A Characteristic Polynomial for The Transition Probability Matrix of A Correlated Random Walk on A Graph
aa r X i v : . [ m a t h . P R ] D ec A CHARACTERISTIC POLYNOMIAL FOR THETRANSITION PROBABILITY MATRIX OF ACORRELATED RANDOM WALK ON A GRAPH
Takashi KOMATSUDepartment of Bioengineering School of Engineering,The University of TokyoBunkyo, Tokyo, 113-8656, JAPANe-mail: [email protected] KONNODepartment of Applied Mathematics, Faculty of Engineering,Yokohama National University,Hodogaya, Yokohama 240-8501, JAPANe-mail: [email protected] SATONational Institute of Technology, Oyama College,Oyama, Tochigi 323-0806, JAPANe-mail: [email protected] 21, 2020 bstract 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 thecharacteristic polynomial of the transition probability matrix of this CRW by using adeterminant expression for the generalized weighted zeta function of G . As applications,we give the spectrum of the transition probability matrices for the CRWs induced fromthe Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore,we consider another type of the CRW on a graph. : 05C50, 15A15. Key words and phrases : zeta function, correlated random walk, transition probabilitymatrix, spectraThe contact author for correspondence:Iwao SatoOyama National College of Technology, Oyama, Tochigi 323-0806, JAPANTel: +81-285-20-2176Fax: +81-285-20-2880E-mail: [email protected] 2
Introduction
Zeta functions of graphs started from the Ihara zeta functions of regular graphs by Ihara[6]. In [6], he showed that their reciprocals are explicit polynomials. A zeta functionof a regular graph G associated with a unitary representation of the fundamental groupof G was developed by Sunada [15,16]. Hashimoto [4] generalized Ihara’s result on theIhara zeta function of a regular graph to an irregular graph, and showed that its reciprocalis again a polynomial by a determinant containing the edge matrix. Bass [1] presentedanother determinant expression for the Ihara zeta function of an irregular graph by usingits adjacency matrix.Morita [12] defined a generalized weighted zeta function of a digraph which contains var-ious zeta functions of a graph or a digraph. Ide et al [5] presented a determinant expressionfor the above generalized weighted zeta function of a graph.The time evolution matrix of a discrete-time quantum walk in a graph is closely relatedto the Ihara zeta function of a graph. A discrete-time quantum walk is a quantum analogof the classical random walk on a graph whose state vector is governed by a matrix calledthe time evolution matrix(see [8]). Ren et al. [13] gave a relationship between the discrete-time quantum walk and the Ihara zeta function of a graph. Konno and Sato [10] obtaineda formula of the characteristic polynomial of the Grover matrix by using the determinantexpression for the second weighted zeta function of a graph.In this paper, we define introduce a new correlated random walk induced from the timeevolution matrix (the Grover matrix) of the Grover walk on a graph, and present a formulafor the characteristic polynomial of its transition probability matrix.In Section 2, we review for the Ihara zeta function and the generalized weighted zetafunctions of a graph. In Section 3, we review for the Grover walk on a graph. In Section4, we define a correlated random walk (CRW) induced from the time evolution matrix (theGrover matrix) of the Grover walk on a graph G , and present a formula for the characteristicpolynomial of the transition probability matrix of this CRW. In Section 5, we give thespectrum 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 semiregularbipartite graph. In Section 7, we present formulas for the characteristic polynomials ofthe transition probability matrices of another type of the CRW on a graph, and give thespectrum of its transition probability matrix. Graphs and digraphs treated here are finite. Let G be a connected graph and D G thesymmetric digraph corresponding to G . Set D ( G ) = { ( u, v ) , ( v, u ) | uv ∈ E ( G ) } . For e = ( u, v ) ∈ D ( G ), set u = o ( e ) and v = t ( e ). Furthermore, let e − = ( v, u ) be the inverse of e = ( u, v ). For v ∈ V ( G ), the degree deg G v = deg v = d v is the number of verticesadjacent to v in G . A graph G is called k -regular if deg v = k for each v ∈ V ( G ).A path P of length n in G is a sequence P = ( e , · · · , e n ) of n arcs such that e i ∈ D ( G ), t ( e i ) = o ( e i +1 )(1 ≤ i ≤ n − e i = ( v i − , v i ) for i = 1 , · · · , n , then we write P =( v , v , · · · , v n − , v n ). Set | P | = n , o ( P ) = o ( e ) and t ( P ) = t ( e n ). Also, P is called an( o ( P ) , t ( P )) -path . We say that a path P = ( e , · · · , e n ) has a backtracking if e − i +1 = e i forsome i (1 ≤ i ≤ n − v, w )-path is called a v -cycle (or v -closed path ) if v = w . The inverse cycle of a cycle C = ( e , · · · , e n ) is the cycle C − = ( e − n , · · · , e − ).We introduce an equivalence relation between cycles. Two cycles C = ( e , · · · , e m )and C = ( f , · · · , 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 cycle of3 is in general not equivalent to C . Let [ C ] be the equivalence class which contains a cycle C . Let B r be the cycle obtained by going r times around a cycle B . Such a cycle is called a multiple of B . A cycle C is reduced if both C and C have no backtracking. Furthermore, acycle C is prime if it is not a multiple of a strictly smaller cycle. Note that each equivalenceclass of prime, reduced cycles of a graph G corresponds to a unique conjugacy class of thefundamental group π ( G, v ) of G at a vertex v of G .The Ihara(-Selberg) zeta function of G is defined by Z ( G, u ) = Y [ C ] (1 − u | C | ) − , where [ C ] runs over all equivalence classes of prime, reduced cycles of G .Let G be a connected graph with n vertices and m edges. Then two 2 m × m matrices B = B ( G ) = ( B e,f ) e,f ∈ D ( G ) and J = J ( G ) = ( J e,f ) e,f ∈ D ( G ) are defined as follows: B e,f = (cid:26) t ( e ) = o ( f ),0 otherwise , J e,f = (cid:26) f = e − ,0 otherwise.The matrix B − J is called the edge matrix of G . Theorem 1 (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 Z ( G, u ) − = det( I m − u ( B − J )) = (1 − u ) m − n det( I n − u A ( G ) + u ( D G − I n )) , where D G = ( d ij ) is the diagonal matrix with d ii = deg G v i ( V ( G ) = { v , · · · , v n } ) . The first identity in Theorem 1 was obtained by Hashimoto [4]. Also, Bass [1] provedthe second identity by using a linear algebraic method.Stark and Terras [14] gave an elementary proof of this formula, and discussed threedifferent zeta functions of any graph. Various proofs of Bass’ Theorem were given by Kotaniand Sunada [11], and Foata and Zeilberger [3].
Let G be a connected graph with n vertices and m edges, and D ( G ) = { e , . . . , e m , e m +1 , . . . ,e m } ( e m + i = e − i (1 ≤ i ≤ m )). Furthermore, we consider two functions τ : D ( G ) −→ C and µ : D ( G ) −→ C . Let θ : D ( G ) × D ( G ) −→ C be a function such that θ ( e, f ) = τ ( f ) δ t ( e ) o ( f ) − µ ( f ) δ e − f . We introduce a 2 m × m matrix M ( θ ) = ( M ef ) e,f ∈ D ( G ) as follows: M ef = θ ( e, f ) . Then the generalized weighted zeta function Z G ( u, θ ) of G is defined as follows(see [12]): Z G ( u, θ ) = det( I m − u M ( θ )) − . We consider two n × n matrices A G ( θ ) = ( a uv ) u,v ∈ V ( G ) and D G ( θ ) = ( d uv ) u,v ∈ V ( G ) asfollows: a uv = (cid:26) τ ( e ) / (1 − u µ ( e ) µ ( e − )) if e ( u, v ) ∈ D ( G ),0 otherwise, d uv = (cid:26) P o ( e )= u τ ( e ) µ ( e − ) / (1 − u µ ( e ) µ ( e − )) if u = v ,0 otherwise.A determinant expression for the generalized weighted zeta function of a graph is givenas follows(see [5]): 4 heorem 2 (Ide, Ishikawa, Morita, Sato and Segawa) Let G be a connected graphwith n vertices and m edges, and let τ : D ( G ) −→ C and µ : D ( G ) −→ C be two functions.Then Z G ( u, θ ) − = m Y j =1 (1 − u µ ( e j ) µ ( e − j )) det( I n − u A G ( θ ) + u D G ( θ )) , where D ( G ) = { e , . . . , e m , e m +1 , . . . , e m } ( e m + j = e − j (1 ≤ j ≤ m )) . Let G be a connected graph with n vertices and m edges, V ( G ) = { v , . . . , v n } and D ( G ) = { e , . . . , e m , e − , . . . , e − m } . Set d j = d v j = deg v j for i = 1 , . . . , n . The Grover matrix U = U ( G ) = ( U ef ) e,f ∈ R ( G ) of G is defined by U ef = /d t ( f ) (= 2 /d o ( e ) ) if t ( f ) = o ( e ) and f = e − ,2 /d t ( f ) − f = e − ,0 otherwise.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: T uv = (cid:26) / (deg G u ) if ( u, v ) ∈ D ( G ),0 otherwise.Note that the matrix T ( G ) is the transition matrix of the simple random walk on G (see[10]). Theorem 3 (Konno and Sato)
Let G be a connected graph with n vertices v , . . . , v n and m edges. Then the characteristic polynomial for the Grover matrix U of G is given by det( λ I m − U ) = ( λ − m − n det(( λ + 1) I n − λ T ( G ))= ( λ − m − n det(( λ +1) D − λ A ( G )) d v ··· d vn . From this Theorem, the spectra of the Grover matrix on a graph is obtained by meansof those of T ( G ) (see [13]). Let Spec ( F ) be the spectra of a square matrix F . Corollary 1 (Emms, Hancock, Severini and Wilson)
Let G be a connected graph with n vertices and m edges. The Grover matrix U has n eigenvalues of the form λ = λ T ± i q − λ T , where λ T is an eigenvalue of the matrix T ( G ) . The remaining m − n ) eigenvalues of U are ± with equal multiplicities. Let G be a connected graph with n vertices and m edges, and U be the Grover matrix of G . Then we define a 2 m × m matrix P = ( P ef ) e,f ∈ D ( G ) as follows: P ef = | U ef | . P ef = /d t ( f ) (= 4 /d o ( e ) ) if t ( f ) = o ( e ) and f = e − ,(2 /d t ( f ) − if f = e − ,0 otherwise.The random walk with the matrix P as a transition probability matrix is called the correlatedrandom walk (CRM) (with respect to the Grover matrix) on G (see [7,9]).Let R = ( R ef ) e,f ∈ D ( G ) be a 2 m × m matrix such that R ef = /d o ( f ) (= 4 /d o ( e ) ) if o ( e ) = o ( f ) and f = e ,(2 /d o ( f ) − if f = e ,0 otherwise.Then we have P = J R . By Theorem 2, we obtain the following formula for P . Theorem 4
Let G be a connected graph with n vertices and m edges, and let P be thetransition probability matrix of the CRW with respect to the Grover matrix. Then det( I m − u P ) = m Y j =1 (1 − u ( 4 d o ( e j ) − d t ( e j ) − I n − u A CRW + u D CRW ) , where ( A CRW ) xy = ( /d x − u (4 /d x − /d y − if ( x, y ) ∈ D ( G ) , otherwise, ( D CRW ) xy = ( P o ( e )= x /d x (4 /d t ( e ) − − u (4 /d x − /d t ( e ) − if x = y , otherwise. Proof . For the matrix P , we have P ef = 4 d o ( e ) δ t ( f ) o ( e ) − ( 4 d o ( e ) − δ f − e . The we let two functions τ : D ( G ) −→ C and µ : D ( G ) −→ C . as follows: τ ( e ) = 4 d o ( e ) and µ ( e ) = 4 d o ( e ) − . Furthermore, let θ ( e, f ) = 4 d o ( f ) δ t ( e ) o ( f ) − ( 4 d o ( f ) − δ e − f . Then we have P = t M ( θ ) . Thus, we obtaindet( I m − u P ) = det( I m − u t M ( θ )) = det( I m − u M ( θ )) = Z G ( u, θ ) − . By Theorem , we havedet( I m − u P ) = m Y j =1 (1 − u ( 4 d o ( e j ) − d t ( e j ) − I n − u A CRW + u D CRW ) , A CRW ) xy = ( /d x − u (4 /d x − /d y − if ( x, y ) ∈ D ( G ),0 otherwise,( D CRW ) xy = ( P o ( e )= x /d x (4 /d t ( e ) − − u (4 /d x − /d t ( e ) − if x = y ,0 otherwise. ✷ By Theorem 4, we obtain the spectrum of the transition probability matrices for theCRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs.
We present spectra for the transition matrix of the correlated random walk on a regulargraph with respect to the Grover matrix.
Theorem 5
Let G be a connected d -regular graph with n vertices and m edges, where d ≥ .Furthermore, let P be the transition probability matrix of the CRW with respect to the Grovermatrix. Then det( I m − u P ) = ( d − u (4 − d ) ) m − n d m det( d ( d + (4 − d ) u ) I n − u A ( G )) . Proof . Let G be a connected d -regular graph with n vertices and m edges, where d ≥ d o ( e ) = d t ( e ) = d f or each e ∈ D ( G ) . Thus, we have 1 − u ( 4 d o ( e ) − d t ( e ) −
1) = d − u (4 − d ) d , ( A CRW ) xy = 4 /d x − u (4 /d x − /d y −
1) = 4 d − u (4 − d ) if ( x, y ) ∈ D ( G )and( D CRW ) xy = X o ( e )= x /d x (4 /d t ( e ) − − u (4 /d x − /d t ( e ) −
1) = d · − d ) d ( d − u (4 − d ) ) = 4(4 − d ) d − u (4 − d ) . Therefore, it follows that A CRW = 4 d − u (4 − d ) A ( G ) and D CRW = 4(4 − d ) d − u (4 − d ) I n . By Theorem 4, we havedet( I m − u P )= ( d − u (4 − d ) ) m d m det( I n − u d − u (4 − d ) A ( G ) + u − d ) d − u (4 − d ) I n )= ( d − u (4 − d ) ) m − n d m det(( d − u (4 − d ) ) I n − u A ( G ) + 4(4 − d ) u I n )= ( d − u (4 − d ) ) m − n d m det( d ( d + (4 − d ) u ) I n − u A ( G )) . ✷ By substituting u = 1 /λ , we obtain the following result.7 orollary 2 Let G be a connected d -regular graph with n vertices and m edges, where d ≥ . Furthermore, let P be the transition probability matrix of the CRW with respect tothe Grover matrix. Then det( λ I m − P ) = ( d λ − (4 − d ) ) m − n d m det( d ( dλ + (4 − d )) I n − λ A ( G ))= ( λ − ( d − ) m − n λ n det(( λ + ( d − λ ) I n − d A ( G )) . 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 withrespect to the Grover matrix on a regular graph. Corollary 3
Let G be a connected d ( ≥ -regular graph with n vertices and m edges. Thenthe transition probability matrix P has n eigenvalues of the form λ = 2 λ A ± p λ A − d (4 − d ) d , where λ A is an eigenvalue of the matrix A ( G ) . The remaining m − n ) eigenvalues of P are ± (4 − d ) /d with equal multiplicities m − n . Proof . By Corollary 2, we havedet( λ I m − P ) = ( d λ − (4 − d ) ) m − n /d m Q λ A ∈ Spec ( A ( G )) ( d ( dλ + 4 − d ) − λ A λ )= ( λ − ( − dd ) ) m − n /d n Q λ A ∈ Spec ( A ( G )) ( d λ − λ A λ + d (4 − d )) . Thus, solving d λ − λ A λ + d (4 − d ) = 0 , we obtain λ = 2 λ A ± p λ A − d (4 − d ) d . ✷ In the case of d = 4, we consider P = ( P ef ) e,f ∈ D ( G ) be the transition probability matrixof the CRW with respect to the Grover matrixon a d -regular graph G . If t ( f ) = o ( e ) and f = e − , then P ef = 4 /d = 4 / = 1 /
4. If f = e − , then P ef = 4 /d − (4 /d −
1) =4 / − (4 / −
1) = 1 /
4. 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. Furthermore,an n × n Hadamard matrix is a unitary matrix whose elements have the absolute value1 / √ n (see [2]). The Grover matrix of a d -regular graph is an Hadamard matrix if and onlyif d = 4. We present spectra for the transition probability matrix of the correlated random walk ona semiregular bipartite graph. Hashimoto [4] presented a determinant expression for theIhara zeta function of a semiregular bipartite graph. We use an analogue of the method inthe proof of Hashimoto’s result.A bipartite graph G = ( V , V ) is called ( q , q ) -semiregular if deg G v = q i for each v ∈ V i ( i = 1 , q + 1 , q + 1)-semiregular bipartite graph G = ( V , V ), let G [ i ] bethe graph with vertex set V i and edge set { P : reduced path | | P | = 2; o ( P ) , t ( P ) ∈ V i } for i = 1 ,
2. Then G [1] is ( q + 1) q -regular, and G [2] is ( q + 1) q -regular.8 heorem 6 Let G = ( V, W ) be a connected ( r, s ) -semiregular bipartite graph with ν verticesand ǫ edges. Set | V | = m and | W | = n ( m ≤ n ) . Furthermore, let P be the transitionprobability matrix of the CRW with respect to the Grover matrix of G , and Spec ( A ( G ) = {± λ , · · · , ± λ m , , . . . , } . Then det( I ǫ − u P ) = (1 − u (4 /r − /s − ǫ − ν (1 − u (4 /r − n − m × m Y j =1 ((1 − u (4 /s − − u (4 /r − − λ j r s u ) . Proof . Let e ∈ D ( G ). If o ( e ) ∈ V , then d o ( e ) = r, d t ( e ) = s. Thus, we have 1 − u ( 4 d o ( e ) − d t ( e ) −
1) = rs − u (4 − r )(4 − s ) rs , ( A CRW ) xy = /d x − u (4 /d x − /d y − = ( srs − u (4 − r )(4 − s ) 1 r if ( x, y ) ∈ D ( G ) and x ∈ V , rrs − u (4 − r )(4 − s ) 1 s if ( x, y ) ∈ D ( G ) and x ∈ W ,and ( D CRW ) xx = P o ( e )= x /d x (4 /d t ( e ) − − u (4 /d x − /d t ( e ) − = ( r · − s ) r ( rs − u (4 − r )(4 − s )) = − s ) rs − u (4 − r )(4 − s )) if x ∈ V , s · − r ) s ( rs − u (4 − r )(4 − s )) = − r ) rs − u (4 − r )(4 − s )) if x ∈ W .Next, let V = { v , · · · , v m } and W = { w , · · · , w n } . Arrange vertices of G as follows: v , · · · , v m ; w , · · · , w n . We consider the matrix A = A ( G ) under this order. Then, let A = (cid:20) t E 0 (cid:21) . Since A is symmetric, there exists an orthogonal matrix F ∈ O ( n ) such that EF = (cid:2) R 0 (cid:3) = µ · · ·
0. . . ... ... ⋆ µ m · · · . Now, let H = (cid:20) I m
00 F (cid:21) . Then we have t HAH = t R 0 00 0 0 . Furthermore, let α = 4 / ( rs − u (4 − r )(4 − s )) . A CRW = (cid:20) αs/r E αr/s t E 0 (cid:21) , and D CRW = (cid:20) α (4 − s ) I m α (4 − r ) I n (cid:21) . Thus, we have t HA CRW H = αs/r R 0 αr/s t R 0 00 0 0 and t HD CRW H = (cid:20) α (4 − s ) I m α (4 − r ) I n (cid:21) . By Theorem 4,det( I ǫ − u P ) = ( rs − u (4 − r )(4 − s )) ǫ r ǫ s ǫ det( I ν − u A CRW + u D CRW )= ( rs − u (4 − r )(4 − s )) ǫ r ǫ s ǫ det I m + α (4 − s ) u I m − αsu/r R 0 − αru/s t R I m + α (4 − r ) u I m
00 0 I n − m + α (4 − r ) u I n − m = ( rs − u (4 − r )(4 − s )) ǫ r ǫ s ǫ (1 + α (4 − r ) u ) n − m × det (cid:18)(cid:20) (1 + α (4 − s ) u ) I m − αsu/r R − αru/s t R (1 + α (4 − r ) u ) I m (cid:21)(cid:19) · det (cid:18)(cid:20) I m α (4 − s ) u αsur R0 I m (cid:21)(cid:19) = ( rs − u (4 − r )(4 − s )) ǫ + m − n r ǫ s ǫ ( rs + u (4 − r ) s ) n − m × det " (1 + α (4 − s ) u ) I m − αru/s t R (1 + α (4 − r ) u ) I m − α u α (4 − s ) u t RR = ( rs − u (4 − r )(4 − s )) ǫ + m − n r ǫ s ǫ ( rs + u (4 − r ) s ) n − m × (1 + α (4 − s ) u ) m det((1 + α (4 − r ) u ) I m − α u α (4 − s ) u t RR )= ( rs − u (4 − r )(4 − s )) ǫ + m − n r ǫ s ǫ ( rs + u (4 − r ) s ) n − m × det((1 + α (4 − s ) u )(1 + α (4 − r ) u ) I m − α u t RR ) . Since A is symmetric, t RR is symmetric and positive semi-definite, i.e., the eigenvaluesof t RR are of form: λ , · · · , λ m ( λ , · · · , λ m ≥ . Furthermore, we have det( λ I ν − A ( G )) = λ n − m det( λ − t RR ) , and so, Spec ( A ( G ) = {± λ , · · · , ± λ m , , . . . , } . I ǫ − u P )= ( rs − u (4 − r )(4 − s )) ǫ + m − n r ǫ s ǫ ( rs + u (4 − r ) s ) n − m × Q mj =1 ((1 + α (4 − s ) u )(1 + α (4 − r ) u ) I m − α λ j u )= ( rs − u (4 − r )(4 − s )) ǫ + m − n r ǫ s ǫ ( rs + u (4 − r ) s ) n − m × Q mj =1 ( rs + u (4 − s ) rrs − u (4 − r )(4 − s ) rs + u (4 − r ) srs − u (4 − r )(4 − s ) − λ j u ( rs − u (4 − r )(4 − s )) )= ( rs − u (4 − r )(4 − s )) ǫ − m − n r ǫ s ǫ ( rs + u (4 − r ) s ) n − m × Q mj =1 ( rs ( s + u (4 − s ))( r + u (4 − r )) − λ j u )= (1 − u (4 /r − /s − ǫ − ν (1 + u (4 /r − n − m × Q mj =1 ((1 + u (4 /s − u (4 /r − − λ j r s u ) . ✷ Now, let u = 1 /λ . Then we obtain the following result. Corollary 4
Let G = ( V, W ) be a connected ( r, s ) -semiregular bipartite graph with ν verticesand ǫ edges. Set | V | = m and | W | = n ( m ≤ n ) . Furthermore, let P be the transitionprobability matrix of the CRW with respect to the Grover matrix and Spec ( A ( G ) = {± λ , · · · , ± λ m , , . . . , } . Then det( λ I ǫ − P ) = ( λ − (4 /r − /s − ǫ − ν ( λ + (4 /r − n − m × m Y j =1 (( λ + (4 /s − λ + (4 /r − − λ j r s λ ) . By Corollary 4, we obtain the spectra for the transition probability matrix P of theCRW with respect to the Grover matrix of a semiregular bipartite graph. Corollary 5
Let G = ( V, W ) be a connected ( r, s ) -semiregular bipartite graph with ν verticesand ǫ edges. Set | V | = m and | W | = n ( m ≤ n ) . Furthermore, let P be the transitionprobability matrix of the CRW with respect to the Grover matrix and Spec ( A ( G ) = {± λ , · · · , ± λ m , , . . . , } . Then the transition matrix P has ǫ eigenvalues of the form1. m eigenvalues: λ = ± vuut r s − rs − r s + 16 λ j ± q (2 r s − rs − r s + 16 λ j ) − r s (4 − r )(4 − s )2 r s ;11 . n − m eigenvalues: λ = ± i r r − ǫ − ν ) eigenvalues: λ = ± r ( 4 r − s − . Proof . Solving ( λ + (4 /s − λ + (4 /r − − λ j r s λ = 0 , i.e., λ + ( 4 r + 4 s − − λ j r s ) λ + ( 4 r − s −
1) = 0 , we obtain λ = ± vuut
12 ((2 − r − s + 16 λ j r s ) ± s (2 − r − s + 16 λ j r s ) −
4( 4 r − s − , i.e., λ = ± vuut r s − rs − r s + 16 λ j ± q (2 r s − rs − r s + 16 λ j ) − r s (4 − r )(4 − s )2 r s ; ✷ The CRW is defined by the following transition probability matrix P on the one-dimensionallattice: P = (cid:20) a bc d (cid:21) , where a + c = b + d = 1 , a, b, c, d ∈ [0 , . As for the CRW, see [7,9], 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 verticesand n edges. Furthermore, let V ( C n ) = { v , . . . , v n } and e j = ( v j , v j +1 )(1 ≤ j ≤ n ),where the subscripts are considered by modulo m . Then we introduce a 2 n × n matrix U = ( U ef ) e,f ∈ D ( C n ) as follows: U ef = d if t ( f ) = o ( e ), f = e − and f = e j , b if f = e − and f = e j , a if t ( f ) = o ( e ), f = e − and f = e − j , c if f = e − and f = e − j ,0 otherwise. 12ote that U is be able to write as follows: U = (cid:20) d Q − c I n b I n a Q (cid:21) , where Q = P σ is the permutation matrix of σ = (12 . . . n ). The CRW with U with atransition probability matrix is called the second type of CRW on C n with respect to theabove matrix P .Now, we define a function w : D ( C n ) −→ R as follows: w ( e ) = (cid:26) d if e = e j (1 ≤ j ≤ n ), a if e = e − j (1 ≤ j ≤ n ).Furthermore, let an n × n matrix W ( C n ) = ( w uv ) u,v ∈ V ( C n ) as folloows: w uv = (cid:26) w ( u, v ) if ( u, v ) ∈ D ( C n ),0 otherwise.The characteristic polynomial of U is given as follows. Theorem 7
Let C n be the cycle graph with n vertices, and U the transition probabilitymatrix of the second type of CRW on C n . Then det( λ I n − U ) = det(( λ + ( ad − bc )) I n − λ W ( C n )) . Proof . At first, we consider two 2 n × n matrices 2 n × n matrices B = ( B ef ) e,f ∈ D ( C n ) and J = ( J ef ) e,f ∈ D ( C n ) as follows: B ef = (cid:26) w ( f ) if t ( e ) = o ( f ),0 otherwise, J ef = b − a if f = e − and e = e j , c − d if f = e − and e = e − j ,0 otherwise.Then we have U = t B + t J . Now, we define two 2 n × n matrices K = ( K ev ) e ∈ D ( C n ); v ∈ V ( C n ) and L = ( L ev ) e ∈ D ( C n ); v ∈ V ( C n ) as follows: K ev = (cid:26) t ( e ) = v ,0 otherwise, L ev = (cid:26) w ( e ) if o ( e ) = v ,0 otherwise.Then we have K t L = B , t LK = W ( C n ) . If A and B are an m × n matrix and an n × m matrix, respectively, then we havedet( I m − AB ) = det( I n − BA ) . Thus, det( I n − u U ) = det( I n − u ( t B + t J ))= det( I n − u ( B + J ))= det( I n − u J − u B )= det( I n − u J − u K t L )= det( I n − u K t L ( I n − u J ) − ) det( I n − u J )= det( I n − u t L ( I n − u J ) − K ) det( I n − u J ) . I n − u J )= (cid:20) I n − ( b − a ) u I n − ( c − d ) u I n I n (cid:21) · (cid:20) I n ( b − a ) u I n n (cid:21) = (cid:20) I n − ( c − d ) u I n I n − u ( b − a )( c − d ) I n (cid:21) = (1 − ( a − c )( d − b ) u ) n . Furthermore, we have( I n − u J ) − = 11 − ( a − b )( d − c ) u ( I n + u J ) . Therefore, it follows thatdet( I n − u U )= (1 − ( a − b )( d − c ) u ) n det( I n − u/ (1 − ( a − b )( d − c ) u ) t L ( I n + u J ) K )= det((1 − ( a − b )( d − c ) u ) I n − u t LK − u t LJK )= det((1 − ( a − b )( d − c ) u ) I n − u W ( C n ) − u t LJK ) . The matrix t LJK is a diagonal, and its ( v i , v i ) entry is equal to( c − d ) w ( e − i − ) + ( b − a ) w ( e i ) = ( c − d ) a + ( b − a ) d = c + bd − ad. That is, t LJK = ( ab + cd − ad ) I n . Thus, det( I n − u U )= det((1 − ( a − b )( d − c ) u ) I n − u W ( C n ) − u ( ac + bd − ad ) I n )= det(((1 + ( ad − bc ) u ) I n − u W ( C n )) . Substituting u = 1 /λ , the result follows. ✷ By Theorem 7, we obtain the spectra for the transition probability matrix U of thesecond type of the CRW on C n . The matrix W ( C n ) is given as follows: W ( C n ) = d . . . aa d . . . . . . dd . . . a , Corollary 6
Let C n be the cycle graph with n vertices, and U the transition probabilitymatrix of the second type of CRW on C n . Then the transition probability matrix U has n eigenvalues of the form λ = µ ± p µ − ad − bc )2 , µ ∈ Spec ( W ( C n )) . roof . At first, we havedet( I n − u U ) = Y µ ∈ Spec ( W ( C n )) ( λ − µλ + ( ad − bc )) . Solving λ − µλ + ( ad − bc ) = 0 , we obtain λ = µ ± p µ − ad − bc )2 . ✷ Now, we consider the case of a = b = c = d = 1 /
2. Then the matrix W ( C n ) is equal to W ( C n ) = 12 A ( C n ) . By Corollary 6, we obtain the spectra for the transition probability matrix U of thesecond type of CRW on C n . Corollary 7
Let C n be the cycle graph with n vertices, and U the transition probabilitymatrix of the second type of the CRW on C n . Assume that a = b = c = d = 1 / . Then thetransition probability matrix U has n eigenvalues of the form λ = cos θ j , θ j = 2 πjn ( j = 0 , , . . . , n −
1) ( ∗ ) . The remaining n eigenvalues of U are with multiplicities n . Proof . It is known that the spectrum of A ( C n ) are2 cos θ j , θ j = 2 πjn ( j = 0 , , . . . , n − . ✷ Note that the spectrum of (*) are those of the transition probability matrix of the simplerandom walk on a cycle graph C n .We can generalize the result for a = b = c = d = 1 / C n to a d -regular graph( d ≥ G be a connected d -regular graph with n vertices and m edges. Furthermore, let P bethe d × d matrix as follows: P = 1 d J d , where J d is the matrix whose elements are all one. Let U = ( U ef ) e,f ∈ D ( G ) be the thetransition probability matrix of a CRW on G with respect to P . Then we have U ef = (cid:26) /d if t ( e ) = o ( f ),0 otherwise,and so, U = 1 d B . Similarly to The proof of Theorem 7, we obtain the following result.
Theorem 8
Let G be a connected d -regular graph with n vertices and m edges. Further-more, let U the transition probability matrix of the CRW on G with respect to P = 1 /d J d .Then det( λ I m − U ) = λ m − n det( λ I n − d A ( G )) . Corollary 8