The discrete-time quaternionic quantum walk and the second weighted zeta function on a graph

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1 The discrete-time quaternionic quantum walk and the second weighted zeta function on a graph Norio Konno arxiv: v1 [quant-ph] 0 Apr 016 Department of Applied Mathematics, Faculty of Engineering, Yokohama National University Hodogaya, Yokohama , Japan Hideo Mitsuhashi Faculty of Education, Utsunomiya University Utsunomiya, Tochigi , Japan Iwao Sato Oyama National College of Technology Oyama, Tochigi , Japan Mathematics Subject Classifications: 60F05, 05C50, 15A15, 11R5 Abstract We define the quaternionic quantum walk on a finite graph and investigate its properties This walk can be considered as a natural quaternionic extension of the Grover walk on a graph We explain the way to obtain all the right eigenvalues of a quaternionic matrix and a notable property derived from the unitarity condition for the quaternionic quantum walk Our main results determine all the right eigenvalues of the quaternionic quantum walk by using complex eigenvalues of the quaternionic weighted matrix which is easily derivable from the walk Since our derivation is owing to a quaternionic generalization of the determinant expression of the second weighted zeta function, we explain the second weighted zeta function and the relationship between the walk and the second weighted zeta function Keywords: Quantum walk; Ihara zeta function; quaternion; quaternionic quantum walk 1

2 1 Introduction The discrete-time quaternionic quantum walk on a graph is a quantum process on a graph which is governed by a unitary matrix The study of quantum walks started in earnest as quantum versions of random walks around the end of the last century, and quantum walks have been developed rapidly for more than two decades in connection with various fields such as quantum information science and quantum physics Detailed information on quantum walks can be found in several books at present, for example, Manouchehri and Wang [11], Portugal [1], Konno [7] An important example of the quantum walk on a graph is the Grover walk which originates from Grover s algorithm Grover s algorithm which was introduced in [4] is a quantum search algorithm that performs quadratically faster than the best classical search algorithm Later various researchers investigated the Grover walk and developed the theory of discrete-time quantum walks intensively Recently, Konno [8] established a quaternionic extension of quantum walks These are a quaternionic extension of quantum walks and can be viewed as quaternionic quantum dynamics One of the important backgrounds of quaternionic quantum walk is quaternionic quantum mechanics The origin of quaternionic quantum mechanics goes back to the axiomatization of quantum mechanics by Birkhoff and von Neumann in 1930s After that, the subject was studied further by Finkelstein, Jauch, and Speiser, and more recently by Adler and others One significant motivation of studying quaternionic quantum mechanics is that physical reality might be described by quaternionic quantum system at the fundamental level, and this dynamics is described asymptotically by the (ordinary) quantum field theory at the level of all presently known physical phenomena A detailed exposition of quaternionic quantum mechanics can be found in Adler [1] On the other hand, Zeta functions of graphs have been investigated for half a century Their origin is the Ihara zeta function which was defined by Ihara [6], and various extensions have appearedso far Among them we focus onthe second weighted zeta function of a graph The second weighted zeta function which was proposed by Sato [13] is a multi-weighted version of the Ihara zeta function, and has several applications in discrete-time quantum walks and quantum graphs For example, the second weighted zeta function played essential roles in the concise proof of the spectral mapping theorem for the Grover walk on a graph in [10] In this paper, we define the discrete-time quaternionic quantum walk on a graph as a quaternionic extension of the Grover walk on a finite graph, and discuss its right spectrum and the relationship between the walk and the second weighted zeta function of a graph Our results can be viewed as a generalization of [9] The Grover walk on 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 undirected edges uv joining two vertices u and v We assume that G is finite connected and has neither loops nor multiple edges throughout For uv E(G), we mean by an arc (u,v) the directed edge from u to v Let D(G) = {(u,v), (v,u) uv E(G)} and V(G) = n, E(G) = m, D(G) = m For e = (u,v) D(G), o(e) = u denotes

3 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 d u = degu = deg G u of a vertex u of G is the number of edges incident to u 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 A path P = (e 1,,e l ) is said to be a cycle if for some i(1 i l 1) 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 We give a definition of the discrete-time quantum walk on G Let H = C e be the finite dimensional Hilbert space spanned by arcs of G The transition matrix U of a discrete-time quantum walk consists of the following two consecutive operations: t(p) = o(p) and to have a backtracking if e i+1 = e 1 i 1 For each u V, we perform a unitary transformation C u on the states f that satisfy t(f) = u For all e D(G), we perform the shift S that is defined by S e = e 1 The transition matrix U Gro of the Grover walk on G is defined by setting the Grover s diffusion matrix as C u : 1+ d u d u d u C u = d u 1+ d u d u, d u d u 1+ d u Then U Gro = (U Gro ef ) e,f D(G) is given by U Gro ef = /d o(e) (= /d t(f) ) if t(f) = o(e) and f e 1, /d o(e) 1 if f = e 1, 0 otherwise U Gro is called the Grover matrix We denote by Spec(A) the multiset of eigenvalues of a complex square matrix A counted with multiplicity We shall give examples of Grover walks and their spectra Example 1 G = K 3 Then d o(e) = for all and U Gro is given as follows: U Gro = e 1 e 1 1 e e 1 e 3 e 1 3 e e e e e e Spec(U Gro ) = { } 1,1, 1± 3i, 1± 3i 3

4 Example G = K 1,3 Then d o(e1 ) = d o(e ) = d o(e3 ) = 1, d o(e 1 1 ) = d o(e 1 ) = d o(e 1 3 ) = 3 and U Gro is given as follows: { Spec(U Gro ) = } ±i,±i,1, 1 U Gro = e 1 e 1 1 e e 1 e 3 e 1 3 e e e e e e Let T = (T uv ) u,v V(G) be the n n matrix defined as follows: T uv = { 1/du if (u,v) D(G), 0 otherwise In [3], Emms et al determined the spectrum of U Gro by using those of T Theorem 3 (Emms, Hancock, Severini and Wilson [3]) Let G be a connected graph with n vertices and m edges The transition matrix U Gro has n eigenvalues of the form: λ = λ T ±i 1 λ T, where λ T is an eigenvalue of the matrix T The remaining (m n) eigenvalues of U Gro are ±1 with equal multiplicities As stated in Section 1, Konno and Sato [10] gave a concise proof of Theorem 3 by using the second weighted zeta function of a graph 3 A quaternionic extension of the Grover walk on a graph In this section, we define a quaternionic extension of the Grover walk on G and discuss some its properties Beforehand, we give a brief account of quaternionic matrices and their right eigenvalues 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: i = j = k = 1, ij = ji = k, jk = kj = i, ki = ik = j For x = x 0 +x 1 i+x j +x 3 k H, x = x 0 x 1 i x j x 3 k denotes the conjugate of x in H x = xx = x x = x 0 +x 1 +x +x 3 is called the norm of x Since x 1 = x / x for a nonzero element x H, H constitutes a skew field Since quaternions do not mutually 4

5 commute in general, we must treat left eigenvalues and right eigenvalues separately In this paper, we concentrate only on right eigenvalues Let Mat(m n,h) be the set of m n quaternionic matrices and Mat(n,H) the set of n n quaternionic square 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) Such an expression is called the symplectic decomposition of M M S and M P are called the simplex part and the perplex part of M respectively The quaternionic conjugate M of a quaternionic square matrix M is obtained from M by taking the transpose and then taking the quaternionic conjugate of each entry A quaternionic square matrix M is said to be quaternionic unitary if M M = MM = I We define ψ to be the map from Mat(m n,h) to Mat(m n,c) as follows: ( ) M S M ψ : Mat(m n,h) Mat(m n,c) M P, where A is the complex conjugate of a matrix A Then ψ is an R-linear map We can easily check that M P Lemma 31 Let M Mat(m n,h) and N Mat(n m,h) Then ψ(mn) = ψ(m)ψ(n) Moreover, if m = n, then ψ is an injective R-algebra homomorphism We consider H n as a right vector space λ H is said to be a right eigenvalue of M and v H n a right eigenvector corresponding to λ if Mv = vλ for M Mat(n,H) Now we state the facts about right eigenvalues of a quaternionic matrix as follows: Theorem 3 For any quaternionic matrix M Mat(n,H), there exist n complex right eigenvalues of M counted with multiplicity, which can be obtained by solving det(λi n ψ(m)) = 0 They appear in complex conjugate pairs λ 1,λ 1,,λ n,λ n The set of right eigenvalues σ r (M) is given by σ r (M) = λ H 1 λ H n where λ H = {h 1 λh h H = H {0}} Remark 33 If Mv = vλ then Mvq = vq(q 1 λq) for every q H and hence vq is a right eigenvector corresponding to the right eigenvalue q 1 λq ( ) λ Example 34 M =, det(λi 0 i 4 ψ(m)) = det 0 λ i λ 1 0 = λ+i λ = 1, ±i, σ r (M) = {1} i H ( ) λ j Example 35 M =, det(λi k i 4 ψ(m)) = det 0 λ i i λ 1 0 = 0 i 0 0 λ+i λ = 1+ 3 ± i, ± 1+ 3 i, ( 1+ 3 σ r (M) = ) H ( 1 3 i ) H i 5 M S

6 Now, we give a quaternionic extension of the Grover walk on G A discrete-time quaternionic quantum walk is a quantum process on G whose state vector, whose entries are quaternions, is governed by a quaternionic unitary matrix called the quaternionic transition matrix Let G be a finite connected graph with n vertices and m edges We define the state space to be the quaternionic right Hilbert space H H = e H We define the quaternionic transition matrix U = (U ef ) e,f D(G) of G as follows: U ef = q(e) if t(f) = o(e) and f e 1, q(e) 1 if f = e 1, 0 otherwise, (31) where q is a map from D(G) to H U can be viewed as the time evolution operator of a discrete-time quaternionic quantum system In [9], we obtained the necessary and sufficient condition for U to be quaternionic unitary as follows: Theorem 36 (Konno-Mitsuhashi-Sato [9]) U is unitary q 0 (e) +q 1 (e) +q (e) +q 3 (e) q 0(e) d o(e) = 0, where q(e) = q 0 (e)+q 1 (e)i+ q (e)j +q 3 (e)k, and q(e) = q(f) for any two arcs e,f D(G) with o(e) = o(f) From Theorem 36, it follows that q 0 (e) must satisfy 0 q 0 (e) d o(e) Furthermore we can readily see if q(e) is positive real for each e D(G), then U must be U Gro 4 The second weighted zeta function of a graph In this section, we give a brief summary of the second weighted zeta functions of a graph We introduce an equivalence relation between cycles in G Two cycles C 1 = (e 1,,e l ) and C = (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 has no backtracking Furthermore, a cycle C is said to be prime if it is not a power of a strictly smaller cycle The Ihara zeta function of a graph G is a function of t C with t sufficiently small, defined by Z(G,t) = Z G (t) = [C](1 t C ) 1, where [C] runs over all equivalence classes of prime, reduced cycles of G Z(G,t)hastwotypesofdeterminantexpressionsasexplainedbelow LetB = (B ef ) e,f D(G) and J 0 = (J ef ) e,f D(G) be m m matrices defined as follows: B ef = { 1 if t(e) = o(f), 0 otherwise, 6 J ef = { 1 if f = e 1, 0 otherwise (41)

7 The matrix B J 0 is called the edge matrix of G Then we can state two determinant expressions of Z(G, t) as follows: Theorem 41 (Hashimoto [5]; Bass []) The reciprocal of the Ihara zeta function of G is given by Z(G,t) 1 = det(i m t(b J 0 )) = (1 t ) r 1 det(i n ta+t (D I n )), (4) where r and A are the Betti number and the adjacency matrix of G respectively, and D = (D uv ) u,v V(G) is the diagonal matrix with D uu = degu for all u V(G) We call the middle formula of (4) the determinant expression of Hashimoto type and the right hand side the determinant expression of Bass type respectively We shall define the second weighted zeta function by using a modification of the edge matrix Consider an n n complex matrix W = (W uv ) u,v V(G) with (u,v)-entry equals 0 if (u,v) / D(G) We call W a weighted matrix of G Let w(u,v) = W uv for u,v V(G) and w(e) = w(u,v) if e = (u,v) D(G) Then W uv is given by { w(e) if e = (u,v) D(G), W uv = 0 otherwise (43) For a weighted matrix W of G, let B w = (B (w) ef ) e,f D(G) be a m m complex matrix defined as follows: { B (w) w(f) if t(e) = o(f), ef = (44) 0 otherwise Then the second weighted zeta function of G is defined by Z 1 (G,w,t) = det(i m t(b w J 0 )) 1 (45) One can consider (45) as a multi-parametrized deformation of the determinant expression of Hashimoto type for Z(G,t) If w(e) = 1 for all e D(G), then the second weighted zeta function of G coincides with Z(G, t) In [13], Sato obtained the determinant expression of Bass type for Z 1 (G,w,t) as follows: Theorem 4 (Sato [13]) Z 1 (G,w,t) 1 = (1 t ) m n det(i n tw+t (D w I n )), where n = V(G), m = E(G) and D w = (D (w) uv ) u,v V(G) is the diagonal matrix defined by D (w) uu = w(e) (46) for all u V(G) e: Remark 43 We mention that taking transpose, the following equation also holds: det(i m t( T B w J 0 )) = (1 t ) m n det(i n t T W+t (D w I n )), (47) where T M denotes the transpose of M We will show a quaternionic generalization of (47) andapplyittothespectralproblemforourquaternionicquantumwalkonginlatersections 7

8 5 A quaternionic generalization of the determinant expressions In this section, we shall give a quaternionic generalization of (47) Assume that w(e) is in H for every e D(G) in (43), (44) and (46) Let K = (K ev ),v V(G) and L = (L ev ),v V(G) be m n matrices defined as follows: K ev = { w(e) if o(e) = v, 0 otherwise, L ev = { 1 if t(e) = v, 0 otherwise, where column index and row index are ordered by fixed sequences v 1,,v n and e 1,,e m such that e r = e 1 r 1 for r = 1,,m respectively Then we canreadily see that T B w = K T L Let K = K S +jk P be the symplectic decomposition Then it follows that ( ) ( ) K S K ψ(k) = P L 0, ψ(l) =, (51) 0 L and by Lemma 31 we obtain K P K S ψ( T B w ) = ψ(k)ψ( T L) In view of (51), rows and columns of ψ(k) and ψ(l) are indexed by the disjoint union of two copies D(G) ± of D(G) and that of two copies V(G) ± of V(G) respectively We denote these disjoint unions by D(G) + D(G) = {e 1+,,e m+,e 1,,e m } (e r± D(G) ± ), V(G) + V(G) = {v 1+,,v n+,v 1,,v n } (v r± V(G) ± ), (5) where e r+ and e r correspond to e r D(G) and v r+ and v r to v r V(G) Orders of indices of ψ(k) and ψ(l) follow the alignment in (5) so that B w turns out to be ψ( T B w ) = e 1+ e 1+ e m+ e 1 e m e m+ T B S w T B P w e 1 e m T B P w T B S w = e 1+ e 1+ e m+ e 1 e m (K S ) T L ( K P ) T L e m+ e 1 (K P ) T L (K S ) T L e m We also notice that T W = T LK Theorem 51 Let t be a complex variable Then det(i 4m tψ( T B w J 0 )) = (1 t ) m n det(i n tψ( T W)+t (ψ(d w ) I n )) Proof We notice T B w = K T L at first It can be shown by direct calculations that det(i 4m tψ(k T L J 0 )) = det(i 4m tψ(k)ψ( T L)+tψ(J 0 )) = det(i 4m tψ(k)ψ( T L)(I 4m +tψ(j 0 )) 1 )det(i 4m +tψ(j 0 )) = (1 t ) m det(i 4m tψ(k)ψ( T L)(I 4m +tψ(j 0 )) 1 ) = (1 t ) m det(i n tψ( T L)(I 4m +tψ(j 0 )) 1 ψ(k)), (53) 8

9 and that ψ( T L)(I 4m +tψ(j 0 )) 1 ψ(k) = ( T ) { L T I L m ( 1 t t 1 t t 1 t 1 1 t ) } ( K S K P Putting X = (I 4m +tψ(j 0 )) 1, we see if f e,e 1 in D(G) + D(G), then X e f = 0 For every u,v V(G) + V(G), the (u,v )-entry of ψ( T L)Xψ(K) is given by (ψ( T L)Xψ(K)) u v = ψ( T L) u e X e f ψ(k) f v e,f D(G) + D(G) Hence if (v,u) = e D(G), then f = e and it follows that (ψ( T L)Xψ(K)) u+ v + = ψ( T L) u+ e + X e+ e + ψ(k) e+ v + = 1 1 t w(e)s = 1 1 t w((v,u))s, (ψ( T L)Xψ(K)) u+ v = ψ( T L) u+ e + X e+ e + ψ(k) e+ v = 1 1 t ( w(e)p ) = 1 1 t w((v,u))p, (ψ( T L)Xψ(K)) u v + = ψ( T L) u e X e e ψ(k) e v + = 1 1 t w(e)p = 1 1 t w((v,u))p, (ψ( T L)Xψ(K)) u v = ψ( T L) u e X e e ψ(k) e v = 1 1 t w(e)s = 1 1 t w((v,u))s Else if u = v, then f = e 1 and it follows that (ψ( T L)Xψ(K)) u+ u + = (ψ( T L)Xψ(K)) u+ u = (ψ( T L)Xψ(K)) u u + = (ψ( T L)Xψ(K)) u u = K P ψ( T L) u+ e 1 + X e 1 + e + ψ(k) e + u + = t 1 t ψ( T L) u+ e 1 + X e 1 + e + ψ(k) e + u = t 1 t ψ( T L) u e 1 X e 1 e ψ(k) e u + = t 1 t ψ( T L) u e 1 X e 1 e ψ(k) e u = t 1 t K S w(e) S, w(e) P, w(e) P, w(e) S Otherwise, we can readily check that all of (u +,v + ),(u +,v ),(u,v + ),(u,v )-entries of ψ( T L)Xψ(K) equal 0 Comparing entries of ψ( T L)Xψ(K) with those of ψ(w) and ψ(d w ), it follows that ψ( T L)Xψ(K) = 1 W) t 1 t ψ(t 1 t ψ(d w) Consequently, we obtain the following equation from (53) as desired det(i 4m tψ(k T L J 0 )) = (1 t ) m det(i n t 1 t ψ(t W)+ t 1 t ψ(d w)) = (1 t ) m n det((1 t )I n tψ( T W)+t ψ(d w )) = (1 t ) m n det(i n tψ( T W)+t (ψ(d w ) I n )) ) 9

10 6 The right spectrum of the quaternionic quantum walk on a graph In this section, we derive the set of right eigenvalues of the quaternionic transition matrix U by using eigenvalues of a complex matrix which can be easily derived from the map q : D(G) H given in (31) Putting t = 1/λ in Theorem 51, we obtain det(λi 4m ψ( T B w J 0 )) = (λ 1) m n det(λ I n λψ( T W)+ψ(D w ) I n ) (61) Setting w(e) = q(e) and comparing (31), (41) and (44), we readily see U = T B w J 0 Thus applying (61), we obtain det(λi 4m ψ(u)) = (λ 1) m n det(λ I n λψ( T W)+ψ(D w ) I n ) (6) Ifψ( T W)andψ(D w )aresimultaneouslytriangularizable, namely, thereexistaregularmatrix P Mat(n,C) such that µ ξ P 1 ψ( T µ 0 W)P =, ξ 0 P 1 ψ(d w )P =, (63) µ n ξ n then by (6) the characteristic equation of ψ(u) turns out to be det(λi 4m ψ(u)) = (λ 1) m n det(λ I n λp 1 ψ( T W)P+P 1 ψ(d w )P I n ) n = (λ 1) m n (λ λµ r +ξ r 1) = 0 r=1 (64) We notice if (µ r, µ s ) (r s) is a complex conjugate pair as stated in Theorem 3 then so is (ξ r, ξ s ) Observing that det(λi 4m ψ(u)) is a polynomial of λ, we obtain by solving (64) that Theorem 61 Spec(ψ(U)) = 4m Suppose that ψ( T W) and ψ(d w ) are simultaneously triangularizable If G is not a tree, then 4n of them are λ = µ r ± µ r 4(ξ r 1) (r = 1,,n), where µ r Spec(ψ( T W)),ξ r Spec(ψ(D w )) as presented in (63) The remaining 4(m n) are ±1 with equal multiplicities If G is a tree, then { µr ± µ Spec(ψ(U)) = r 4(ξ r 1) r = 1,,n } {1,1, 1, 1} σ r (U) = λ Spec(ψ(U)) λh 10

11 Finally, we discuss quaternionic quantum walks which satisfy the following condition on q(e) : q(e) does not depend on u (65) e: We have investigated this case in [9] In short, this condition says that the sum of the entries in each column of U does not depend on the column Quaternionic quantum walks need not satisfy (65) in general, however, the Grover walk satisfies (65) by definition Hence we can consider this condition is inherited from the Grover walk Since ψ is injective, T WD w = D w T W is equivalent to ψ( T W)ψ(D w ) = ψ(d w )ψ( T W) In this case, ψ( T W) and ψ(d w ) are simultaneously triangularizable Proposition6 (65) implies T WD w = D w T W Moreover, if w(e) 0 for everye D(G), then (65) T WD w = D w T W Proof If (v,u) D(G), then by Theorem 36 we have ( T WD w ) uv = w((v,u)) w(e) = d v w((v,u))w((v,u)), (D w T W) uv = o(e)=v w(e)w((v,u)) = d u w((u,v))w((v,u)) Therefore (65) implies d v w((v,u))w((v,u))= d u w((u,v))w((v,u)) for all (v,u) D(G) and thereby T WD w = D T w W If w(e) 0 for every e D(G), then d v w((v,u))w((v,u)) = d u w((u,v))w((v,u)) implies d v w((v,u)) = d u w((u,v)) Hence w(e) = w(e) for u, v V(G),,o(e)=v, with (v,u) D(G) Since G is connected, the equation just before holds for every pair of vertices and thereby (65) holds Hence one can view Theorem 61 as a generalization of [9] By (65), we may put α = e: q(e) independently of u Then we immediately see that q(e) = α/d o(e) for every e D(G) For α H R, it is known that there exist nonzero quaternions h ± H such that h 1 ± αh ± = α ± are complex numbers which are complex conjugate with each other (For the details, see [9]) Then we readily see U ± = h 1 ± Uh ± = ((U ± ) ef ) e,f D(G) Mat(m,C), where (U ± ) ef = α ± d o(e) if t(f) = o(e) and f e 1, α ± d o(e) 1 if f = e 1, 0 otherwise 11

12 It follows that det(λi 4m ψ(u)) = det(ψ(h + I m ) 1 (λi 4m ψ(u))ψ(h + I m )) = det(λi 4m ψ(h + I m ) 1 ψ(u)ψ(h + I m )) = det(λi 4m ψ(h 1 + Uh +)) = det(λi 4m ψ(u + )) = λi m U λi m U = det(λi m U + )det(λi m U ) = det(λi m U + )det(λi m U + ) Therefore, we can calculate all right eigenvalues of U by calculating eigenvalues of U + since eigenvalues of U = U + are complex conjugates of those of U + Accordingly, W ± = h 1 ± Wh ± and B w± = h 1 ± B w h ± = (B (w ±) ) ef are given by (W ± ) uv = { α± d u if (u,v) D(G), 0 otherwise, B (w ±) ef = { α ± d o(f) if t(e) = o(f), 0 otherwise Inthis case, we canapply (47) to obtainthenext theorem which isaspecial case oftheorem 61 Theorem 63 (Konno-Mitsuhashi-Sato [9]) Spec(ψ(U)) = 4m Suppose that (65) holds If G is not a tree, then 4n of them are λ = µ + ± µ + 4(α + 1), µ ± µ 4(α 1), where µ ± Spec( T W ± ) The remaining 4(m n) are ±1 with equal multiplicities If G is a tree, then Spec(ψ(U)) { µ+ ± µ + 4(α + 1) =, µ ± µ 4(α 1) } µ± Spec( T W ± ) {1,1, 1, 1} σ r (U) = λ Spec(U + ) λh We will show an example which does not satisfy (65) 1

13 Example 64 G = K 1,3 Let w(e 1 ) = 1+i, w(e ) = 1 j, w(e 3 ) =, w(e 1 1 ) = w(e 1 ) = w(e 1 3 ) = 0 v 1 v v 3 v 4 v T v W = v , v 4 1+i 1 j 0 ψ( T W) = 1+i i 1 0 Then U, D w and ψ(d w ) are given by 0 i i U = j ,D w = 0 1 j , i ψ(d w ) = i Let P be defined by P = i i

14 Then P 1 ψ( T W)P =, 1+i 1+i 1 i i 1 i 1 i i i i P 1 ψ(d w )P = i Hence it follows that Spec(ψ( T W)) = {µ 1,µ,,µ 8 } = {0,0,0,0,0,0,0,0}, Spec(ψ(D w )) = {ξ 1,ξ,,ξ 8 } = {1+i,1+i,1 i,1 i,,,0,0} Applying Theorem 61, we obtain Spec(ψ(U)) = {± i,± i,± i,± i,± 1,± 1,±1,±1} {1,1, 1, 1} = {± 1 i,± 1 i,± 1+i,± 1+i,±i,±i} Since i = j 1 ij i H, an eigenvalue and its complex conjugate belong to the same set of ( 1+i ) H ( all quaternionic conjugations of the eigenvalue Thus σ r (U) = i H 1+i ) H Acknowledgments The first author is partially supported by the Grant-in-Aid for Scientific Research (Challenging Exploratory Research) of Japan Society for the Promotion of Science (Grant No 15K13443) The second author is partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No 16K0549) The third author is partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No 15K04985) We are grateful to K Tamano, S Matsutani and Y Ide for some valuable comments on this work References [1] Adler, SL: Quaternion Quantum Mechanics and Quantum Fields, Oxford University Press (1995) 14

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