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Let $G$ be a finite connected graph with minimum degree $\delta$. The leaf number $L\left(G\right)$ of $G$ is defined as the maximum number of leaf vertices contained in a spanning tree of $G$. We prove that if $\delta \ge \frac{1}{2}(L\left(G\right)+1)$, then $G$ is 2-connected. Further, we deduce, for graphs of girth greater than 4, that if $\delta \ge \frac{1}{2}\left(L\right(G)+1)$, then $G$ contains a spanning path. This provides a partial solution to a conjecture of the computer program Graffiti.pc [DeLaVi na and Waller, Spanning trees with many leaves and average distance, Electron. J. Combin. 15 (2008),...

The Gutman index and the edge-Wiener index have been extensively investigated particularly in the last decade. An important stream of re- search on graph indices is to bound indices in terms of the order and other parameters of given graph. In this paper we present asymptotically sharp upper bounds on the Gutman index and the edge-Wiener index for graphs of given order and vertex-connectivity κ, where κ is a constant. Our results substantially generalize and extend known results in the area.