The chain graph shown above has only one centre, $d$. The
largest distance from $d$ to any other vertex is $3$, from $d$
to $a$ and from $d$ to $g$. The largest distance from $c$ to any
other vertex is $4$, to vertex $g$, which is greater than $3$,
so $c$ is not a centre. Similarly, the greatest distance from
$e$ to any other vertex is $4$, to $a$, and you can check that
the other vertices are even worse. The tree above has two
centres; can you find them? HINT for the proof: Let $T$ be a
tree. If $u$ and $v$ are any vertices of $T$, there is a unique
path from $u$ to $v$ in $T$. (You’ve probably proved this
already; if not, you’ll want to do so now.) Call the length of
this path the distance between $u$ and $v$ in $T$. Among all
pairs of vertices of $T$ pick two, $u$ and $v$, with the largest
possible distance between them. If that distance is even,
there’s a vertex smack in the middle of that path; prove that
it’s the unique centre of $T$. If the distance between $u$ and
$v$ is odd, the path looks, for examp