Vol. 31,No JOURNAL OF CHINA UNIVERSITY OF SCIENCE AND TECHNOLOGY Feb

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1 Ξ 31 Vol 31,No JOURNAL OF CHINA UNIVERSITY OF SCIENCE AND TECHNOLOGY Feb : (2001) (, ) : Q ( m 1, m 2,, m n ) k = m 1 + m m n - n : Q ( m 1, m 2,, m n ) k n + 1 : ; ; ; :O157 9 ; TP302 7 :A AMS( 1991) : 05C12, 05C40, 68M10 1,, ( ), Q ( m 1, [1,2 m 2,, m n ) ] G, G, ( ), ( ) [3 ],,,,,, [4 ],, n de Bruijn Kautz [4 9 ] Q ( m 1, m 2,, m n ) k = m 1 + m m n - n, k Ξ : : ( ) :,, ,,

2 1 17 n + 1, k n + 1 :,, 2 G = G( V, E) ( ) G k, G k x y P ( x, y) P P, P ( x, y) x y, d ( G; x, y) d ( G) = max{ d ( G; x, y) Π x, y V ( G) } G G n ( x, y) P 1, P 2,, P n, Π i, j (0 i j < n), V ( P i P j ) = { x, y} G, ( G), G G k, ( G) k, [10 ] G k,, G x y k d k ( G; x, y) d G k ( x, y), d G k ( k, ) d k ( G), d k ( G) = max{ d k ( G; x, y) Π x, y V ( G) }, G k, d k ( G), d k ( G), d k ( G) k G ( G), 1 [6,7 ] G k k, k 2, d k ( G) d ( G) + 1 ( x 1 x 2 x n ) n, x l ( l = 1,2,, n) l m 1, m 2,, m n n 2 n ( Generalized Hypercube) [2 ], Q ( m 1, m 2,, m n ),, V = { x = ( x 1 x 2 x n ),0 x l m l - 1, 1 l n}, m 1 = = m n = m m2ary n2cube [2 ], Q n ( m) ; n = 1, m K m ; m = 2, n, Q n, [9,11 ], 2 Q ( m 1, m 2,, m n ) m 1 + m m n - n, m 1 m 2 m n, n, Q n ( m) n ( m - 1), m n, n 3 Q ( m 1, m 2,, m n ) l (1 l n) m l n - 1 Q i ( m 1,, m i - 1, i, m l +2, m n ),0 i m l - 1, Q i ( m 1,, m i - 1, i, m l +2, m n ) u, Q ( m 1, m 2,, m n ) m l - 1 Q i ( m 1,, m i - 1, i, m l +2, m n ), m l - 1 Q j ( m 1,, m i - 1, i, m l +2, m n ),0 j i m l - 1, Q n ( m) m m2ary n2cube Q i n- 1 ( m),0 i m - 1, u V ( Q i n - 1 ( m) ),0 i m - 1, u Q n ( m) m - 1 Q i n - 1 ( m), m - 1 Q j n - 1 ( m),0 j i m - 1

3 d n ( m - 1) ( Q n ( m) ) = n + 1, m = 2, n 2 d n ( m - 1) ( Q n ( m) ) n + 1 (1) n 1 n = 1, Q 1 ( m) m K m d m - 1 ( K m ) 2, m = 2, m 3, d n ( m - 1) ( Q l ( m) ) = d m - 1 ( K m ) = 2 = n + 1, m2ary n2cube Q n- 1 ( m) d ( n - 1) ( m - 1) ( Q n - 1 ( m) ) n Q n ( m) m2ary n2cube, n 2, x y Q n ( m) Q n ( m) n ( m - 1) n + 1 ( x, y), x = (0 x 2 x 3 x n ), y = ( ly 2 y 3 y n ) x 0 = x, y l x i = ( ix 2 x 3 x n ), y i = ( iy 2 y 3 y n ), i = 0,1,, m - 1 = y 3, Q 0 n - 1 ( m), Q 1 n - 1 ( m),, Q m - 1 n - 1 ( m) Q n ( m) 1, x i Q i n - 1 ( m), i = 0,1,, m - 1, d ( n - 1) ( m - 1) ( Q i n- 1 ( m) ) n, 0 i m - 1 P 0 i, P1 i,, P i ( n - 1) ( m - 1) - 1 Q i n- 1 ( m) ( n - 1) ( m - 1) n ( x i, y i ), P i Q i n - 1 ( m) ( x i, y i ) 2, P i dq i n- 1 ( m) = n - 1, 0 i m - 1 Q i n - 1 ( m) ( n - 1) ( m - 1), P0 i x i P i, j = 1,, ( n - 1) ( m - 1) - 1, Pj i 2 i, j = 0,1,, ( n - 1) ( m - 1) - 1, w j i P j i x i, Pj i ( wj i, y i ) P j i ( wj i, y i ), i ( i = 0,1,, m - 1) j ( j = 1,2,, ( n - 1) ( m - 1) ) - 1, P i j P i j = x i w i j P i j ( w i j, y i ) y i Q n ( m) n ( m - 1) ( n + 1) ( x 0, y l ) 1 ( x 2 x 3 x n ) ( y 2 y 3 y n ) l = 0, y = y 0 R i - 1 = x 0 x i P i y i y 0, i = 1,, m - 1 ; R m - 1+ j = x 0 P 0 j y 0, j = 0,1,2,, ( n - 1) ( m - 1) - 1, { R j 0 j n ( m - 1) - 1} Q n ( m) n ( m - 1) ( x 0, y 0 ), R i - 1 = 1 + P i + 1 ( n - 1) + 2 = n + 1, i = 1,, m - 1 ; R m j = P 0 j n, j = 0,1,2,, ( n - 1) ( m - 1) ( x 2 x 3 x n ) ( y 2 y 3 y n ) l 0 R 0 = x 0 P 0 y 0 y l ; R l = x 0 x l P l y l ; R i = x 0 x i P i y i y l, i = 1,, m - 1 ; i l ; R m + j = x 0 w 0 j w l j P l j ( w l j, y l ) y l, j = 1,2,, ( n - 1) ( m - 1) - 1

4 1 19, { R j 0 j n ( m - 1) - 1} Q n ( m) n ( m - 1) ( x 0, y l ), R 0 = P ( n - 1) + 1 = n ; R l = P l + 1 ( n - 1) + 1 = n ; R i = 1 + P i + 1 ( n - 1) + 2 = n + 1, i = 1,, m - 1, i l ; R m + j = 1 + P l j 1 + n, j = 1,2,, ( n - 1) ( m - 1) ( x 2 x 3 x n ) ( y 2 y 3 y n ), l 0, y = x l R 0 = x 0 x l ; R i = x 0 x i x l, i = 1,2,, m - 1, i l ; R m j = x 0 w i j w l j x l, j = 0,1,2,, ( n - 1) ( m - 1) - 1, { R j 0 j n ( m - 1) - 1} Q n ( m) n ( m - 1) ( x 0, x l ), j, (0 j n ( m - 1) - 1), R i 3 n + 1, (1) 2, Q n ( m) n ( m - 1), d ( Q n ( m) ) = n, (1), Q n ( m) n ( m - 1) 1, (1) (2), 1 m 1 2 [9 ] d n ( Q n ) = n + 1, n 2 d n ( m - 1) ( Q n ( m) ) n + 1 (2) 3 k = m 1 + m m n - n, d k ( Q ( m 1, m 2,, m n ) ) = n + 1, = 2, n 2 3, 1, 4 ( Q ( m 1, m 2,, m n ) ) = m 1 + m m n - n ( Q n ( m) ) = n ( m - 1) 3 Q ( m 1, m 2,, m n ) m 1 + m m n - n, ( Q ( m 1, m 2,, m n ) ) m 1 + m m n - n, Q ( m 1, m 2,, m n ) m 1 + m m n - n, ( Q ( m 1, m 2,, m n ) ) m 1 + m m n - n, ( Q ( m 1, m 2,, m n ) ) = m 1 + m m n - n [ 1 ] Bhuyan L N, Agrawal D P Generalized hypercube and hyperbus structures for a computer network [J ] IEEE Trans Comput 1984, 32(4) : [2 ] Wu J, Guo G H Fault tolerance measures for m2ary n2dimensional hypercubes based on forbid2 den faulty sets [J ] IEEE Trans Comput 1988, 47 (8) : [3 ] Bermond J C, Homobono N, Peyrat C Large fault2tolerant interconnection networks [ J ] Graphs and Combinatorics, 1989, 5 : [4 ] Hsu D F, Lyuu Y D A graph2theoretical study of transmission delay and fault tolerance [M] Proc of 4th ISMM International Conference on Parallel and Distributed Computing and Systems, 1991 [ 5 ] Du D Z, Lyuu YD, Hsu D F Line digraph itera2 tions and connectivity analysis of de Bruijn and Kautz graphs [J ] IEEE Trans Comput 1993, 42 (5) : [6 ] Hsu D F, Luczak T Note on the k2diameter of k2

5 regular k2connected graphs [J ] Discrete Math 1994, 132 : [7 ] Ishigami Y The wide2diameter of the n2dimen2 sional toroidal mesh [J ] Networks 1996, 27 : [8 ] Li Q Sotteau D, Xu J M 22diameter of de Bruijn graphs [J ] Networks 1996, 28 : 7 14 [9 ] Saad Y, Schultz M H Topological properties of hypercubes [J ] IEEE Trans Comput 1988, 37 (7) : [10 ] [M] :, 1998 [11 ] Leighton F T Introduction to Parallel Algorithms and Architectures : Array Trees Hypercubes [M] San Mateo : Morgan Kaufwann Publishers, 1992 Fault Tolerance and Transmission Delay of Generalized Hypercube Net works XU Jun2ming ( Department of Mathematics, USTC, Hefei , China) Abstract : The wide diameter is an important measure for fault tolerance and transmission delay of a parallel processing computer network The generalized hypercube Q ( m 1, m 2,, m n ) is an important network topology for parallel processing computer systems In this paper, it is shown that the diameter with width k = m 1 + m m n - n of Q ( m 1, m 2,, m n ) is equal to n + 1 Key words : generalized hypercube ; fault tolerance ; transmission delay ; connectivity ; wide2diameter ( 49 ) The Synchro2curvature Spectra of Electrons with a Power2la w Energy Distribution XIA Tong2sheng, ZHANGJia2lu, CHEN Ci2xing ( Center for Astrophysics, USTC, Hefei , China) Abstract : Based on the new synchro2curvature radiation mechanism by J L Zhang et al, if the mag2 netic field of a radiation region is not flat and straight, the synchro2curvature radiation, not the syn2 chrotron radiation, should be the basis on which a real description is to be achieved In this paper, the synchro2curvature radiation spectrum of electrons is calculated with a power2law energy distribution and it is found that in a curved magnetic field, the resulting spectrum of electrons could be clearly distin2 guished from power2law This means that the previous conclusion is open to question Key words : curved magnetic field ; synchro2curvature mechanism ; radiation spectrum