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On the minimum number of spanning trees in $$k$$-edge-connected graphs. (English) Zbl 1359.05023
Summary: 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 $$c_k^{n-1}$$, where $$c_k>k/2$$. Specifically, $$c_3>1.77$$ and $$c_5>2.75$$. Not surprisingly, $$c_3$$ is smaller than the corresponding number for 4-edge-connected graphs. Examples show that $$c_3\leq\sqrt{2+\sqrt{3}}\approx1.93$$. However, we have no examples of 5-edge-connected graphs with fewer spanning trees than the $$n$$-cycle with all edge multiplicities (except one) equal to 3, which is almost 6-regular. We have no examples of 5-regular 5-edge-connected graphs with fewer than $$3.09^{n-1}$$ spanning trees, which is more than the corresponding number for 6-regular 6-edge-connected graphs. The analogous surprising phenomenon occurs for each higher odd edge connectivity and regularity.

##### MSC:
 05C05 Trees 05C40 Connectivity
##### Keywords:
spanning tree; cubic graph; edge connectivity
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##### References:
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