| 05C12 | Distance in graphs |
| 05C90 | Applications of graph theory |
| 92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |
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