For the ring graph, the only diﬀerence is that the ﬁrst and last vertices are
neighbors, so
L
=
2

1

1

1
2
.
.
.

1
.
.
.

1
.
.
.
2

1

1

1
2
(b) If
v
∈
R
N
is an eigenvector for the path graph, then
L
p
v
=
λv
If
w
= (
v
1
,v
2
,...,v
n
,v
n
,v
n

1
,...,v
1
)
>
, then (
L
r
v
)
i
=
λv
i
for
i
=
{
2
,...,N

1
,N
+ 2
,...,
2
N

1
}
since the ring graph Laplacian is the same on these
entries as the path graph Laplacian, and the vectors are the same (up to a
reversal of order on the last half of the vector, which doesn’t matter). On the
ﬁrst coordinate, we know that since
v
is an eigenvector of
L
p
,
v
1

v
2
=
λv
1
, so
(
L
r
w
)
1
= 2
w
1

w
2
N

w
2
= 2
v
1

v
2

v
1
=
v
1

v
2
=
λv
1
The argument is similar for the other endpoints, so
w
is an eigenvector of
L
r
.
Likewise, if