Extremum problems for eigenvalues of discrete Laplace operators

8 REN GUO
Q(2) ≤ 0 implies Q ′ (2) ≤ 0. Hence the only possibility is Q ′ (2) = 0. This requires
that a 1 + a 2 + a 3 + a 4 = 4. Therefore we must have θ 1 = θ 2 = θ 3 = θ 4 = π 4 .
4. general cyclic polygons
4.1. Assume n ≥ 5. The vertices of a cyclic n-gon decompose its circumcircle into
n arcs. We assume the radius of the circumcircle is 1 and the lengths of the n arcs
are 2θ 1 ,2θ 2 ,...,2θ n .
The discrete Laplace operator of a cyclic n-gon is independent of the choice of a
triangulation. It is
⎛
L n =
⎜
⎝
⎞
a 1 + a n −a 1 0 0 ... −a n
−a 1 a 1 + a 2 −a 2 0 ... 0
0 −a 2 a 2 + a 3 −a 3 ... 0
0 0 −a 3 a 3 + a 4 ... 0
0 0 0 −a 4 ... 0
. ..
⎟
⎠
−a n 0 0 0 ... a n−1 + a n
The eigenvalues are 0 = λ 0 ≤ λ 1 ≤ ... ≤ λ n−1 .
4.2. We have ∑ n−1
i=1 λ i = 2 ∑ n
i=1 a i. By the similar argument in the case of triangles
and cyclic quadrilaterals, we can show that ∑ n
i=1 a i has the unique critical point
(θ 1 ,...,θ n ) = ( π n ,..., π n ).
Since there is at most one non-positive number in a 1 ,...,a n , without loss of
∑
generality, we may assume a 1 > 0,...,a n−1 > 0. Since a n−1 + a n > 0, we have
n
i=1 a i > 0.
We investigate the behavior of ∑ n
i=1 a i when the variable approaches the boundary
of the domain
Ω n = {(θ 1 ,...,θ n ) | θ 1 + ... + θ n = π,θ i > 0,i = 1,...,n}.
Let (θ 1 (t),θ 2 (t),θ 3 (t),θ(t)), t ∈ [0, ∞), be a path in the domain Ω 4 . Let a i (t) =
cot θ 1 (t) for i = 1,2,3,4.
Without loss of generality, we have
lim (θ 1(t),...,θ n (t)) = (0,s 2 ,...,s n )
t→∞
where s i ≥ 0 for i = 2,...,n and s 2 + ... + s n = π. And we can assume that
s 2 < π 2 ,...,s n−1 < π 2 . Since a i(t) > 0 for i = 2,...,n − 1 and a n−1 + a n > 0 when t
is sufficiently large, lim t→∞ a 1 = ∞ implies that lim t→∞
∑ n
i=1 a i = ∞.
Thus ∑ n
i=1 a i achieved the absolute minimum at ( π n ,..., π n ).
4.3. In this subsection we verify the statement about ∏ n−1
i=1 λ i.
The weighted matrix-tree Theorem. Let M be an n by n matrix. If the sum
of the entries of each row or each column of M vanishes, all principle n−1 by n−1
submatrices of M have the same determinant, and this value is equal to 1 n times
the product of all nonzero eigenvalues of M.
For the reference of the weighted matrix-tree Theorem, for example, see [LW],
page 450, Problem 34A or [DKM], Theorem 1.2.
.