Liang, M., Cheng, B., Liu, J. (2018). Solution to the minimum harmonic index of graphs with given minimum degree. Transactions on Combinatorics, 7(2), 25-33. doi: 10.22108/toc.2017.101076.1462

Meili Liang; Bo Cheng; Jianxi Liu. "Solution to the minimum harmonic index of graphs with given minimum degree". Transactions on Combinatorics, 7, 2, 2018, 25-33. doi: 10.22108/toc.2017.101076.1462

Liang, M., Cheng, B., Liu, J. (2018). 'Solution to the minimum harmonic index of graphs with given minimum degree', Transactions on Combinatorics, 7(2), pp. 25-33. doi: 10.22108/toc.2017.101076.1462

Liang, M., Cheng, B., Liu, J. Solution to the minimum harmonic index of graphs with given minimum degree. Transactions on Combinatorics, 2018; 7(2): 25-33. doi: 10.22108/toc.2017.101076.1462

Solution to the minimum harmonic index of graphs with given minimum degree

The harmonic index of a graph $G$ is defined as $ H(G)=\sum\limits_{uv\in E(G)}\frac{2}{d(u)+d(v)}$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. Let $\mathcal{G}(n,k)$ be the set of simple $n$-vertex graphs with minimum degree at least $k$. In this work we consider the problem of determining the minimum value of the harmonic index and the corresponding extremal graphs among $\mathcal{G}(n,k)$. We solve the problem for each integer $k (1\le k\le n/2)$ and show the corresponding extremal graph is the complete split graph $K_{k,n-k}^*$. This result together with our previous result which solve the problem for each integer $k (n/2 \le k\le n-1)$ give a complete solution of the problem.

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