The revised edge-Szeged index of a connected graph $G$ is defined as Sze*(G)=∑e=uv∊E(G)( (mu(e|G)+(m0(e|G)/2)(mv(e|G)+(m0(e|G)/2) ), where mu(e|G), mv(e|G) and m0(e|G) are, respectively, the number of edges of G lying closer to vertex u than to vertex v, the number of ed...

Theorem 1 was the starting point for many other suÆ ient onditions for = Æ. Lesniak [9℄ weakened the ondition Æ bn=2 to deg(u) + deg(v) n for all pairs of nonadja ent verti es u and v in G. Plesn k [10℄ showed that every graph of diameter 2 satis es = Æ and thus generalized Lesniak's result. Improvements of Plesn k's result an be found in Plesn k, Zn am [11℄ and Dankelmann, Volkmann [4℄ (see al...

We show that a k-edge-connected graph on n vertices has at least n(k/2)n−1 spanning trees. This bound is tight if k is even and the extremal graph is the n-cycle with edge-multiplicities k/2. For k odd, however, there is a lower bound cn−1 k where ck > k/2. Specifically, c3 > 1.77 and c5 > 2.75. Not surprisingly, c3 is smaller than the corresponding number for 4-edge-connected graphs. Examples ...