# Minimal Spanning Tree vs Minimum Spanning Tree

I got confused about minimal and minimum in context of graph theory. Although, I have understanding that minimal means more than one minimum i.e. none qualifies as actual minimum so we say them minimal.

What exactly does minimal spanning tree mean? Also, what exactly does 'minimal edge' mean?

How do they differ from minimum spanning tree? How are 'minimal edge' different from minimum edge?

This doubt arose due to the statement:

S: There exists a minimum weight edge in G, which exists in every minimum spanning tree of G.

Can we say on using word 'minimum' it's implied that there is only one minimum weight edge. Besides, statement used "exists" which is singular.

• Where did you encounter "minimal spanning tree" and "minimal edge"? This statement doesn't mention them. Minimum here means a global minimum: a tree/edge with the smallest weight (there is no other tree/edge with a smaller weight).
– user114966
Mar 2, 2021 at 22:44
• Today, here: link. Although, someone else pointed out too, but he seem to have deleted his comment. Mar 2, 2021 at 22:47
• @Dmitry, Sir, Does all edges with minimum weight qualifies as Global Minimum, or Do we say Global minimum in that case don't exist? Mar 2, 2021 at 22:55
• I agree with this answer: math.stackexchange.com/a/2142655/743044 . Note that "minimum" implies "minimal". So it's not incorrect to use "minimal" instead of "minimum", but no reason to do that.
– user114966
Mar 2, 2021 at 22:55
• All such edges are global minima. (I use word "global" only to stress that they are minima across the set of all edges)
– user114966
Mar 2, 2021 at 22:57

• For example, the minimum value attained by the sine function is $-1$. It is not attained uniquely. Mar 3, 2021 at 12:08