In particular, at q = 1, the Perron-Frobenius eigenvalue of A1, v is
λ1, v = 1 + v
1 k
1 + v
2 k
, with the corresponding eigenvector as Z0; 1,
v. Proof: To show the transfer matrix Aq, v given in 25 is primitive, it is enough to show it is
irreducible and has all positive entries on the diagonal, see Theorem 8.5.5 in [HJ86]. Both of these two properties are easy to verify, because it is enough to show these are true for Aq,
v at
q = v
1
= v
2
= 1. In this case, our transfer matrix A1, v is reduced to the transfer matrix in
[HL07] at x − 1 y − 1 = 1, which was shown to be irreducible and have positive diagonal entries. In addition, since Aq,
v is clearly nonnegative, by Perron-Frobenius theorem, it has the Perron-
Frobenius eigenvalue, which we denote as λq, v.
Since graph Γ0, k = V, {E
1
, E
2
} has edge count |E
1
| = 0, |E
2
| = k and graph Γ1, k = V
′
, {E
′ 1
, E
′ 2
} has |E
′ 1
| = k, |E
′ 2
| = 2k, from the simplification formula 8 of the multivariate Tutte polynomial at q = 1, we have
ZΓ0, k; 1, v = 1 + v
2 k
, and
ZΓ1, k; 1, v = 1 + v
2 2k
1 + v
1 k
= 1 + v
1 k
1 + v
2 k
ZΓ0, k; 1, v.
Moreover, Z1; 1, v is a vector of ZΓ1, k; 1, v for all noncrossing partitions Γ ∈ G , thus by
Theorem 5, Z1; 1,
v = Aq, vZ0; 1, v = 1 + v
1 k
1 + v
2 k
Z0; 1, v.
Therefore, Z0; 1, v is a positive eigenvector of the nonnegative matrix A1, v, with the corre-
sponding positive eigenvalue 1 + v
1 k
1 + v
2 k
. Thus by Theorem 8.1.30 in [HJ86], at q = 1, the eigenvalue of maximum modulus of Aq,
v, which is the Perron-Frobenius eigenvalue, is λ1, v = 1 + v
1 k
1 + v
2 k
. This completes the proof of Lemma 2.
Next we show that
Lemma 3. For v = {v
1
, v
2
}, if we let Z
q
P
n
× C
k
; 1, v be the derivative of the multivariate Tutte
polynomial of the cylinder graph P
n
× C
k
with respect to q and evaluated at q = 1, then Z
q
P
n
× C
k
; 1, v
n1 + v
2 n+1k
1 + v
1 nk
→ λ
q
1, v λ1, v
, as n → ∞.
Proof: The proof of this lemma goes along the similar line as the proof of Theorem 7.1 in [HL07]. Adopt the notations there, we let ξ
q, v be the Perron-Frobenius eigenvector. Choose a scaling of
ξ q,
v such that ξ1, v = Z0; 1, v. If we let
rq, v = Z0; q, v − ξq, v,
then Zn; q,
v = A
n
q, vZ0; q, v = A
n
q, vξq, v + rq, v
= λ
n
q, vξq, v + A
n
q, vrq, v. 31
129
Therefore, by taking derivative on both sides of equation 31 with respect to q, Z
q
n; q, v = nλ
n−1
q, vξq, vλ
q
q, v + λ
n
q, vξ
q
q, v
+nA
n−1
q, vrq, vA
q
q, v + A
n
q, vr
q
q, v.
We are only interested in the above quantity at q = 1, at which ξ1, v = Z0; 1, v and r1, v = 0.
This shows that, by dividing nλ
n
1, v, we obtain
Z
q
n; 1, v
nλ
n
1, v
= λ
q
1, v λ1, v
Z0; 1, v +
ξ
q
1, v
n +
A1, v λ1, v
n
r
q
1, v
n .
As n → ∞, obviously ξ
q
1, vn → 0. Since the matrix Aq, v is primitive A1, v
λ1, v
n
→ D1, v, as n → ∞,
for some matrix D1, v, according to Theorem 8.5.1 in [HJ86]. Hence
A1, v λ1, v
n
r
q
1, v
n → 0, as n → ∞.
Consequently, Z
q
n; 1, v
nλ
n
1, v
→ λ
q
1, v λ1, v
Z0; 1, v, as n → ∞.
Since Z
q
n; 1, v is a vector of Z
q
Γn, k; 1, v for all Γ ∈ G , we have for any noncrossing partition
Γ of C
k
, Z
q
Γn, k; 1, v
nλ
n
1, vZΓ0, k; 1, v
→ λ
q
1, v λ1, v
. as n → ∞. If Γ is the partition consisting of k isolated vertices, then Γn, k is the cylinder graph P
n
× C
k
and Γ0, k is the cycle C
k
with only type 2 edges. Therefore, by Lemma 2 and the fact that ZC
k
; 1, v = 1 + v
2 k
, Lemma 3 is proved.
Now by dominated convergence theorem, Theorem 3 follows easily from the next lemma, details are omitted.
Lemma 4.
Z
q
P
n
× C
k
; 1, v
n1 + v
2 n+1k
1 + v
1 nk
≤ n + 1k
nk .
Proof of Lemma 4: Let the edge subset A = A
1
∪ A
2
, where the set A
1
, A
2
consists of type 1 and type 2 edges in A respectively. By the definition of the multivariate Tutte polynomial,
ZG
nk
; q, v =
X
A⊆E
q
kA
v
|A
1
| 1
v
|A|−|A
1
| 2
, where 0 ≤ |A
1
| ≤ nk and 0 ≤ |A
2
| ≤ n + 1k. As such, by taking derivative on both sides of the above,
Z
q
G
nk
; q, v =
X
A⊆E
kAq
kA−1
v
|A
1
| 1
v
|A|−|A
1
| 2
, 130
then Z
q
G
nk
; 1, v =
|E|
X
m=0
X
|A|=m
kAv
|A
1
| 1
v
m−|A
1
| 2
. 32
Since the number of components in a graph is bounded by the number of vertices the graph has, for the cylinder graph P
n
× C
k
, 1 ≤ kA ≤ n + 1k.
Then from equation 32 and the fact that |E| = 2n + 1k, Z
q
P
n
× C
k
; 1, v ≤ n + 1k
2n+1k
X
m=0
X
|A|=m
v
|A
1
| 1
v
m−|A
1
| 2
= n + 1k
2n+1k
X
m=0 nk
X
i=0
X
|A
1
|=i,|A|=m
1
v
i 1
v
m−i 2
= n + 1k
2n+1k
X
m=0 nk
X
i=0
nk i
n + 1k m − i
v
i 1
v
m−i 2
= n + 1k1 + v
1 nk
1 + v
2 n+1k
, which completes the proof of Lemma 4.
Theorem 3 shows that to calculate the asymptotic value of EL
M S T
G
nk
n for a specific value of k, one only needs to find λ
q
1, v. The calculation of λ
q
q, v in general is hard, since we do not have
enough information for λq, v at q 6= 1. However, following the idea of using the characteristic
polynomial of Aq, v mentioned in [HL07], one can find the following without much difficulty
λ
q
1, v = −
P
q
λ1, v; 1, v
P
λ
λ1, v; 1, v
, 33
where Pλ; q, v is the characteristic polynomial of Aq, v, P
q
λ1, v; 1, v and P
λ
λ1, v; 1, v are
obtained by taking derivative of Pλ; q, v with respect to q and λ respectively, then evaluated at
q = 1. Finally, recall that we used ELn as a shorthand for EL
M S T
G
n,k
in 30, where G
n,k
denotes the cylinder graph P
n
× C
k
. With the calculation of the transfer matrix for k = 2, 3 at the end of Section 4.2.1, we compute the asymptotic values of n
−1
E Ln by applying Theorem 3 and formula
33. The results are shown in Table 3. This table shows that for k = 2 and k = 3, as n → ∞,
n
−1
E L
e
n ≥ n
−1
E L
eu
n ≥ n
−1
E L
ue
n ≥ n
−1
E L
u
n. One may notice that in Table 3, the difference between n
−1
E L
eu
n and n
−1
E L
ue
n is bigger than e
−k
k. From Theorem 3, this means that for the cases of k = 2, 3 the integral Z
1
λ
q
1, vt λ1, vt
dt 131
Table 3: Asymptotic Values of the Ratio of the Expected Length of the Cylinder Graph P
n
× C
k
to the Length n
k = 2 k = 3
n
−1
E L
e
n 0.87790
1.14579 n
−1
E L
eu
n 0.80005
0.96989 n
−1
E L
ue
n 0.65564
0.91171 n
−1
E L
u
n 0.61661
0.84085
is bigger if vt = {v
1
t, v
2
t}, v
1
t = e
t
− 1 and v
2
t = t1 − t. We expect the same inequality holds for larger k too.
Conjecture 1. For any k ≥ 2, as n → ∞,
n
−1
E L
e
n ≥ n
−1
E L
eu
n ≥ n
−1
E L
ue
n ≥ n
−1
E L
u
n.
4.3 Discussions on The Complete Graph
Since the multivariate Tutte polynomial enables us to control the length of each edge, we expect a lot more applications of the general formula 4. For example, it is known that EL
M S T
K
n
converges to a finite number as long as the identical edge distribution F satisfies F
′
0 6= 0, but diverges if F
′
0 = 0. For a complete graph with certain portion of edge lengths following distribution U0, 1 and the other edge lengths following a distribution F , with F
′
0 = 0, it would be interesting to know for what portion and configuration of edges with distribution F
′
0 0 that the expected length of MST of this complete graph still converges to a constant. In this section, we discuss the
simplest problem in this category, that is, the expected lengths of MSTs of a complete graph with a portion of edges removed.
While the complete graph is the most dense simple graph, removing one edge should not change the value of expected length of MST too much for large n. Heuristically, removing two edges should
not make much difference either. Then, an intriguing problem is to find the maximum number mK
n
of edges that can be removed arbitrarily from K
n
without changing the asymptotic value of the expected length of the MST.
Let us first consider two extreme cases. Since the degree of each vertex is n − 1, it is trivial to see that mK
n
n − 1. Otherwise, the remaining graph may become disconnected and the length of the MST is infinity. Moreover, if n − 2 edges incident to a common vertex, say v
1
are removed, then the vertex v
1
is incident to only one edge, which becomes a bridge and has to be included in the MST by the cut property. By properties of the MST, the remaining graph G
n
satisfies E
L
M S T
G
n
= Eξ
e
+ EL
M S T
K
n−1
, whose asymptotic value is obviously different from the asymptotic value of EL
M S T
K
n
by the quan- tity Eξ
e
. Moreover, it is easy to see that for both of these cases, if the m edges are not removed from a single vertex, then the MST of the remaining graph may be much shorter. In particular, the MST
in the first case at least has finite length. Therefore, we formulate the following conjecture.
132
Conjecture 2. Removing arbitrary m ≤ n − 1 edges from the complete graph K
n
, the expected length of the MST of the remaining graph is the largest if these m edges share a common end point.
In fact, by examining the number of spanning trees of a complete graph with m n − 1 edges re- moved, we suspect that the smallest number of spanning trees appear in the resulting graph if these
m edges share a common end point. This might be a possible approach to proving Conjecture 2.
Let K
−m n
= V, E
−m
denote the graph obtained by removing m edges which share a common vertex from the complete graph K
n
, with edge set E
−m
and same vertex set as K
n
. Based on Conjecture 2, to find the maximal number mK
n
, it suffices to examine graph K
−m n
. In [LZ09], Li and Zhang showed the difference between EL
e M S T
K
n
and EL
u M S T
K
n
to be of size ζ3n. More precisely, the following equation is shown that for m = 1.
E L
e M S T
K
n
− EL
u M S T
K
n
= Z
1
1 1 − t
m
T
x
K
n
; 1t, 11 − t T K
n
; 1t, 11 − t dt ∼ ζ3n.
34 Actually, by a careful argument, one can show that the above equation holds for m up to at least
n3. Then we obtain the next lemma by applying Steele’s formula 2 and comparing the difference of EL
M S T
K
−m n
and EL
u M S T
K
n
. The Tutte polynomial of K
−m n
is found to be related to T K
n
; x, y by some algebraic manipulation.
Lemma 5. In a complete graph K
n
, for both the cases of U0, 1 and exp1 edge length distributions, we can remove at least on edges which share a common vertex from K
n
, such that the asymptotic value of the expected length of the MST of K
−m n
is the same as that of K
n
. That is for any m = on, lim
n→∞
E L
u M S T
K
−m n
= lim
n→∞
E L
u M S T
K
n
= ζ3, and
lim
n→∞
E L
e M S T
K
−m n
= lim
n→∞
E L
e M S T
K
n
= ζ3.
Proof: First, we prove the case of U0, 1 distribution. It is well known that the standard Tutte polynomial is simplified on H
1
= {x − 1 y − 1 = 1} as in formula 9. Thus, if we let N =
n 2
denote the number of edges in the complete graph K
n
, then T K
n
; 1t, 11 − t = t
1−n
1 − t
n−N −1
, and
T K
−m n
; 1t, 11 − t = 1 − t
m
T K
n
; 1t, 11 − t. This shows that
E L
u M S T
K
−m n
− EL
u M S T
K
n
35 =
Z
1
1 − t t
1 T K
n
; 1t, 11 − t T
x
K
−m n
; 1t, 11 − t 1 − t
m
− T
x
K
n
; 1
t ,
1 1 − t
dt.
By Definition 3 of the Tutte polynomial, T
x
K
n
; 1t, 11 − t = X
A⊆E
kA − 1t
|A|−n+2
1 − t
n−|A|−2
, 133
and T
x
K
−m n
; 1t, 11 − t = X
A⊆E
−m
kA − 1t
|A|−n+2
1 − t
n−|A|−2
. Since K
n
and K
−m n
share the same vertex set, then given the set A, which is a subset of both E and E
−m
, kA is the same in both K
n
and K
−m n
. In addition, since E
−m
⊂ E, we have T
x
K
−m n
; 1t, 11 − t T
x
K
n
; 1t, 11 − t. By 36, this implies
E L
u M S T
K
−m n
− EL
u M S T
K
n
≤ Z
1
1 − 1 − t
m
t1 − t
m−1
T
x
K
n
; 1t, 11 − t T K
n
; 1t, 11 − t dt
≤ Z
1
m 1 − t
m−1
T
x
K
n
; 1t, 11 − t T K
n
; 1t, 11 − t dt.
36 As one can show 34 holds for any m = on, then as n → ∞,
Z
1
m 1 − t
m−1
T
x
K
n
; 1t, 11 − t T K
n
; 1t, 11 − t dt → 0.
In addition, it is obvious that for any m ≤ n − 1, E
L
u M S T
K
−m n
≥ EL
u M S T
K
n
. Therefore, by squeeze theorem, for any m = on,
E L
u M S T
K
−m n
− EL
u M S T
K
n
→ 0, as n → ∞, which completes the proof of the first part.
The proof for the case of exp1 distribution follows similar steps as above. Details are omitted. Note that we have one less factor 1 − t in the representation of EL
e M S T
K
−m n
− EL
e M S T
K
n
. So the maximal number edges allowed to remove is one less than that in the uniform distribution case.
The proof of Lemma 5 is thus complete.
However, we actually believe that the result could be sharpened to εn for a small constant ε. The proof of this might require a much tighter bound than 36. For more detailed discussions, refer to
[Zha08].
5 Appendix: Proof of Theorem 5
The procedure of finding a general recursive relation between the multivariate Tutte polynomials of F
n,k
in the case of two different types of edges goes similar to the one obtaining the standard Tutte polynomial in [HL07], but more complicated and requires precise identification of various
quantities. For the completeness of this paper, we first recall a few definitions and lemmas from [HL07] and adopt their notations.
134
Assume C
k
has vertices {0, ..., k − 1}, for each i, let vi denote the set of vertices that are in the same block with i in the noncrossing Γ. To each partition Γ of C
k
, k + 1 cap graphs Γ , . . . , Γ
k
are associated. They all have the same vertex set, namely the set of blocks of Γ, but different edge set.
The edge set of Γ
i
includes all edges connecting j to j + 1, for j = k − i..k − 1. Thus a loop is created if j + 1 and j are in the same block.
Definition 5 Capped Cylinder Graph. The capped cylinder graph Γn, j is obtained by attaching the cap graph Γ
j
to the ends of cylinders P
n−1
× C
k
, where each vertex n − 1, l of P
n−1
× C
k
is joined to vl of Γ
j
by a type 1 edge, for each l ∈ 0, ..., k − 1. Note that Γn, k is the same as the one defined in Section 4.2.1.
Definition 6 Pinched Capped Cylinder Graph. For n ≥ 1, let πΓn, j be the pinched capped cylinder graph obtained from first switching the edge types in Γn, j, then contracting all the edges in
C
k
at level 0. Thereby, in πΓn, j the edges on cycles are of type 1 and the edges on paths are of type 2. Note that this definition is different from the one in [HL07], in that the edge types are specified
explicitly here.
Lemma Hutson-Lewis. Let Γ be a partition of C
k
, and let Φ be the partition obtained from Γ by conflating the blocks of a pair of adjacent vertices of C
k
. If Γ is noncrossing, then Φ is noncrossing.
Lemma Hutson-Lewis. The dual partition of a noncrossing partition is noncrossing.
Let Γ
∗
denote the dual of the noncrossing partition Γ of C
k
and G
∗
denote the dual graph of a planar graph G. It is easy to see that the cylinder graph, the capped cylinder graph and the pinched capped
cylinder graph are all planar graphs. The dual of a partition and the dual of a planar graph is related by the following theorem.
Theorem Hutson-Lewis: Duality. For n ≥ 1,
Γn, 0
∗
= πΓ
∗
n, k, 37
and πΓn, 0
∗
= Γ
∗
n − 1, k. 38
Now given noncrosssing partitions Γ and Φ of C
k
, we define the matrices Θ, ∆, M and B used in the transfer matrix in Section 4.2.1. Define function φ as φx = x + 1 mod k and let φΓ denote the
noncrossing partition obtained by applying φ to each of the vertices of Γ. Then the rotation matrix Θ is defined as
Θ
ΓΦ
= ¨
1 if φΦ = Γ, 0 otherwise,
39 and the dual matrix ∆ as
∆
ΓΦ
= ¨
1 if Φ = Γ
∗
, 0 otherwise.
40 The multivariate Tutte matrix M u is defined as the following:
• If 0 ∼ 1, i.e., vertices 0 and 1 are in the same block in Γ, then
M u
ΓΦ
= ¨
1 + u if Φ = Γ, otherwise.
41 135
• If 0 ≁ 1, i.e., vertices 0 and 1 are not in the same block of vertices, let Σ be the partition obtained
by conflating the two blocks containing 0 and 1 in Γ , then
M u
ΓΦ
=
1 if Φ = Γ, u
if Φ = Σ, 0 otherwise.
42
Let m
G
= min{|V Γn, 0|, Γ ∈ G }, then the matrix B
ΓΦ
= ¨
q
|V Γn,0|−m
G
if Φ = Γ, otherwise.
43 For a more detailed discussion of noncrossing partitions and capped graphs, we refer to [HL07].
Proof of Theorem 5: Given an ordering G of the noncrossing partitions of C
k
, we want to find a recursive relation between the family of {Γn, k, Γ ∈ G } and {Γn − 1, k, Γ ∈ G }. As for the
computation of the Tutte polynomial for any graph, we first choose an edge and apply the deletion- contraction identity 12.
For a noncrossing partition Γ, if 0 ∼ 1 in Γ, then there exists a loop edge of type 2 at vertex v0 in graph Γ
k
. Therefore, ZΓn, k; q,
v = 1 + v
2
ZΓn, k − 1; q, v.
If 0 ≁ 1 in Γ, then there exists an edge of type 2 between vertex v0 and v1 in graph Γ
k
. Since we have not removed any edge from the path P
n
, the graph remains connected even if the edge e is deleted. Therefore, the edge e is not a bridge, and
ZΓn, k; q, v = ZΓn, k − 1; q, v + v
2
ZΣn, k − 1; q, v.
By the definition of the multivariate Tutte matrix M u, we have Zn; q,
v = M v
2
[ZΓn, k − 1; q, v]
G
. Based on whether 1 ∼ 2 in Γ, as discussed above, we have a similar relation between [ZΓn, k −
1; q,
v]
G
and [ZΓn, k − 2; q, v]
G
, but with a different coefficient matrix. To make the represen- tation easier, note that 1 ∼ 2 in Γ if and only if 0 ∼ 1 in φ
−1
Γ. Therefore,
[ZΓn, k − 1; q, v]
G
= ΘM v
2
Θ
−1
[ZΓn, k − 2; q, v]
G
. In general, considering the edge between v j and v j + 1, we have
[ZΓn, k − j; q, v]
G
= Θ
j
M v
2
Θ
− j
[ZΓn, k − j − 1; q, v]
G
. Hence,
Zn; q, v =
M v
1
ΘM v
2
Θ
−1
· · · Θ
k−1
M v
2
Θ
−k+1
[ZΓn, 0; q, v]
G
= M v
2
Θ
k
[ZΓn, 0; q, v]
G
, 44
since Θ
−k+1
= Θ. 136
Now that for a given noncrossing partition Γ, ZΓn, k; q, v is related to ZΓn, 0; q, v, we are left
