Number of perfect matchings in grids and matching polynomial
Péter Csikvári
This is some additional text to Miniature 21 of Matousek’s book. In this
text we mainly concentrate on the number of perfect matchings of grids of size
m × n.
Theorem 1.1 (Kasteleyn and independently Fisher and Temperley). Let Zm,n
be the number of perfect matchings of the grid of size m × n. Then
(m n (
(
)
(
)))1/4
∏∏
πj
πk
4 cos2
Zm,n =
+ 4 cos2
.
m
+
1
n+1
j=1 k=1
Proof. Note that the grid is a bipartite graph, so we can color the vertices of the
grid by black and white such that only vertices of different color are adjacent.
Let S be the incidence matrix (also called bipartitie adjacency matrix) of the
bipartite graph: Sij = 1 if black and white vertices bi and wj are adjacent,
and 0 otherwise. Then the number of perfect matchings is exactly per(S), the
permanent of S. We will give a "signing" σ of S such that per(S) = | det(S σ )|.
σ
σ
Let S(x,y),(x,y±1)
= i and S(x,y),(x±1,y)
= 1, and 0 otherwise. We claim that
σ
per(S) = | det(S )|. One way to see it is the following: from any perfect
matching M1 we can arrive to any other perfect matching M2 by a sequence
of moves of the following type: choose two edges of the form e = ((x, y), (x +
1, y)), f = ((x, y + 1), (x + 1, y + 1) and replace them by e′ = ((x, y), (x, y +
1)), f ′ = ((x + 1, y), (x + 1, y + 1), or do the reverse of this operation (why?). In
det(S σ ) this operation does the following thing: the sign of th corresponding
permutation changes because we did a transposition, but also the weight of the
perfect matching changes since we changed two edges of weight 1 to two edges
of weight i or vice versa. So every expansion term of det(S σ ) corresponding to
a perfect matching will give the same quantity, hence per(S) = | det(S σ )|.
Next we will compute det(S σ ). It will be more convenient to work with the
matrix
)
(
0
Sσ
.
A=
(S σ )T 0
Clearly, det(A) = det(S σ )2 . It turns out that we can give all eigenvectors and
eigenvalues explicitly. Note that the vector consisting of the values f (x, y) is
an eigenvector of A belonging to the eigenvalue λ if
λf (x, y) = f (x + 1, y) + f (x − 1, y) + if (x, y + 1) + if (x, y − 1),
1
2
where f (r, t) = 0 if r ∈ {0, m + 1} or t ∈ {0, n + 1}. Let 1 ≤ j ≤ m, 1 ≤ k ≤ n,
j
k
and z = e2πi m+1 and w = e2πi n+1 . Let us consider the vector fj,k defined as
follows:
(
)
(
)
πjx
πky
x
−x
y
−y
fj,k (x, y) = (z − z )(w − w ) = −4 sin
sin
.
m+1
n+1
(
)
Then with λj,k = z + z1 + i w + w1 we have
λj,k fj,k (x, y) = fj,k (x + 1, y) + fj,k (x − 1, y) + ifj,k (x, y + 1) + ifj,k (x, y − 1).
Indeed,
fj,k (x+1, y)+fj,k (x−1, y) = (z x+1 −z −x−1 )(wy −w−y )+(z x−1 −z −x+1 )(wy −w−y ) =
= (z + z −1 )(z x − z −x )(wy − w−y ) = (z + z −1 )fj,k (x, y),
and
ifj,k (x, y+1)+ifj,k (x, y−1) = i((z x −z −x )(wy+1 −w−y−1 )+(z x −z −x )(wy−1 −w−y+1 ) =
= i(w + w−1 )(z x − z −x )(wy − w−y ) = i(w + w−1 )fj,k (x, y),
It is easy to see that the vectors fj,k are pairwise orthogonal to each other,
consequently they are linearly independent. Since there nm eigenvalues of A,
we have found all of them. Note that
)
(
)
(
πk
πj
+ i2 cos
.
λj,k = 2 cos
m+1
n+1
Hence
Zm,n =
(
=
(m n
∏∏
)1/2
λj,k
=
j=1 j=1
(m n
∏∏
)1/4
|λj,k |2
=
j=1 j=1
(
)
(
)))1/4
m ∏
n (
∏
πj
πk
4 cos2
+ 4 cos2
.
m
+
1
n+1
j=1 k=1
Corollary 1.2. We have
4
1
log Zm,n = 2
lim
m,n→∞ mn
π
∫
π/2
∫
π/2
log(4 cos2 (x) + 4 cos2 (y)) dx dy.
0
0
Remark 1.3. Surprisingly, there is a nice expression for the above integral:
∫ π/2 ∫ π/2
∞
1 ∑ (−1)k
4
2
2
log(4 cos (x) + 4 cos (y)) dx dy =
.
π2 0
π k=0 (2k + 1)2
0
3
2. Matching polynomial
Let G be a graph on v(G) = n vertices and let mk (G) denote the number
of matchings of size k. Then the matching polynomial µ(G, x) is defined as
follows:
⌊v/2⌋
∑
µ(G, x) =
(−1)k mk (G)xn−2k .
k=0
Note that m0 (G) = 1.
Proposition 2.1. (a) Let u ∈ V (G) then
∑
µ(G − {u, v}, x).
µ(G, x) = xµ(G − u, x) −
v∈N (u)
(b) For e = (u, v) ∈ E(G) we have
µ(G, x) = µ(G − e, x) − µ(G − {u, v}, x).
(c) For G = G1 ∪ G2 ∪ · · · ∪ Gk we have
µ(G, x) =
k
∏
µ(Gi , x).
i=1
Theorem 2.2 (Heilmann and Lieb). All zeros of the matching polynomial
µ(G, x) are real. Furthermore,
√ if the
√ largest degree ∆ is at least 2 then all
zeros lie in the interval (−2 ∆ − 1, ∆ − 1).
Proof. Currently we will only prove the real-rootedness. We will sketch the
proof the second claim later.
We will prove the following two statements by induction on the number of
vertices.
(i) All zeros of µ(G, x) are real.
(ii) For an x with ℑx > 0 we have
µ(G, x)
ℑ
>0
µ(G − u, x)
for all u ∈ V (G).
Note that in (ii) we already use the claim (i) inductively, namely that µ(G −
u, x) doesn’t vanish for an x with ℑx > 0. On the other hand, claim (ii) for G
implies claim (i). So we need to check claim (i).
By the recursion formula we have
∑
∑ µ(G − {u, v}, x)
xµ(G − u, x) − v∈N (u) µ(G − {u, v}, x)
µ(G, x)
=
= x−
.
µ(G − u, x)
µ(G − u, x)
µ(G − u, x)
v∈N (u)
4
By induction we have
ℑ
for ℑx > 0. Hence
µ(G − u, x)
>0
µ(G − {u, v}, x)
−ℑ
µ(G − {u, v}, x)
>0
µ(G − u, x)
which gives that
ℑ
µ(G, x)
> 0.
µ(G − u, x)
Remark 2.3. One can also deduce from this proof that the zeros of µ(G, x) and
µ(G − u, x) interlace each other just like the zeros of a real-rooted polynomial
and its derivative.
Definition 2.4. Let G be graph with a given vertex u. The path-tree T (G, u)
is defined as follows. The vertices of T (G, u) are the paths1 in G which start
at the vertex u and two paths joined by an edge if one of them is a one-step
extension of the other.
1
2
4
1
3
12
13
14
123
143
132
134
1234
1432
Figure 1. A path-tree from the vertex 1.
Proposition 2.5. Let G be a graph with a root vertex u. Let T (G, u) be the
corresponding path-tree. Then
µ(T (G, u) − u, x)
µ(G − u, x)
=
,
µ(G, x)
µ(T (G, u), x)
and µ(G, x) divides µ(T (G, u), x).
1In
statistical physics, paths are called self-avoiding walks.
5
The proof of this proposition is again by induction using part (a) of Proposition 2.1.
Proposition 2.6. For a tree T , the matching polynomial µ(T, x) coincides
with the characteristic polynomial ϕ(T, x)√= det(xI − AT ). Furthermore, the
largest eigenvalue of a tree T is at most 2 ∆ − 1 if ∆ ≥ 2.
Remark 2.7. Clearly, propositions 2.5 and 2.6 together give a new proof of
the Heilmann-Lieb theorem.
Proposition 2.8. The following identities hold true.
(a) We have
∑
µ(G − u, x)µ(G, y) − µ(G − u, y)µ(G, x)
=
µ(G − P, x)µ(G − P, y).
y−x
P ∈P
u
(b) We have
µ(G − u, x)µ(G − v, x) − µ(G, x)µ(G − {u, v}, x) =
∑
µ(G − P, x)2 .
P ∈Pu,v
Proof. Both statements follows from part (a) Proposition 2.1 and using induction.
Remark 2.9. From part (a) one can gives a third proof for the real-rootedness
of the matching polynomial. Namely, suppose for contradiction that there is
a graph G with non-real zeros z and z, take the smallest such graph. Apply
the identity of part (a) to x = z and y = z, then the left hand side is 0 while
the right hand side is the sum of terms |µ(G − P, z)|2 , and they are non-zero,
because G − P has no non-real zero, contradiction.
Proposition 2.10. Let A be the adjacency matrix of a graph G. For an s :
E(G) → {±1}, let As denote the signed adjacency matrix: As (u, v) = s(u, v)
if (u, v) ∈ E(G), otherwise 0. Then
1 ∑
µ(G, x) = e(G)
det(xI − As ).
2
s
(The summation goes for all signings.)
Remark 2.11. This identity played a major role in the proof of Marcus,
Spielman and Srivastava "constructing" infinitely many d–regular bipartite
Ramanujan-graphs.