Let 𝐺(𝑉,𝐸) be a simple, undirected, edge-weighted graph with unique edge…

Given:

G(V,E) is a simple undirected weighted graph
with unique edge weights.

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Statement 1
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"In every cycle C of G,
the edge with the largest weight in C
is not in any MST."

This statement is TRUE.

By the Cycle Property of MSTs:

The maximum-weight edge in a cycle
can never belong to a Minimum Spanning Tree.

Since all edge weights are unique,
the largest edge is uniquely determined.

Therefore Statement 1 is TRUE.

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Statement 2
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"In every cycle C of G,
the edge with the smallest weight in C
is in every MST."

This statement is FALSE.

The minimum edge of a cycle is preferred,
but it is not guaranteed to appear
in every MST.

Hence Statement 2 is FALSE.

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Statement 3
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"For every vertex v in V,
the edge with the largest weight incident on v
is not in any MST."

This statement is FALSE.

The largest incident edge may still be required
to connect that vertex to the graph.

Hence Statement 3 is FALSE.

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Statement 4
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"For every vertex v in V,
the edge with the smallest weight incident on v
is in every MST."

This statement is TRUE.

Consider the cut separating vertex v
from all remaining vertices.

The minimum-weight edge incident on v
is the unique lightest edge crossing that cut.

By the Cut Property of MSTs,
this edge must belong to every MST.

Hence Statement 4 is TRUE.

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Final Answer
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Statements 1 and 4 are TRUE.
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