A conjecture on the number of Hamiltonian cycles on thin grid cylinder graphs

We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph C m × P n +1 . We distinguish two types of Hamiltonian cycles depending on their contractibility (as Jordan curves) and denote their numbers h ncm ( n ) and h cm ( n ) . For ﬁxed m , both of them satisfy linear homogeneous recurrence relations with constant coefﬁcients. We derive their generating functions and other related results for m ≤ 10 . The computational data we gathered suggests that h ncm ( n ) ∼ h cm ( n ) when m is even.


Introduction
A Hamiltonian path of a simple graph is a path that visits each vertex exactly once.A closed Hamiltonian path is called a Hamiltonian cycle or Hamiltonian circuit, which we shall abbreviate as HC.The enumeration of Hamiltonian cycles on rectangular grid graphs P m × P n had been studied extensively in, among others, [2,4,9,15,10,13,14,17,19,20].In contrast, little work [2,9,11,17] was devoted to enumerate Hamiltonian cycles on rectangular grid cylinders C m × P n .
In this paper we investigate, for each fixed m ≥ 2, the generation and enumeration of Hamiltonian cycles on C m × P n+1 , where n ≥ 1.Since n grows while m is fixed, such graphs are called thin grid cylinders in the literature.In [2], vertices were encoded.We adopt a different approach by coding the cells or squares on the cylindrical surface, along with the so-called k-SIST equivalence relation.This equivalence relation was formerly called k-SISET, and was first used in [4] to enumerate Hamiltonian cycles on P m × P n .A very similar approach for the same enumeration was implemented in [19] using the language of finite automa.
We distinguish two different types of HCs.In the sense of homotopy: one type of HCs are contractible (as Jordan curves) to a point, and the other type of HCs are not.We denote them HC c and HC nc , respectively.Simply put, a HC nc is one that "perches" on or wraps around the cylinder likes a bracelet on an arm, and a HC c can be "pasted" on the cylindrical surface.Let h nc m (n) and h c m (n) be the number of HC nc s and HC c s, respectively, on C m × P n+1 , Our objective is to determine, for each fixed m, the sequences h nc n = {h nc m (n)} n≥1 and h c m = {h c m (n)} n≥1 .It is obvious that the number of HCs on C m × P n+1 is given by h m (n) = h nc m (n) + h c m (n).We characterize both types of HCs, and use it to define, for each fixed m ≥ 2, a digraph D m .The original enumeration problem is equivalent to counting oriented walks of length n − 1 in this digraph with first and last vertices from two special sets.Using the transfer matrix method [5,18], we obtain the generating functions for the sequences h nc m and h c m , thereby proving that they both satisfy some linear homogeneous recurrence relations with constant coefficients.
For each fixed m, these two generating functions share the same denominator, hence the same recurrence relation.We used Pascal programs and Mathematica 6 to carry out the computation.Our results agree with those reported in [2,11], which used a different approach.The computational data from m = 2, 4, 6, 8, 10 suggest that h nc m (n) and h c m (n) have the same number of digits and start with the same sequence of digits.For example, h nc 10 (100) = 106189661997982901262641694866260787081353490654045349773784 008483411988691035247114502475722767402987233190282387756909 3701143503070291097245473763298031619982266082, h c 10 (100) = 106189661997133629777153967627991207437193145571362259096752 805056007992463634686046052605540587643324294617040045670714 1143497346647742593316608877569233239238111440.
Both numbers have 166 digits, and their first 12 digits are identical.Why is this happening?

Preliminaries
The graph C m × P n+1 can be drawn on a cylindrical surface in such a way that no edges cross each other, see Figure 1.There are mn squares (4-cycles) called windows.Label the vertices (i, j) and the windows w i,j , where 1 ≤ i ≤ m, 1 ≤ j ≤ n + 1 for vertices, and 1 ≤ j ≤ n for windows, as shown in Figure 1.Construct a window lattice graph W m,n with vertices representing the windows of C m × P n+1 , and two vertices are adjacent if and only if their corresponding windows in C m × P n+1 share a common edge.It should be clear that W m,n is isomorphic to C m × P n .We distinguish two types of closed Jordan curves on a cylindrical surface: those that divide the surface into two infinite regions (image the cylinder being extended indefinitely in both directions to the left and to the right), see the curve K nc in Figure 2, and those that divide the surface into one finite and one infinite region, see the curve K c in Figure 2. The first type (non-contractible HC) wraps around the cylindrical surface, hence divides the cylindrical surface into the left half and the right half, it resembles a bracelet around an arm.The second type (contractible HC) encloses a finite region and leaves an infinite region on the outside.One could imagine it being pasted onto the cylindrical surface.
We abbreviate these two types of Hamiltonian cycles as HC nc and HC c , respectively.We use the following convention to name the two regions separated by a HC:  • For a non-contractible HC: all edges that connect two adjacent vertices from {(i, 1) | 1 ≤ i ≤ m}, but do not lie on the HC, belong to the same region.We call this region (on the left of the HC) the zero region, and the other region (on the right of the HC) the positive region.
• For a contractible HC: the windows within the bounded region are marked with 1s, hence the bounded region is the positive region, which makes the exterior unbounded region the zero region.
Alternatively, the orientation of the HC is chosen such that the zero region is always on our left as we traverse through the HC (see Figure 3).For HC c this orientation is in the clockwise direction.We use h nc m (n) and h c m (n) to indicate the number of HC nc s and HC c s. Their respective generating functions are written as H nc m (x) and H c m (x).Using a standard parity argument (likes the one used on a checkerboard), it is easy to tell which thin grid cylinders have a Hamiltonian cycle.
Theorem 2.1 For m ≥ 2 and n ≥ 1, we have h nc m (n) = 0 if and only if both m and n are odd, and h c m (n) = 0 if and only if m is odd and n is even.
Proof: It is straightforward to construct a HC nc for even m or even n, and a HC c for even m or odd n (see Figure 4).It remains to establish the condition under which no HC exists.
Consider the "vertical" edges joining vertices (m, i) to (1, i) for 1 ≤ i ≤ n + 1, see Figure 3. Any HC may contain some of these vertical edges, and the number of such edges is odd for a HC nc , and even for a HC c .
As we travel along a non-contractible Hamiltonian cycle, the number of steps "to the left" and "to the right" must be equal, while the difference between the "up" and "down" steps is m.Since the HC contains m(n + 1) edges, we deduce that m(n + 1) ≡ m (mod 2).Thus, a HC nc does not exist if both m and n are odd.
Similarly, if there exists a contractible Hamiltonian cycle, then m(n + 1) must be even, because there is an equal number of left and right steps, and an equal number of up and down steps.Hence, there is no HC c if m is odd and n is even. 2 Hamiltonicity of a graph has both a local (every vertex is visited exactly once) and a global (the subgraph is connected) aspect.For a HC nc , the windows belonging to any one of the two regions induce a forest in the window lattice graph W m,n .We call the trees in these forests zero trees (abbreviated ZTs) or positive trees (abbreviated PTs) depending on which region they belong to.Accordingly, their respective windows are called zero windows or positive windows.Every zero tree contains exactly one window on the first column of W m,n from the set {w i,1 | 1 ≤ i ≤ m} called the left root, and every positive tree contains exactly one window on the last column of W m,n from the set {w i,n | 1 ≤ i ≤ m} called the right root.For example, the HC nc in Figure 3 has three zero trees with left roots w 1,1 , w 3,1 , and w 7,1 (striped), and two positive trees with right roots w 7,10 , and w 10,10 (striped).
For a HC c , the interior windows (they are marked with 1s in the HC c in Figure 3) form a tree in W m,n , but the exterior windows form a forest of exterior trees (abbreviated ETs).Note that only one ET from this forest contains exactly one window on the first column of W m,n (the left root), and also exactly one window on the last column of W m,n (the right root).We call this ET the split tree of the HC.Any ET different from the split tree contains either exactly one left root or exactly one right root, but not both.For example, the HC c in Figure 3 has a split tree with the left root w 1,1 and the right root w 3,10 , one ET with the left root w 7,1 , and one ET with the right root w 9,10 .For the purpose of this study, interior tree and exterior trees are also called positive tree and zero trees, and their windows are labeled by 1 and 0, respectively.
We need a few additional definitions to facilitate our discussion.
Definition 1 Given a nonnegative integer word d 1 d 2 . . .d m , its support is defined as the binary word d1 d2 . . .dm , where The support of a nonnegative integer matrix [d i,j ] is defined in a similar manner.

Definition 2
The factor u of a word v is called a b-factor if it is a block of consecutive letters all of which equal to b.A b-factor of v is said to be maximal if it is not a proper factor of another b-factor of v.
The approach described in the next section allows us to simultaneously analyze both types of Hamiltonian cycles.

First Characterization of HC
We associate with each Hamiltonian cycle of C m × P n+1 , for both types, a binary matrix [a i,j ] m×n , denoted A nc for HC nc , and A c for HC c , according to a i,j = 1 if w i,j is a positive window, 0 otherwise.This matrix satisfies the following necessary conditions which are easy to verify (we adopt the convention that a m+1,j = a 1,j , for 1 ≤ j ≤ n).
[A3] (Root condition): Each connected component of the subgraph of W m,n induced by the 1-windows has a tree structure, and • For HC nc , every positive tree has exactly one square from the last column of W m,n .
• For HC c , there is exactly one positive tree.
• For HC c , at least one 0, and does not contain the factor 00.
It is clear that every HC nc (HC c , resp.) yields exactly one matrix A nc (A c resp.) that satisfies conditions [A1]-[A4].The converse is also true.Proof: The entries in the matrix A can be used to label the windows of C m ×P n+1 with 0 and 1. Construct a subgraph on C m × P n+1 by forming its edges as follows.Any edge neighboring a 0-window and a 1window is selected.For A nc , a left edge that joins the vertices (m, 1) and (1, 1), or the vertices (i, 1) and (i + 1, 1), for 1 ≤ i ≤ m − 1, is selected if it is adjacent to a 1-window, and a right edge that joins (m, n + 1) to (1, n + 1) or (i, n + 1) to (i + 1, n + 1), for 1 ≤ i ≤ m − 1, is selected if it is adjacent to a 0-window.For A c , an edge on the left or right boundary is selected if it adjacent to a 1-window.For example, for the matrices in Figure 3, the edge between the vertices (3, n + 1) and (4, n + 1) is selected for A nc but not for A c .
The We note that every possible first column in both A nc and A c and last column in A c is a circular binary words of length m with no consecutive 0's, and is different from the word 1 m .Likewise, every possible last column in A nc is a circular binary words of length m with no consecutive 1's, and is different from the word 0 m .It is well-known that the number of such binary words is L m − 1, where L m is the mth Lucas numbers with L 0 = 2, L 1 = 1, and L k+1 = L k+1 + L k for k ≥ 0. See, for example, [1].

Second Characterization of HC
In this section, we propose an alternate characterization of the HCs on C m × P n+1 .Although it is more complicated, it leads to an effective way to compute the generating functions H nc m (x) and H c m (x).In the following discussion, A denotes either A nc or A c .Definition 3 Given a fixed positive integer k, two windows w i,l and w j,s that satisfy a i,l = a j,s = 1 (from either A nc or A c ) and l, s ≤ k are said to be k-SIST (surely in the same tree looking from the k-th column) if and only if they belong to the same component in the subgraph of W m,n induced by {w p,t | a p,t = 1 and t ≤ k}.
For fixed k, being k-SIST is an equivalence relation on the set {w i,k | a i,k = 1 and 1 ≤ i ≤ m} and it has at most m/2 equivalence classes.It is possible that two different classes eventually belong to the same positive tree of a Hamiltonian cycle on the entire cylindrical surface of C m × P n+1 .In other words, two windows that are not k-SIST could become -SIST for some integer > k.However, we cannot tell whether it is true just from the first k columns of the matrix A.
Let C + = {2, 3, . . ., m/2 + 1}.For any HC nc or HC c , we associate to the matrix A nc or A c from the first characterization a second matrix [b i,j ] m×n , denoted B nc or B c , where b i,j ∈ C + ∪ {0}, in the following way (see Figure 5).For each j: where p ≥ 1 is the number of 0s and [B2] The support of the matrix B, that is, the matrix [a i,j ] m×n , satisfies the adjacency condition [A2].
[B3] For 1 ≤ k ≤ n, the kth column of the matrix B satisfies these conditions: , where l ≤ m/2 , and p 1 < p 2 < • • • < p l , are the first appearance of the elements from C + in the kth column, then b pi,k = i + 1. (g) Every column has both positive and zero entries.
[B4] The last column b 1,n b 2,n . . .b m,n is where p ≥ 1 is the number of positive integers and d i > 0 for 1 ≤ i ≤ p.
• For HC c , where p ≥ 1 is the number of 0s and Proof: First, a few remarks.
• [B1] and [B2] follow from the definition of the matrix B.
• [B3a]: Two windows belonging to the same equivalence class must be associated with the same number.• [B3b]: This follows from the definition of the matrix B.
• [B3c]: If w i,k−1 and w j,k−1 are (k − 1)-SIST, and if the windows w i,k , w j,k , w i,k−1 and w j,k−1 are from the positive region, then the windows w i,k and w j,k must be k-SIST.• [B3d]: If the opposite is true, we would obtain a cycle in a positive tree, which is impossible.
• [B3e]: If we can conclude by knowing the first k columns that v and u are in the same tree, then there is exactly one path from v to u in their positive tree via some windows from the previous column, that is, the (k − 1)th column.• [B3f]: Every positive tree must "reach" the last column.
• [B3g]: For a HC nc , the unique path in W m,n starting in a positive window from the first column and finishing in the last column must cross every column.For a HC c , the unique split tree must cross every column as well.Furthermore, the occurrence of a column with no zero window would imply that the corresponding subgraph in C m × P n+1 is not connected, which is impossible.• [B4]: This follows from the definition of the matrix B.
Based on these remarks, it is not difficult to verify the properties listed in the theorem. 2 We note that Faase [7] used a similar method to enumerate spanning subgraphs of G × P n that meet certain conditions.
Because of the rotational symmetry and reflection symmetry of C m × P n , we can further simplify the digraph D m by identifying some of its vertices, hence reducing its adjacency (transfer) matrix T m to a smaller size.By doing so, we obtain the multidigraph D * m instead of D m with transfer matrix T * m .
The computation of the generating functions is rather routine (see Theorem 4.7.2 in [18]).It is obvious that These generating functions are rational functions.Their denominators are determined by the characteristic polynomials of the adjacency matrices.Table 1 displays, for 3 ≤ m ≤ 10, the numbers of vertices in F m , D m and D * m , as well as the degrees of the denominators in these generating functions, which determine the orders of the recurrence relations for h nc m and h c m .We find an interesting upper bound of |V (D m |.A column in the matrix [b i,j ] m×n can be viewed as a word.Let its maximal nonzero b-factors, in the order of their appearance, be p 1 -factor, p 2 -factor, . . ., p kfactor.Call p 1 p 2 . . .p k a positive truncated word.For example, the positive truncated words correspond to the 2nd and 6th columns of B c in Figure 5 are 22233 and 2322, respectively.Every truncated word v has two properties: • If a letter s ≥ 3 appears in v, then, in accordance with the property [B3b], each number from {2, . . ., s − 1} must have appeared at least once before it.In other words, if we remove the duplicated letters, the remaining letters will form the word 234. . . . • If abab is a subsequence of the word v, then a = b (because of the properties [B3e] and [B1]).
A word over the alphabet {2, . . ., k + 1} that possesses the above-mentioned properties is called a color word.The number of color words of length k is the Catalan number C k = 1 k+1 2k k , see [3,16].Using a relation between Catalan and Motzkin numbers described in [6], we obtain the following corollary.

Corollary 5.1 An upper bound on the number of vertices of digraph
where C m is the mth Catalan number and M m is mth Motzkin number.
In light of Corollay 5.1, we would like to remark that we could use Motzkin words to encode the columns.See, for example, [19].

Computational Results
Based on the discussion in the previous section, we use Pascal programs to compute the adjacency matrices of the multidigraphs D * m , from which we obtain H nc m (x) and H c m (x).The results are summarized in Since H nc m (x) and H c m (x) are derived from the same transfer matrix, their denominators are identical.After adding the two rational functions to form H m (x), the new denominator may have a lesser degree.In fact, numerical data reveal that the degree is reduced by roughly one-half, see Table 1.
Upon further examination of the factorization of the denominator, we conclude that a better way to study them is to introduce the function such that, together with ( 1), Since both H m (x) and K m (x) are rational functions, we can express them as for some polynomials H m (x), K m (x), p m (x), q m (x), r m (x) and s m (x), such that deg(p m ) < deg(q m ) and deg(r m ) < deg(s m ).
The denominator q m (x) of the generating function H m (x) provides important information about the numbers h m (n).Let its degree be d m .Then χ m (t) = t dm q m (1/t) is the characteristic polynomial which determines the recurrence relation that h m (n) satisfies.It has d m nonzero roots (the characteristic roots) over C, name them λ m,i so that Note that the zeros of q m (x) are λ −1 m,i .For the sake of brevity, we shall still call λ m,i s the characteristic roots of q m (x).It is a routine exercise to show that, if λ m,i s are simple (hence distinct) roots, then so that for sufficiently large n where The solution is more complicated if some of the λ m,i s are repeated roots.Nonetheless, if λ m,1 is a simple positive root such that λ m,1 > |λ m,2 |, then , in which the formula for α 1 given above still holds.See the following sections for illustrations of our discussion.

The Thin Grid Cylinder C 2 × P n+1
We find , for all n ≥ 1.

The Thin Grid Cylinder
We obtain the following: The characteristic polynomial of T * 3 is x 2 − 4. Because of Cayley-Hamilton theorem, we obtain the recurrence relations . The generating functions are Therefore, .
The denominator of H 3 (x) yields the recurrenece relation x k , we obtain the following simple formula for h 3 (n).

The Thin Grid Cylinder
The denominator q 6 (x) has seven simple roots, three real and four complex, and λ 6,1 ≈ 9.07807.

Asymptotic Values
Let ρ m be the radius of convergence for H m (x).The coefficients of H m (x) are non-negative, Pringsheim's Theorem (see, for example, [8]) states that it has a singularity at x = ρ m .Since we assume that q m (x) = m is nonnegative points to the Perron-Frobenius theorem for an answer.Let M be a nonnegative square matrix.We say that M is irreducible if, for every i and j, there exists a positive integer k = k(i, j) such that (M k ) ij > 0. This is equivalent to saying that the multidigraph G M with adjacency matrix M is strongly connected.The matrix M is said to be primitive if A γ > 0 for some positive integer γ.For example, the matrix is irreducible but not primitive, and T * 4 is primitive because T * 4 2 > 0. The period or order of cyclicity of M , labeled by g, can be defined as the greatest common divisor of the lengths of the directed cycles in G M [12].From the Perron-Frobenius theory (see, for example, [12]), g is the number of eigenvalues of M having the largest modulus.In particular, a primitive matrix is an irreducible nonnegative matrix with g = 1, it has exactly one dominant characteristic root.We find that, for 3 ≤ m ≤ 10, the transfer matrix T * m is irreducible, but it is primitive only when m is even.Accordingly, we shall study the cases of odd and even m separately.
When m is odd, Theorem 2.1 implies that Hence H nc m (x) comes from the odd terms of H m (x), and H c m (x) from the even terms.This means Hence, K m (x) = H m (−x) when m is odd.Since H nc m (x) and H c m (x) share the same denominator q m (x)q m (−x), both sequences h nc m and h c m satisfy a linear recurrence relation of order 2d m .However, q m (x)q m (−x) is an even function, so it is a polynomial of degree d m in x 2 .Thus, the subsequences of nonzero terms {h nc m (2n)} n≥1 and {h c m (2n − 1)} n≥1 satisfy a linear recurrence relation of order d m .Because of (5), it is clear that, for the nonzero terms, the asymptotic behavior of h nc m (n) and h c m (n) is same as that of h m (n).More precisely, h nc m (2n) ∼ h m (2n), and h c m (2n + 1) ∼ h m (2n + 1).If the transfer matrix D * m is irreducible, then it would have g m dominant characteristic roots, where g m denotes the period of D * m .The fact that q m (x)q m (−x), the denominator that H nc m (x) and H c m (x) share, is a polynomial in x 2 suggests that D * m is a bipartite graph.If this can be confirmed, then g m must be even.In fact, it contains the following directed cycle of length 2: u 1 u 2 u 1 , where u For even m, we note that D * m contains loops.For example, there is a loop around the vertex representing the word 2030 • • • ((m + 2)/2)0, see Figure 4. We conclude that, if T * m is irreducible (recall that our computational data confirm that the matrix T * m is indeed irreducible for m ≤ 10), then g m = 1.But the dominant characteristic root can come from either H m (x) or K m (x).Our computational data reveal that, for m ≤ 10, the radius of convergence for K m (x) is greater than that of H m (x).This, together with (3) and ( 4), imply that the dominant characteristic root of both H nc (x) and H c m (x) comes from H m (x).This immediately proves that h nc m (n) ∼ h c m (n) when m = 2, 4, 6, 8, 10.Is it always true when m is even?
8 Concluding Remarks and Open Problems Our computational data affirm that for 3 ≤ m ≤ 10, the denominator q m (x) has only one real positive dominant characteristic root θ m , see where a m = −θ m p m (θ −1 m )/q m (θ −1 m ).As we have discussed in the previous section, the validity this conjecture, and other related asymptotic relations, can be completely resolved if we can settle the following open problems: 1. Is T * m irreducible for all m ≥ 3? Note that this has been confirned for m ≤ 10.

Is D *
m bipartite when m is odd?Again, this has been confirmed up to m = 9. 3. Does the dominant characteristic root remain in H m when m is even?This is equivalent to showing that the radius of convergence for K m (x) is greater than that of H m (x).
Our computational data suggest further problems for investigation: 4. Is the sequence generated by K m (x) always alternating?In other words, do we always have h c m (2k) < h nc m (2k) and h c m (2k + 1) > h nc m (2k + 1)?We close our discussion with three more interesting questions: 6.What is an appropriate combinatorial interpretation for K m (x)?
7. Can we define a labeling of the windows such that a single transfer matrix can be used to obtain the generating function H m (x) directly?If such a transfer matrix does exist, its characteristic polynomial should be χ m (t) mentioned in Section 6. (Recall that the matrices obtained in [2] for graph C m ×P n are transfer matrices for sequences h m (n), but obtained by a labeling of the vertices of C m × P n .We additionally verified that they are indeed primitive for m ≤ 12.) 8. Can we find some similar properties of sequences h m (n), h nc m (n) and h c m (n) for the case of thick cylinder P m × C n (m is kept constant, whereas n grows)?

Fig. 2 :
Fig. 2: Two types of closed Jordan curves on a cylindrical surface.

Fig. 4 :
Fig. 4: The construction of the two types of Hamiltonian cycles.
conditions [A1], [A2] and [A4] imply that this subgraph of C m × P n+1 is a 2-factor.The global aspect of Hamiltonicity is provided by condition [A3].The boundary of the positive region determines the uniqueness of the HC. 2 (a) If a i,j = 0, then b i,j = 0. (b) Partition the positive windows in the jth column into j-SIST equivalence classes, label all the windows within each equivalence class 2, 3, . . ., according to the order in which the equivalence classes first appear within the jth column, from top to bottom.

Fig. 5 :
Fig. 5: The labeling of the windows of a HC nc on C10 × P11, and a HC c on C10 × P11.

Theorem 4 . 1
The matrix B = [b i,j ] m×n (either B nc or B c ) satisfies the following properties (we adopt the convention b m+1,j = b 1,j , and b 0,j = b m,j , for 1 ≤ j ≤ n): [B1] The first column b 1,1 b 2,1 . . .b m,1 is either 02 d1 03 d2 04 d3 . . .0(p + 1) dp , p + p i=1 d i = m, or 2 d1 03 d2 04 d3 . . .0(p + 1) dp 02 (m−p−d1−d2−...−dp) , p + p i=1 then the kth column does not contain any b-factor that contains both b i,k and b j,k .(e) If k ≥ 2 and if v and u are two different maximal nonzero b-factors in the kth column, then there is exactly one sequence v = v 1 , v 2 , . . ., v p = u of p > 1 different maximal b-factors in the kth column with the property that for every i with 1 ≤ i ≤ p − 1, in the (k − 1)th column, there exists exactly one letter b ji,k−1 with a ji,k−1 = a ji,k for which b ji,k ∈ v i , and there exists exactly one letter b si+1,k−1 with a si+1,k−1 = a si+1,k for which b si+1,k ∈ v i+1 and b ji,k−1 = b si+1,k−1 ; and j i = s i for 1 < i < p (see Figure 6).(f) For k ≥ 2 and for each number b ∈ C + that appears in the (k − 1)th column, there must exist an integer i, where 1 ≤ i ≤ m, for which b i,k−1 = b and b i,k > 0.

1 and u 2
are vertices (written as words) in D * m u 1 = 222 • • • 20, u 2 = 000 • • • 02.See Figure 4 and note that, because of the rotational symmetry, the vertex 22 • • • 202 is identified to 22 • • • 220, as well 20 • • • 00 to 00 • • • 02.We conclude that g m = 2, so the two dominant characteristic roots of D * m must be ±θ m .This in turn implies that θ m is the sole dominant characteristic root of q m (x).Consequently, we deduce that, for odd m, h nc m (2n) ∼ a m θ 2n−1 m and h c m (2n + 1) ∼ a m θ 2n m , provided that T * m is irreducible, and D * m is a bipartite graph.Our computational data reveal that D * 3 , D * 5 , D * 7 , D * 9 are bipartite multidigraphs.

Table 1 .
Notice that the numbers |V (D m )| and 2(M m − 1) are equal when m is odd.