A characteristic polynomial for the transition probability matrix of correlated random walks on a graph

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph G, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of G. As an application, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph. Mathematics Subject Classifications: 05C50, 15A15


Introduction
Zeta functions of graphs started from the Ihara zeta functions of regular graphs by Ihara [7]. In [7], he showed that their reciprocals are explicit polynomials. 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 [5] 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 by a determinant containing the edge matrix. Bass [1] presented another determinant expression for the Ihara zeta function of an irregular graph by using its adjacency matrix.
Cooper [2] treated various properties of a graph G determined by the Ihara zeta function of G. Morita [13] defined a generalized weighted zeta function of a digraph which contains various zeta functions of a graph or a digraph. Ide et al. [6] presented a determinant expression for the generalized weighted zeta function of a graph.
The time evolution matrix of a discrete-time quantum walk in a graph is closely related to the Ihara zeta function of a graph. A discrete-time quantum walk is a quantum analog of the classical random walk on a graph whose state vector is governed by a matrix called the time evolution matrix(see [9]). Ren et al. [14] gave a relationship between the discretetime quantum walk and the Ihara zeta function of a graph. Konno and Sato [11] obtained a formula of the characteristic polynomial of the Grover matrix by using the determinant expression for the second weighted zeta function of a graph. Thus, the relation between the Grover walk and a simple random walk on a graph was established. Konno [10] treated the one-dimensional correlated random walk derived from one-dimensional quantum walk.
In this paper, we present an analogue of the above relation for the correlated random walk derived from the Grover walk on a graph. We introduce a new correlated random walk induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph, and present a formula for the characteristic polynomial of its transition probability matrix.
In Section 2, we review the Ihara zeta function and the generalized weighted zeta functions of a graph. In Section 3, we review the Grover walk on a graph. In Section 4, we define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph G, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW. In Section 5, we give the spectrum of the transition probability matrix for this CRW of a regular graph. In Section 6, we present the spectrum for the transition probability matrix of this CRW of a semiregular bipartite graph. In Section 7, we present formulas for the characteristic polynomials of the transition probability matrices of another type of the CRW on a graph, and give the spectrum of its transition probability matrix.

Zeta functions of graphs
Graphs and digraphs treated here are finite. Let G be a connected graph and D G the symmetric digraph corresponding to G.
A walk P of length n in G is a sequence P = (e 1 , · · · , e n ) of n arcs such that e i ∈ D(G), t(e i ) = o(e i+1 )(1 i n − 1)(see [2]). If e i = (v i−1 , v i ) for i = 1, · · · , n, then we write P = (v 0 , v 1 , · · · , v n−1 , v n ). Set | P |= n, o(P ) = o(e 1 ) and t(P ) = t(e n ). Also, P is called an (o(P ), t(P ))-walk. We say that a walk P = (e 1 , · · · , e n ) has a backtracking if e −1 i+1 = e i for some i (1 i n − 1). A (v, w)-walk is called a closed walk if v = w. The inverse closed walk of a closed walk C = (e 1 , · · · , e n ) is the closed walk C −1 = (e −1 n , · · · , e −1 1 ). We introduce an equivalence relation between closed walks. Two closed walks C 1 = (e 1 , · · · , e m ) and C 2 = (f 1 , · · · , f m ) are called equivalent if there exists a positive number k such that f j = e j+k for all j, where the subscripts are considered by modulo m. The inverse closed walk of C is in general not equivalent to C. Let [C] be the equivalence class which contains a closed walk C. Let B r be the closed walk obtained by going r times around a closed walk B. Such a closed walk is called a multiple of B. A closed walk C is reduced if both C and C 2 have no backtracking. Furthermore, a cclosed walk C is prime if it is not a multiple of a strictly smaller closed walk. Note that each equivalence class of prime, reduced closed walks of a graph G corresponds to a unique conjugacy class of the fundamental group π 1 (G, v) of G at a vertex v of G.
The Ihara(-Selberg) zeta function of G is defined by Theorem 1 (Ihara; Hashimoto; Bass). Let G be a connected graph with n vertices and m edges. Then the reciprocal of the Ihara zeta function of G is given by The first identity in Theorem 1 was obtained by Hashimoto [5]. Also, Bass [1] proved the second identity by using a linear algebraic method.
Stark and Terras [15] gave an elementary proof of this formula, and discussed three different zeta functions of any graph. Various proofs of Bass' Theorem were given by Kotani and Sunada [12], and Foata and Zeilberger [4].

The Grover walk on a graph
Let G be a connected graph with n vertices and m edges, The discrete-time quantum walk with the matrix U as a time evolution matrix is called the Grover walk on G.
Let G be a connected graph with n vertices and m edges. Then the n × n matrix T(G) = (T uv ) u,v∈V (G) is given as follows: Note that the matrix T(G) is the transition matrix of the simple random walk on G(see [11]).
Theorem 3 (Konno and Sato). Let G be a connected graph with n vertices v 1 , . . . , v n and m edges. Then the characteristic polynomial for the Grover matrix U of G is given by From this Theorem, the spectra of the Grover matrix on a graph is obtained by means of those of T(G) (see [3]). Let Spec(F) be the spectra of a square matrix F. Corollary 4 (Emms, Hancock, Severini and Wilson). Let G be a connected graph with n vertices and m edges. The Grover matrix U has 2n eigenvalues of the form where λ T is an eigenvalue of the matrix T(G). The remaining 2(m − n) eigenvalues of U are ±1 with equal multiplicities.

A correlated random walk on a graph
Let G be a connected graph with n vertices and m edges, and U be the Grover matrix of G. Then we define a 2m × 2m matrix P = (P ef ) e,f ∈D(G) as follows: the electronic journal of combinatorics 28(4) (2021), #P4.21 Note that The random walk with the matrix P as a transition probability matrix is called the correlated random walk (CRW) (with respect to the Grover matrix) on G(see [8,10]). otherwise.
Let G be a d-regular graph. In the case of d = 4, we consider P = (P ef ) e,f ∈D(G) be the transition probability matrix of the CRW with respect to the Grover matrix on a Thus, this CRW is considered to be a simple random walk on G which the particle moves over each arc in terms of the same probability.
By Theorem 2, we obtain the following formula for P.
Theorem 5. Let G be a connected graph with n vertices and m edges, and let P be the transition probability matrix of the CRW with respect to the Grover matrix. Then Proof. For the matrix P, we have Furthermore, let Then we have P = t M(θ).
Thus, we obtain By Theorem 2, we have By Theorem 4, we obtain the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs.

An application to the correlated random walk on a regular graph
We present spectra for the transition probability matrix of the correlated random walk on a regular graph with respect to the Grover matrix.
Theorem 6. Let G be a connected d-regular graph with n vertices and m edges, where d 2. Furthermore, let P be the transition probability matrix of the CRW with respect to the Grover matrix. Then Proof. Let G be a connected d-regular graph with n vertices and m edges, where d 2.
Then we have d o(e) = d t(e) = d f or each e ∈ D(G).
Thus, we have Therefore, it follows that By Theorem 4, we have By substituting u = 1/λ, we obtain the following result.
). The second identity of Corollary 2 is considered as the spectral mapping theorem for P.
By Corollary 2, we obtain the spectra for the transition matrix P of the CRW with respect to the Grover matrix on a regular graph.
Corollary 8. Let G be a connected d( 2)-regular graph with n vertices and m edges. Then the transition probability matrix P has 2n eigenvalues of the form where λ A is an eigenvalue of the matrix A(G). The remaining 2(m − n) eigenvalues of P are ±(4 − d)/d with equal multiplicities m − n.
Proof. By Corollary 2, we have Thus, solving 6 An application to the correlated random walk on a semiregular bipartite graph We present spectra for the transition probability matrix of the correlated random walk on a semiregular bipartite graph. Hashimoto [5] presented a determinant expression for the Ihara zeta function of a semiregular bipartite graph. We use an analogue of the method in the proof of Hashimoto's result.
By Corollary 4, we obtain the spectra for the transition probability matrix P of the CRW with respect to the Grover matrix of a semiregular bipartite graph.
the electronic journal of combinatorics 28(4) (2021), #P4.21 7 Another type of the correlated random walk on a cycle graph The CRW is defined by the following transition probability matrix P on the one dimensional lattice: As for the CRW, see [8,10], for example. We formulate a CRW on the arc set of a graph with respect to the above matrix P. The cycle graph is a connected 2-regular graph. Let C n be the cycle graph with n vertices and n edges. Furthermore, let V (C n ) = {v 1 , . . . , v n } and e j = (v j , v j+1 )(1 j n), where the subscripts are considered by modulo n. Then we introduce a 2n × 2n matrix U = (U ef ) e,f ∈D(Cn) as follows: Note that U can be written as follows: where Q = P σ is the permutation matrix of σ = (12 . . . n). The CRW with U as a transition probability matrix is called the second type of CRW on C n with respect to the above matrix P. Now, we define a function w : D(C n ) −→ R as follows: Furthermore, let an n × n matrix W(C n ) = (w uv ) u,v∈V (Cn) as follows: The characteristic polynomial of U is given as follows.
Theorem 12. Let C n be the cycle graph with n vertices, and U the transition probability matrix of the second type of CRW on C n . Then det(λI 2n − U) = det((λ 2 + (ad − bc))I n − λW(C n )).
Proof. At first, we consider two 2n×2n matrices B = (B ef ) e,f ∈D(Cn) and J = (J ef ) e,f ∈D(Cn) as follows: Then we have Now, we define two 2n × n matrices K = (K ev ) and L = (L ev ) as follows: where e ∈ D(C n ), v ∈ V (C n ). Then we have But, we have det(I 2n − uJ) Furthermore, we have Therefore, it follows that det(I 2n − uU) The matrix t LJK is diagonal, and its (v i , v i ) entry is equal to That is, t LJK = (ab + cd − 2ad)I n . Thus, = det(((1 + (ad − bc)u 2 )I n − uW(C n )).
Substituting u = 1/λ, the result follows. By Theorem 7, we obtain the spectra for the transition probability matrix U of the second type of the CRW on C n . The matrix W(C n ) is given as follows: , Corollary 13. Let C n be the cycle graph with n vertices, and U the transition probability matrix of the second type of CRW on C n . Then the transition probability matrix U has 2n eigenvalues of the form Proof. At first, we have det(I 2n − uU) = µ∈Spec(W(Cn)) (λ 2 − µλ + (ad − bc)). Solving Now, we consider the case of a = b = c = d = 1/2. Then the matrix W(C n ) is equal to W(C n ) = 1 2 A(C n ).
By Corollary 6, we obtain the spectra for the transition probability matrix U of the second type of CRW on C n . Corollary 14. Let C n be the cycle graph with n vertices, and U the transition probability matrix of the second type of the CRW on C n . Assume that a = b = c = d = 1/2. Then the transition probability matrix U has n eigenvalues of the form λ = cos θ j , θ j = 2πj n (j = 0, 1, . . . , n − 1) ( * ).
The remaining n eigenvalues of U are 0 with multiplicities n.
Note that the spectrum of (*) are those of the transition probability matrix of the simple random walk on a cycle graph C n .
We can generalize the result for a = b = c = d = 1/2 on C n to a d-regular graph(d 2). Let G be a connected d-regular graph with n vertices and m edges. Furthermore, let P be the d × d matrix as follows: where J d is the matrix whose elements are all one. Let U = (U ef ) e,f ∈D(G) be the the transition probability matrix of a CRW on G with respect to P. Then we have U ef = 1/d if t(e) = o(f ), 0 otherwise, and so, Similarly to The proof of Theorem 7, we obtain the following result.
Theorem 15. Let G be a connected d-regular graph with n vertices and m edges. Furthermore, let U the transition probability matrix of the CRW on G with respect to P = 1/dJ d . Then det(λI 2m − U) = λ 2m−n det(λI n − 1 d A(G)).
Thus, Corollary 16. Let G be a connected d-regular graph with n vertices and m edges. Furthermore, let U the transition probability matrix of the CRW on G with respect to P = 1/dJ d . Then the transition probability matrix U has n eigenvalues of the form λ = 1 d λ A , λ A ∈ Spec(A(G)).
The remaining 2(m − n) eigenvalues of U are 0 with multiplicities 2m − n.

Future work
In this paper, we presented the spectrum of the transition probability matrix P of the CRW induced from the time evolution matrix U of the Grover walk on a regular graph and a semiregular bipartite graph by using a determinant expression for the generalized weighted zeta function of a graph. Here, the transition probability matrix P is the Hadamard product U • U of U and itself. Thus, we can propose the following problem.
Problem 17. Let a matrix U be the time evolution matrix of any discrete-time quantum walk on a graph. Then, what is the spectrum of the doubly stochastic matrix P = U • U?
From now on, we shall study the above problem.