Table 1: Exact Values of P
n
× C
2
for Different Edge Distributions 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 1
1.35000 1.13134
1.06472 0.94286
2 1.11349
0.96567 0.85633
0.77958 3
1.03495 0.91046
0.78916 0.72525
4 0.99568
0.88286 0.75576
0.69809 5
0.97212 0.86630
0.73573 0.68179
6 0.95642
0.85526 0.72238
0.67093 7
0.94520 0.84737
0.71285 0.66317
8 0.93679
0.84145 0.70570
0.65735 9
0.93025 0.83685
0.70013 0.65282
10 0.92501
0.83317 0.69568
0.64920 k = 2
Table 2: Exact Values of P
n
× C
3
for Different Edge Distributions 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 1
2.17421 1.71031
1.82033 1.56310
2 1.64128
1.33985 1.35248
1.20091 3
1.46555 1.21652
1.20432 1.08083
4 1.37779
1.15487 1.13103
1.02083 5
1.32515 1.11787
1.08715 0.98483
6 1.29005
1.09321 1.05790
0.96084 7
1.26499 1.07559
1.03702 0.94370
8 1.24619
1.06238 1.02136
0.93084 9
1.23156 1.05210
1.00917 0.92084
10 1.21986
1.04388 0.99943
0.91284 k = 3
4.2.3 Asymptotic Values of the Expected Lengths of MSTs of Cylinder Graphs
Tables 1 and 2 show that the values of EL
M S T
G
n,k
n are decreasing for all the four cases as n grows larger. Hutson and Lewis [HL07] showed that EL
u M S T
G
n,k
n converges to a number that can be represented by the dominant eigenvalue of their transfer matrix. For the cylinder graph with
mixed edge weight distributions, we prove a similar asymptotic result in Theorem 3. Recall that Perron-Frobenius theorem says a nonnegative primitive matrix has a unique eigenvalue of maximum
modulus, which is called the Perron-Frobenius eigenvalue, see Th2.1 in [Var62].
To prove Theorem 3, we first show three basic lemmas.
Lemma 2. For k ≥ 2, the transfer matrix Aq, v is primitive, i.e., there exists k such that A
k
q, v ≥ 0.
128
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
