Abstract
Abstract. We establish the quaternionic weighted zeta function of a graph and its Study determinant expressions. For a graph with quaternionic weights on arcs, we define a zeta function by using an infinite product which is regarded as the Euler product. This is a quaternionic extension of the square of the Ihara zeta function. We show that the new zeta function can be expressed as the exponential of a generating function and that it has two Study determinant expressions, which are crucial for the theory of zeta functions of graphs.
Introduction
Zeta functions of graphs started from Ihara zeta functions of regular graphs by Ihara [8] . Originally, Ihara [8] presented p-adic Selberg zeta functions of discrete groups, and showed that its reciprocal is an explicit polynomial. Ihara also showed the logarithm of Ihara zeta function has an expression in the form of a generating function. Serre [13] pointed out that the Ihara zeta function is the zeta function of the quotient T /Γ (a finite regular graph) of the one-dimensional Bruhat-Tits building T (an infinite regular tree) associated with GL(2, k p ). A zeta function of a regular graph G associated with a unitary representation of the fundamental group of G was developed by Sunada [16] , [17] . Hashimoto [6] treated multivariable zeta functions of bipartite graphs. For a general graph, Hashimoto [6] gave a determinant expression for its Ihara zeta function by its edge matrix. Bass [2] generalized Ihara's result on the Ihara zeta function of a regular graph to an irregular graph, and showed that its reciprocal is again a polynomial. Various proofs of Bass' Theorem have been given by Stark and Terras [14] , Foata and Zeilberger [5] , Kotani and Sunada [9] . For the weighted type of zeta function of graphs, Hashimoto [7] introduced a zeta function of a graph with weights assigned to its edges. Stark and Terras [14] defined the edge zeta function of a graph with weights assigned to its arcs, and gave its determinant expression by using its edge matrix. Mizuno and Sato [11] introduced a special version of the edge zeta function of a graph, and defined the weighted zeta function of a graph by using arc weights and the variable t counting the length of cycles. Later, they called this zeta function the first weighted zeta function in order to distinguish it from another zeta function which Sato defined after. It is noteworthy that the expressions of those zeta functions by the exponentials of generating functions were important to deriving the determinant expressions in the studies above. Recently, Watanabe and Fukumizu [18] presented a determinant expression for the edge zeta function of a graph by using matrices with size of the number of its vertices. As described above, zeta functions of graphs have been investigated for half a century.
Introduction
On the other hand, the quaternions were discovered by Hamilton in 1843. It can be considered as an extension of the complex numbers. However, quaternions do not commute mutually in general. For many years, a number of people, for example Cayley, Study, Moore, Dieudonné, Dyson, Mehta, Xie, Chen, have given different definitions of determinants of quaternionic matrices. Detailed accounts on the determinants of quaternionic matrices can be found in, for example, [1, 19] . In order to extend the zeta function of a graph to the case of quaternions, we will use the approach developed by Study [15] . The Study determinant is the unique, up to a real power factor, functional which satisfies the following three axioms Therefore it is essential to investigate the relation between the Study determinant and the quaternionic zeta function of a graph. The advantage of this approach is that one can reduce a calculation of the Study determinant to that of the ordinary determinant. Its disadvantage is that the Study determinant is not an exact extension of the determinant but is rather that of the square of it. Our aim in this paper is to define the quaternionic zeta function of a graph by using an infinite product, namely the Euler product, and to show its essential characteristics. Specifically, we derive its expression in the form of the exponential of a generating function and determine its two types of Study determinant expressions such as in [6] (Hashimoto type) and in [2] (Bass type). Both types are crucial for a zeta function of a graph. In order to obtain these results, we explain the formal power series on a monoid algebra for preparations. The formal series, which includes the formal power series, is treated in detail in [3] Our approach in this material follows the manner in [5, 12] . Furthermore, we give an account of some relations between their exponentials and logarithms which can be found in Chapter IV of [4] . Up to the present, no quaternionic extension of zeta functions of graphs has been appeared. Hence our results can be regarded as the first successful construction of a quaternionic zeta function of a graph.
Introduction
The rest of the paper is organized as follows. Section 2 treats noncommutative formal series and monoids. The notion of Lyndon word is given, and some application of the factorization theorem is discussed. The conclusion (Proposition 2.3) plays an essential role in our investigation. In Section 3, we define the exponential and the logarithm of noncommutative formal power series and give some formulas of which we make use in Section 6. In Section 4, we explain the Study determinant of a quaternionic matrix and extend it to the matrix whose entries are formal power series with quaternionic coefficients. In Section 5, we provide a summary of the Ihara zeta function and the various zeta function of a graph, and present their determinant expressions. In Section 6, we define a quaternionic zeta function of a graph by using an infinite product, and give its expression in the form of the exponential of a generating function. In Section 7, we show that the quaternionic zeta functions has two types of Study determinant expressions. One (Theorem 7.1) is an analogy to Hashimoto type expression [6] and another (Theorem 7.2) is to Bass type [2] .
Noncommutative formal power series and monoids
Let R be a commutative ring with unity, and A an algebra over R. A [[t] ] denotes the ring of formal power series in t with coefficients in A.
Noncommutative formal power series and monoids
] can be equipped with the topology defined by the following manner. Let ω be the function defined as follows:
Noncommutative formal power series and monoids
is the set of formal sums z = g ∈ G z g g, where z g ∈ R for each g ∈ G and z g = 0 for all but finitely many g. The addition in R[G] is coefficient-wise, and the elements of R commute with the elements of G in the multiplication.
Noncommutative formal power series and monoids
Let X = {x 1 , · · ·, x N } be a finite nonempty totally ordered set in which elements are arranged ascendingly. X * denotes the free monoid generated by X. Let < be the lexicographic order on X * derived from the total order on X. For a word w = x i 1 x i 2 · · ·x ir ∈ X * , r is called the length of w which is denoted by |w|. The length of the empty word is defined to be 0. A nonempty word w in X * is called a Lyndon word if w is prime, namely, not a power w ′r of any other word w ′ for any r ≥ 2, and is minimal in the cyclic rearrangements of w. We denote by L X the set of Lyndon words in X * . It is well known that any nonempty word w can be formed uniquely as a nonincreasing sequence of Lyndon words. 
Noncommutative formal power series and monoids
Proof. For the proof, see for example [10] .
Noncommutative formal power series and monoids
], Theorem 2.1 implies: 
Noncommutative formal power series and monoids
(2.1) and (2.2) imply the following equation:
Noncommutative formal power series and monoids
From (2.3), we obtain: Proposition 2.2. Proof. In order to show that 5) we check that the coefficient of t r of the left hand side is equal to 1 if r = 0 and 0 if r > 0. Since
Noncommutative formal power series and monoids
, the coefficient of at most r-th power of t in the left hand side of (2.5) is the same as that of the product:
Noncommutative formal power series and monoids
for every nonnegative integer r ≥ 0. This is a finite product since |X| < ∞ and therefore is equal to 1. Thus (2.5) holds. 
Noncommutative formal power series and monoids
However we easily see that
Noncommutative formal power series and monoids
and that A(1, 1) + A(1, 2) + · · · + A(n, n) = A, hence we obtain:
Noncommutative formal power series and monoids
3 Exponentials and Logarithms of noncommutative formal power series
Noncommutative formal power series and monoids
], the ring of formal power series with coefficients in Q[X * ] where t is considered as a central
Noncommutative formal power series and monoids
For an α which has zero constant term, we define the exponential of α to be the element of
Noncommutative formal power series and monoids
as follows:
Noncommutative formal power series and monoids
Similarly, for an α whose constant term equals 1, we define the logarithm of α as follows:
Noncommutative formal power series and monoids
] with zero constant term. Then we may write log α = log(1+β) = n≥1 (−1) n−1 n −1 β n and log α has zero constant term. Conversely, if β has zero constant term, then the constant term of exp β equals 1.
Noncommutative formal power series and monoids
(1) If α 0 = 1, then exp log α = α.
Noncommutative formal power series and monoids
(2) If β has zero constant term, then log exp β = β.
Noncommutative formal power series and monoids
Proof. We shall prove (1). Letting α = 1 + β, we have:
Noncommutative formal power series and monoids
We consider the partial sum of (3.10) as follows:
Noncommutative formal power series and monoids
If k, ℓ ≥ r, then the coefficient of at most r-th power of β in (3.10) is equal to that in (3.11).
Noncommutative formal power series and monoids
On the other hand, by the Taylor expansion we have:
Noncommutative formal power series and monoids
for a real number x with |x| < 1. Similar to (3.11), if k, ℓ ≥ r, then the coefficient of at most r-th power of x in (3.12) is equal to that in (3.13):
Noncommutative formal power series and monoids
Hence the coefficient of at most r-th power of β in (3.10) is equal to that of x in (3.12). Since r is an arbitrary nonnegative integer, the coefficient of any power of β in (3.10) coincide with the coefficient of the same power of x in (3.12). Moreover β = n≥1 α n t n implies that the coefficient of at most r-th power of t is determined by the terms of at most r-th power of β in (3.10). Since r is arbitrary, we obtain that exp log(1
Noncommutative formal power series and monoids
. (2) can be proven in the same manner.
Noncommutative formal power series and monoids
Proof. By easy calculations, we have:
Noncommutative formal power series and monoids
. Both p and q have zero constant term. By
Noncommutative formal power series and monoids
Proof. It follows from Proposition 3.3 that:
Noncommutative formal power series and monoids
} in the same manner as in the Section 2. For each matrix A = (a ij ) ∈ Mat(n, A), we define σ A to be the Q-algebra 
The Study determinant of a quaternionic matrix
Let H be the set of quaternions. H is a noncommutative associative algebra over R, whose underlying real vector space has dimension 4 with a basis 1, i, j, k which satisfy the following relations:
The Study determinant of a quaternionic matrix
and Re x = x 0 the real part of x. One can easily check xx
The Study determinant of a quaternionic matrix
3 the norm of x. Indeed, | · | satisfies (1) |x| ≥ 0, and moreover |x| = 0 ⇔ x = 0,
The Study determinant of a quaternionic matrix
Any quaternion x can be presented by two complex numbers x = a + jb uniquely. Such a presentation is called the symplectic decomposition. Two complex numbers a and b are called the simplex part and the perplex part of x respectively. The symplectic decomposition is also valid for a quaternionic matrix, namely a matrix whose entries are quaternions. Mat(m, n, H) denotes the set of m×n quaternionic matrices and Mat(m, H) the set of m×m quaternionic matrices. For M ∈ Mat(m, n, H), we can write M = M S + jM P uniquely where M S , M P ∈ Mat(m, n, C). M S and M P are called the simplex part and the perplex part of M respectively. We define ψ to be the map from Mat(m×n, H) to Mat(2m×2n, C) as follows:
The Study determinant of a quaternionic matrix
where A is the complex conjugate of a complex matrix A. Then ψ is an R-linear map. We also have:
The Study determinant of a quaternionic matrix
Proof. Let M = A + jB and N = C + jD be the symplectic decompositions of M and N. Then, MN = (A + jB)(C + jD) = AC + AjD + jBC + jBjD.
The Study determinant of a quaternionic matrix
Since Xj = jX for every complex matrix X, we obtain:
The Study determinant of a quaternionic matrix
and therefore
The Study determinant of a quaternionic matrix
On the other hand,
The Study determinant of a quaternionic matrix
Proof. By Lemma 4.1, ψ is an R-algebra homomorphism. Injectivity of ψ is clear.
The Study determinant of a quaternionic matrix
As in [1] , one can characterize the image of ψ in Mat(2n, C) as follows.
The Study determinant of a quaternionic matrix
Lemma 4.3. Let J be the 2n×2n matrix defined as follows:
The Study determinant of a quaternionic matrix
where N denotes the complex conjugate of N.
The Study determinant of a quaternionic matrix
Proof. For the proof, see [1] .
The Study determinant of a quaternionic matrix
In [15] , Study defined a determinant of a n×n quaternionic matrix which we denote by Sdet as follows:
The Study determinant of a quaternionic matrix
where det is the ordinary determinant. Sdet is called the Study determinant. Sdet can be extended to the one for matrices whose entries are formal power series with coefficients in H in the following way. Let H
The Study determinant of a quaternionic matrix
One can extend ψ to the injective homomorphism ψ t of R-algebras from Mat(n, H) Proof. For the proof, see for example [20] .
Remark 4.5. We notice that the Lemma 4.4 can be extended to matrices whose entries belong to C[[t]] immediately.
Lemma 4.6.
Remark 4.5. We notice that the Lemma 4.4 can be extended to matrices whose entries belong to C[[t]] immediately.
Proof. By Lemma 4.3, it follows that
Remark 4.5. We notice that the Lemma 4.4 can be extended to matrices whose entries belong to C[[t]] immediately.
Hence, the assertion holds.
Remark 4.5. We notice that the Lemma 4.4 can be extended to matrices whose entries belong to C[[t]] immediately.
Proof. By a direct calculation, we have:
Remark 4.5. We notice that the Lemma 4.4 can be extended to matrices whose entries belong to C[[t]] immediately.
. Similarly, the same conclusion holds for α * α.
Remark 4.5. We notice that the Lemma 4.4 can be extended to matrices whose entries belong to C[[t]] immediately.
Lemma 4.7 immediately implies:
Remark 4.5. We notice that the Lemma 4.4 can be extended to matrices whose entries belong to C[[t]] immediately.
(ii) The constant term of Sdet t (M) is equal to 0 ⇔ M is singular.
(iii) Sdet t (MN) = Sdet t (M) Sdet t (N) for M, N ∈ Mat(n, H[[t]]). (iv) If N is obtained from M by adding a left-multiple of a row to another row or a rightmultiple of a column to another column, then Sdet t (N) = Sdet t (M). (v) If N is obtained from M by exchanging a row with another row or a column with another column, then Sdet t (N) = Sdet t (M). (vi) Sdet
is of the form:
(iii) Sdet t (MN) = Sdet t (M) Sdet t (N) for M, N ∈ Mat(n, H[[t]]). (iv) If N is obtained from M by adding a left-multiple of a row to another row or a rightmultiple of a column to another column, then Sdet t (N) = Sdet t (M). (v) If N is obtained from M by exchanging a row with another row or a column with another column, then Sdet t (N) = Sdet t (M). (vi) Sdet
(viii) Let A be an m × n matrix and B an n × m matrix. Then 
Proof. (i) It follows from Lemma 4.6 that
Hence by Lamma 4.6,
Proof. (i) It follows from Lemma 4.6 that
) and it follows that ψ t (M) is singular ⇔ M is singular. (iii) Since ψ t is an algebra homomorphism, (iii) immediately holds.
Proof. (i) It follows from Lemma 4.6 that
. It suffice to show that for k = ℓ, the n×n matrix B kℓ (b) = I n + bE kℓ satisfies Sdet t (B kℓ (b)) = 1. Since four blocks of
Proof. (i) It follows from Lemma 4.6 that
commute with each other, we can apply Remark 4.5 and obtain:
Proof. (i) It follows from Lemma 4.6 that
(v) We prove for row exchange. Putting N = PM for the permutation matrix P corresponding to some transposition, it follows that:
Proof. (i) It follows from Lemma 4.6 that
) be the symplectic decomposition. Then by using Lemma 4.1, Remark 4.5 and Lemma 4.7, we have:
Proof. (i) It follows from Lemma 4.6 that
In the same way, we can deduce Sdet t (Mα) = (αα * ) n Sdet t (M). (vii) For a 2n×2n matrix N and any two subsets
Proof. (i) It follows from Lemma 4.6 that
IJ denotes the submatrix obtained from N by deleting i 1 , i 2 , · · ·, i r th rows and j 1 , j 2 , · · ·, j s th columns. Then by definitions of Sdet t and ψ t , we get the following:
Proof. (i) It follows from Lemma 4.6 that
(viii) Let A be an m × n matrix and B an n × m matrix. Then, by the definition of the Study determinant, we have: 
The Ihara zeta function of a graph
Let G = (V (G), E(G)) be a finite connected graph with the set V (G) of vertices and the set E(G) of unoriented edges uv joining two vertices u and v. We assume that G has neither loops nor multiple edges throughout. For uv ∈ E(G), an arc (u, v) is the oriented edge from
The Ihara zeta function of a graph
For e = (u, v) ∈ D(G), o(e) = u denotes the origin and t(e) = v the terminal of e respectively. Furthermore, let e −1 = (v, u) be the inverse of e = (u, v). The degree deg v = deg G v of a vertex v of G is the number of edges incident to v. For a natural number k, a graph G is called k-regular if deg G v = k for each vertex v of G. A path P of length ℓ in G is a sequence P = (e 1 , · · · , e ℓ ) of ℓ arcs such that e i ∈ D(G) and t(e i ) = o(e i+1 ) for i ∈ {1, · · · , ℓ − 1}. We set o(P ) = o(e 1 ) and t(P ) = t(e ℓ ). |P | denotes the length of P . We say that a path P = (e 1 , · · · , e ℓ ) has a backtracking if e i+1 = e −1 i for some i(1≤i≤ℓ − 1). A path P is said to be a cycle if t(P ) = o(P ). The inverse of a path P = (e 1 , · · · , e ℓ ) is the path (e −1 ℓ , · · · , e −1 1 ) and is denoted by P −1 . An equivalence relation between cycles is given as follows. Two cycles C 1 = (e 1 , · · · , e ℓ ) and C 2 = (f 1 , · · · , f ℓ ) are said to be equivalent if there exists k such that f j = e j+k for all j where indices are treated modulo ℓ. Let [C] be the equivalence class which contains the cycle C. Let B r be the cycle obtained by going r times around a cycle B. Such a cycle is called a power of B. A cycle C is said to be reduced if both C and C 2 have no backtracking. Furthermore, a cycle C is said to be prime if it is not a power of a strictly smaller cycle. Note that each equivalence class of prime, reduced cycles of a graph G corresponds to a unique conjugacy class of the fundamental group
The Ihara zeta function of a graph
The Ihara zeta function of a graph G is a function of t ∈ C with |t| sufficiently small, defined by
The Ihara zeta function of a graph
where [C] runs over all equivalence classes of prime, reduced cycles of G. Let B = (B ef ) e,f ∈D(G) and J 0 = (J ef ) e,f ∈D(G) be 2m × 2m matrices defined as follows:
The Ihara zeta function of a graph
Then the matrix B − J 0 is called the edge matrix of G.
The Ihara zeta function of a graph
Theorem 5.1 (Hashimoto [6] ; Bass [2] ). Let G be a finite connected graph. Then the reciprocal of the Ihara zeta function of G is given by
The Ihara zeta function of a graph
where r and A are the Betti number and the adjacency matrix of G, respectively, and
The Ihara zeta function of a graph
has an expression as the exponential of a generating function as follows:
The Ihara zeta function of a graph
where N k is the number of equivalence classes of reduced cycles of length k.
The Ihara zeta function of a graph
The matrix D is called the degree matrix of G.
The Ihara zeta function of a graph
A weighted zeta function of a graph was first defined in Hashimoto [6] by giving a weight for each edge of a graph. After that, it was generalized in Stark and Terras [14] by giving a weight for each arc of a graph as follows. Let D(G) = {e 1 , · · · , e m , e m+1 , · · · , e 2m } where
The Ihara zeta function of a graph
where [C] runs over all equivalent classes of prime, reduced cycles of G. [14] ). Let G be a connected graph with m edges. Then
Theorem 5.2 (Stark and Terras
where
Theorem 5.2 (Stark and Terras
In [14] , the multivariable analogue N k (u) of N k was defined and used to obtain the determinant expression. Mizuno and Sato [11] defined the weighted zeta function of a graph G, and gave a determinant expression by the weighted matrix of G. Later, they called this zeta function the first weighted zeta function in order to distinguish it from another zeta function which Sato defined after. Assigning a complex number w(e) to each arc e ∈ D(G), we define W = (w uv ) u,v∈V (G) to be the n × n matrix as follows:
Theorem 5.2 (Stark and Terras
where [C] runs over all equivalent classes of prime, reduced cycles of G. [11] ). Let G be a connected graph with n vertices. Suppose that w(e −1 ) = w(e) −1 for e ∈ D(G). Then
Theorem 5.3 (Mizuno and Sato
Z(G, w, t) −1 = (1 − t 2 ) m−n det(I n − tW + t 2 (D − I n )).
Z(G, w, t) has an expression as the exponential of a generating function as follows:
where [C] runs over all equivalent classes of prime, reduced cycles of G.
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Later, Watanabe and Fukumizu [18] gave a determinant expression for the edge zeta function of a graph with no additional condition w(e −1 ) = w(e) −1 . Rearranging arcs so that . . .
Z(G, w, t) has an expression as the exponential of a generating function as follows:
In [18] , two n × n matricesÂ = (Â uv ) u,v∈V (G) andD = (D uv ) u,v∈V (G) were defined as follows:
Z(G, w, t) has an expression as the exponential of a generating function as follows:
w(e)w(e −1 ) 1 − w(e)w(e −1 )
Z(G, w, t) has an expression as the exponential of a generating function as follows:
. where δ uv denotes the Kronecker's delta. They derived their determinant expression withÂ andD instead of the weighted matrix and the degree matrix of G.
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Theorem 5.4 (Watanabe and Fukumizu [18] ).
Z(G, w, t) has an expression as the exponential of a generating function as follows:
The quaternionic weighted zeta function of a graph
Z(G, w, t) has an expression as the exponential of a generating function as follows:
We continue with the notations in former sections. We shall define the quaternionic weighted zeta function of a graph. First we extend w(e) to a quaternion. Let w be a map from
Z(G, w, t) has an expression as the exponential of a generating function as follows:
We may write w(e) instead of w(u, v) if e = (u, v) ∈ D(G). We call w(e) the quaternionic weight of e ∈ D(G). For a path P = (e 1 , · · · , e ℓ ) of G, the norm (or the quaternionic weight) w(P ) of P is defined by w(P ) = w(e 1 )w(e 2 )· · ·w(e ℓ ). We arrange the arcs e 1 , e 2 , · · ·, e 2m such that e m+k = e 15) where C runs over all reduced cycles C = e i 1 e i 2 · · ·e ir such that i 1 i 2 · · ·i r ∈L [2m] . If w(e) = 1 for any e ∈ D(G), then the quaternionic weighted zeta function of G is reduced to the square of the Ihara zeta function of G. (ii) (6.15) does not depend on the order in which inverses of
Z(G, w, t) has an expression as the exponential of a generating function as follows:
We shall show that Z H (G, w, t) can be expressed as the exponential of a generating function. We recall that Theorem 3. 
Z(G, w, t) has an expression as the exponential of a generating function as follows:
and has zero constant term,
Z(G, w, t) has an expression as the exponential of a generating function as follows:
is also an element of R [[t] ] and β C = k≥1 α k C has zero constant term. Since the constant term of Z H (G, w, t) is equal to 1, we can derive the following:
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Therefore log Z H (G, w, t) has zero constant term, and it follows that: Theorem 6.2. Z H (G, w, t) has the expression as the exponential of a generating function as follows:
Z(G, w, t) has an expression as the exponential of a generating function as follows:
where C runs over all reduced cycles C = e i 1 e i 2 · · ·e ir such that
Z(G, w, t) has an expression as the exponential of a generating function as follows:
We can also express Z H (G, w, t) as the square of the exponential of a generating function as follows:
Z(G, w, t) has an expression as the exponential of a generating function as follows:
7 Study determinant expressions for Z H (G, w, t)
Z(G, w, t) has an expression as the exponential of a generating function as follows:
For a quaternionic weighted matrix W of G, we define two 2m × 2m matrices B w = (B ef ) e,f ∈D(G) as follows:
Z(G, w, t) has an expression as the exponential of a generating function as follows:
We call B w − J w the quaternionic weighted edge matrix. Then the Study determinant expression of Z H (G, w, t) of Hashimoto type by use of the quaternionic weighted edge matrix is stated as follows:
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Beforehand, we mention that the right hand side does not depend on the way of arranging arcs which determines row and column indexes. Indeed, if we rearrange arcs, then it turns into Sdet t P(I 2m − (B w − J w )t)P −1 for some permutation matrix P. However, from Proposition 4.9 (v), this is equal to Sdet t (I 2m − (B w − J w )t).
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Proof. Set A = H in Proposition 2.3. Then we can take Study determinants of both sides in (2.7), so that 
Z(G, w, t) has an expression as the exponential of a generating function as follows:
and otherwise,
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Putting A = B w − J w , then a ij = a e i e j = w(e i ) if t(e i ) = o(e j ) and e j =e
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Therefore, (7.16) yields: (1 − w(e i 1 )· · ·w(e ir )t r )(1 − w(e i 1 )· · ·w(e ir )t r ) * .
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Remark 6.1 (ii) implies the inverse of the right hand side is Z H (G, w, t). Hence we obtain the conclusion as desired.
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Finally, we shall show the Study determinant expression of Bass type. We define n × n matricesW = (W uv ) u,v∈V (G) andD = (D uv ) u,v∈V (G) to be as follows:
Z(G, w, t) has an expression as the exponential of a generating function as follows:
(1 − w(e)w(e −1 )t 2 ) −1 w(e) if e = (u, v)∈D(G) 0 otherwise,
Z(G, w, t) has an expression as the exponential of a generating function as follows:
(1 − w(e)w(e −1 )t 2 ) −1 w(e)w(e −1 ).
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Then the Study determinant expression of Bass type is stated as follows: Proof. We readily see that:
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Sdet t (I 2m − t(B w − J w )) = Sdet t (I 2m + tJ w − tB w ) = Sdet t (I 2m − tB w (I 2m + tJ w ) −1 ) × Sdet t (I 2m + tJ w ). Now, let K = (K ev ) e∈D(G);v∈V (G) and L = (L ev ) e∈D(G);v∈V (G) be two 2m × n matrices defined as follows: Let X(e) be the 2×2 matrix defined as follows:
Z(G, w, t) has an expression as the exponential of a generating function as follows:
X(e) = 1 w(e)t w(e −1 )t 1 .
Z(G, w, t) has an expression as the exponential of a generating function as follows:
Then, the inverse of X(e) is given by Now, the assertion follows from Theorem 7.1.

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