Gutman index‎, ‎edge-Wiener index and edge-connectivity

Document Type: Research Paper

Authors

1 Department of Mathematics, University of Zimbabwe, P. O. Box MP 167, Mount Pleasant, Harare, Zimbabwe

2 School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa

3 Department of Mathematics and Applied Mathematics, University of the Free State, P. O. Box 339, Bloemfontein, 9300, South Africa

10.22108/toc.2020.124104.1749

Abstract

‎We study the Gutman index ${\rm Gut}(G)$ and the edge-Wiener index $W_e (G)$ of connected graphs $G$ of given order $n$ and edge-connectivity $\lambda$‎. ‎We show that the bound ${\rm Gut}(G) \le \frac{2^4 \cdot 3}{5^5 (\lambda+1)} n^5‎ + ‎O(n^4)$ is asymptotically tight for $\lambda \ge 8$‎. ‎We improve this result considerably for $\lambda \le 7$ by presenting asymptotically tight upper bounds on ${\rm Gut}(G)$ and $W_e (G)$ for $2 \le \lambda \le 7$‎.

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