RELIABILITY is an important concept in the design

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1 Poceedings of the Wold Congess on Engineeing 0 Vol I WCE 0, July -, 0, London, U.K. Reliability Measues in Ciculant Netwok *Inda Rajasingh, Bhaati Rajan, and R. Sundaa Rajan Abstact Reliability and efficiency ae impotant citeia in the design of inteconnection netwoks. Connectivity is a widely used measuement fo netwok fault-toleance capacities, while diamete detemines outing efficiency along individual paths. In pactice, we ae inteested in high-connectivity, smalldiamete netwoks. Recently, the w-wide diamete d w(g), (w 1)-fault diamete D w (G) and the w-rabin numbe w (G), fo w k(g) have been used to measue netwok eliability and efficiency. In this pape, in addition to these paametes we intoduce (w 1)-fault wide diamete ρ w(g) and study these paametes fo the undiected ciculant netwok. Index Tems wide diamete, fault diamete, fault wide diamete, Rabin numbe, ciculant netwok. I. INTRODUCTION RELIABILITY is an impotant concept in the design of netwoks. Connectivity and edge connectivity ae widely used as eliability measues. In pactice, we ae often inteested in the collection of multipaths between a pai of nodes []. The distance d G (x, y) fom a vetex x to anothe vetex y in a netwok G is the minimum numbe of edges of a path fom x to y. The diamete d(g) of a netwok G is the maximum distance fom one vetex to anothe. The connectivity k(g) of a netwok G is the minimum numbe of vetices whose emoval esults in a disconnected o tivial netwok. Accoding to Menge s theoem, thee ae atleast k (intenally) vetex-disjoint paths fom a vetex x to anothe vetex y in a netwok of connectivity k [1]. The classical appoach to study outing in inteconnection netwoks is to ty to find the shotest path between the sending station and the eceiving station. Wheneve some stations ae faulty on the path between the sending station and the eceiving station, the management potocol has to find a way to bypass those faulty stations and set up a new path between them. Similaly, if this new path is disconnected again, a thid path needs to be set up, if it is possible []. In this context, diamete is the measuement fo maximum tansmission delay and connectivity is a good paamete to study the toleance of the netwok on occasions when nodes fail. Fault toleant inteconnection netwoks can be found in Hsu [1]. Fo a gaph (netwok) G with connectivity k(g), the paametes w-wide diamete d w (G), (w 1)-fault diamete D w (G) and the Rabin numbe w (G) fo any w k(g) aise fom the study of paallel outing, fault-toleant systems and andomized outing espectively [], [], [], [1]. Due to the widespead use of eliable, efficient and fault-toleant netwoks, these thee paametes have been the subject of extensive study ove the past decade []. In the sequal, (x 1, x,..., x n ) denotes a path fom x 1 to x n. Manuscipt eceived Decembe 0, 0; evised Decembe 1, 0. *This wok is suppoted by DST Poject No.SR/S/MS: /0. Inda Rajasingh, Bhaati Rajan, and R. Sundaa Rajan ae with the Depatment of Mathematics, Loyola College, Chennai 00 0, Tamilnadu, India ( hellosunda@ediffmail.com). II. PRELIMINARIES Definition 1. [] A containe C(x, y) between two distinct nodes x and y in a netwok G is a set of node-disjoint paths between x and y. The numbe of paths in C(x, y) is called the width of C(x, y). A C(x, y) containe with width w is denoted by C w (x, y). The length of C w (x, y), witten as l(c w (x, y)), is the length of a longest path in C w (x, y). Definition. [] Fo w k(g), the w-wide distance fom x to y in a netwok G is defined as d w (x, y) = min{l(c w (x, y))/c w (x, y) is a containe with width w between x and y} The w-wide diamete of G is defined as d w (G) = max x,y V (G) {d w (x, y)}. In othe wods, fo w k(g), the w-wide diamete d w (G) of a netwok G is the minimum l such that fo any two distinct vetices x and y thee exist w vetex-disjoint paths of length at most l fom x to y. The notion of w-wide diamete was intoduced by Hsu [] to unify the concepts of diamete and connectivity. It is desious that an ideal inteconnection netwok G should be one with connectivity k(g) as lage as possible and diamete d(g) as small as possible. The wide-diamete d w (G) combines connectivity k(g) and diamete d(g), whee 1 w k(g). Hence d w (G) is a moe suitable paamete than d(g) to measue fault-toleance and efficiency of paallel pocessing compute netwoks., detemining the value of d w (G) is of significance fo a given gaph G and an intege w. Hsu [] poved that this poblem is NP -complete []. Remak 1. If thee exists a containe C w(x, y) such that each of the w paths in C w(x, y) is a shotest path between x and y in G, then d w (x, y) = l(c w(x, y)) Definition. [] Fo w k(g), the (w 1)-fault distance fom x to y in a netwok G is D w (x, y) = max{d G S (x, y) : S V with S = w 1 and x, y ae not in S} whee d G S (x, y) denotes the shotest distance between x and y in G S. The (w 1)-fault diamete of G is D w (G) = max{d w (x, y) : x and y ae in G} The notion of D w (G) was defined by Hsu [] and the special case in which w = k(g) was studied by Kishnamoothy et al. []. Motivated by the definitions of wide diamete and fault diamete in a netwok, we intoduce a new paamete in this pape known as fault wide diamete and define it as follows: ISBN: ---- ISSN: 0-0 (Pint); ISSN: 0-0 (Online) WCE 0

2 Poceedings of the Wold Congess on Engineeing 0 Vol I WCE 0, July -, 0, London, U.K. 0 1 n- n n- +1 n- n Fig. 1. Ciculant gaph G(; ±{1,, }) Fig.. -wide diamete of G Definition. Fo w k(g), the (w 1)-fault wide distance fom x to y in a netwok G is ρ w (x, y) = max{d k(g) S (x, y) : S V with S = w 1 and x, y ae not in S} The (w 1)-fault wide diamete of G is ρ w (G) = max{ρ w (x, y) : x and y ae in G} Definition. [] The w-rabin numbe w (G) of a netwok G is the minimum l such that, fo any w +1 distinct vetices x, y 1,..., y w thee exist w vetex-disjoint (except at x) paths of length at most l fom x to y i, 1 i w. This concept was fist defined by Hsu []. It is clea that when w = 1, d 1 (G) = D 1 (G) = ρ k (G) = 1 (G) = d(g) fo any netwok G. The following ae basic popeties and elationships among d w (G), D w (G), ρ w (G) and w (G). Lemma 1. [] The following statements hold fo any netwok G of connectivity k: 1. D 1 (G) D (G) D k (G). d 1 (G) d (G) d k (G). 1 (G) (G) k (G). D w (G) d w (G) and D w (G) w (G) fo 1 w k It is easy to check that the new paamete ρ w (G) exhibits the following popeties. Lemma. Let G be a netwok of connectivity k. Then 1. ρ 1 (G) ρ (G) ρ k (G). D w (G) d w (G) ρ w (G) w (G) fo 1 w k In, Chen et al. detemined the wide diamete of the cycle pefix netwok []. In, Liaw et al. found faulttoleant outing in ciculant diected gaphs and cycle pefix netwoks [1]. The line connectivity and the fault diametes in pyamid netwoks wee studied by Cao et al. in []. In the same yea Liaw et al. detemined the Rabin numbe and wide diamete of buttefly netwoks [], []. In 00, Liaw et al. found the wide diametes and Rabin numbes of genealized folded hypecube netwoks [1]. In 00, Jia and Zhang found the wide diamete of Cayley gaphs of Z m, the cyclic goup of esidue classes modulo m and they poved that the k-wide diamete of the Cayley gaph Cay(Z m, A) geneated by a k-element set A is d fo k = and is bounded above by d fo k =, whee d is the diamete of Cay(Z m, A) []. In this pape we compute the w-wide diamete d w (G), (w 1)-fault diamete D w (G), (w 1)-fault wide diamete ρ w (G), and the w-rabin numbe w (G) fo w k(g) when G is an undiected ciculant gaph with connectivity k. III. MAIN RESULTS The ciculant is a natual genealization of the double loop netwok and was fist consideed by Wong and Coppesmith [1]. Ciculant gaphs have been used fo decades in the design of compute and telecommunication netwoks due to thei optimal fault-toleance and outing capabilities []. It is also used in VLSI design and distibuted computation [1], [], [1]. The tem ciculant comes fom the natue of its adjacency matix. A matix is ciculant if all its ows ae peiodic otations of the fist one. Ciculant matices have been employed fo designing binay codes []. Theoetical popeties of ciculant gaphs have been studied extensively and suveyed by Bemond et al. [1]. Evey ciculant gaph is a vetex tansitive gaph and a Cayley gaph [1]. Most of the ealie eseach concentated on using the ciculant gaphs to build inteconnection netwoks fo distibuted and paallel systems [1], []. Definition. A ciculant undiected gaph, denoted by G(n; ±{1,,..., j}), 1 j n/, n is defined as an undiected gaph consisting of the vetex set V = {0, 1,..., n 1} and the edge set E = {(i, j) : j i s(mod n), s {1,,..., j}}. The ciculant gaph shown in Fig. 1 is G(; ±{1,, }). It is clea that G(n; ±1) is the undiected cycle C n and G(n; ±{1,,..., n/}) is the complete gaph K n. Theoem 1. If G is an undiected ciculant gaph G(n; ±{1,,..., }), then { n d (G) = +, if n 1(mod ) n +, othewise fo all = 1,,..., n/. Poof. Since G is -egula, by Menge s theoem thee ae exactly vetex disjoint paths between any two vetices u and v in G. We begin with the case when u and v ae adjacent in C n. Since G is vetex-tansitive, without loss of geneality we assume that u = 0 and v = 1. Since the neighbouhood of 0 namely N(0) is the set N(0) = {1,,...,, n 1,..., n }, each path in any containe C (0, 1) contains exactly one membe fom N(0). Conside a path P in C (0, 1) passing though n. Since V (P ) (N(0)\(n )) = ϕ, in ode to compute d (G) we choose P to be a shotest path between 0 and 1. P = (0, n, n,..., 1 +, 1 +, 1) ISBN: ---- ISSN: 0-0 (Pint); ISSN: 0-0 (Online) WCE 0

3 Poceedings of the Wold Congess on Engineeing 0 Vol I WCE 0, July -, 0, London, U.K Fig.. -fault diamete of G(; ±{1,, }) is Fig.. -fault wide diamete of G(; ±{1,, }) is of length n +, when n 1 (mod ) and P = (0, n, n,..., j +, 1 +, 1), j 1 + of length n + in all othe possibilities. evey containe C (0, 1) contains a path of length n + if n 1 (mod ) and n + if n 1 (mod ). We obseve that the edge (0, 1) is a path of length 1 and the paths (0, j, 1) and (0, n +1+j, 1), 0 j ae paths of length between 0 and 1 passing though vetices in N(0)\{n }. thee exist a containe C (0, 1) in which the longest path is P and all othe paths ae of length 1 o. See Fig.. The case when u and v ae not adjacent on C n natually yields paths between u and v of length atmost n +, when n 1 (mod ) and atmost n +, when n 1 (mod ). In the following theoem we obtain the ( +1)-fault diamete fo the undiected ciculant gaph G(n; ±{1,,..., }). Theoem. Let G be the ciculant gaph G(n; ±{1,,..., }). Then n D +1 (G) = fo all =,..., n/. Poof. We begin with faulty nodes which ae consecutive vetices in C n. Without loss of geneality let 1,,..., be the faulty nodes. Let G = G\{1,,..., }. It is easy to see that d G (0, ) d G (i, j) fo all vetices i, j taken modulo n. Now d G (0, ) = n. On the othehand, suppose the faulty nodes ae not consecutive vetices of C n, then d G (i, j) is bounded above by n. See Fig.. Theoem. If G is an undiected ciculant gaph, then n D (G(n; ±{1,,..., })) = fo all = 1,,..., n/. Poof. Since G is vetex-tansitive, let us assume that the vetex labeled be the faulty node. Let G = G\{}. We assume that u = 0 and v = n/. In ode to compute D (G) we choose P to be shotest path between 0 and n/. Then P = (0, n, n,..., n/) of length n if n 0 (mod ). Othewise, we choose the shotest path P between 0 and n/ as P = (0, 1, 1,..., n/) of length n. It is easy to see that dg (0, n/) = n d G (i, j) fo all vetices i, j taken modulo n. In the next esult we obtain the (w 1)-fault wide diamete fo an undiected ciculant gaph. Theoem. Let G be the ciculant gaph G(n; ±{1,,..., }). Then n ( ) ρ (G) = + fo all = 1,,..., n/. Poof. Since G is vetex-tansitive, let us assume that 1 be the faulty node. Let G = G\{1}. Then G is ( 1)-connected. We choose u = 0 and v =. Since the neighbouhood of 0 is the set N(0) = {,,...,, n 1,..., n }, each path in the containe C 1 (0, ) contains exactly one membe fom N(0). Conside a path P in C 1 (0, ) passing though n +1. Since V (P ) (N(0)\(n )) = ϕ, in ode to compute ρ (G) we choose P to be a shotest path between 0 and. P = (0, n, n,..., ) of length n (+1) othe paths ae less than o equal to easy to see that d 1 (0, ) = +. Again by Theoem 1, the length of +. It is also n (+1) n (+1) + d 1 (i, j) fo all vetices i, j taken modulo n in G. Theoem. Let G be the ciculant gaph G(n; ±{1,,..., }). Then ρ (G) = n ( ) +, if n ( ) x (mod ) whee x {,,..., 1} n ( ) +, othewise fo all =,,..., n/. Poof. Since G is vetex-tansitive, let us assume that 1, be the faulty nodes. Let G = G\{1, }. Then G is ( )- connected. We assume that u = 0 and v =. Since the neighbouhood of 0 is the set N(0) = {,...,, n 1,..., n }, each path in the containe C (0, ) contains exactly one membe fom N(0). Conside a path P in C (0, ) passing though n +. Since V (P ) (N(0)\(n + )) = ϕ, in ode to compute ρ (G) we choose P to be a shotest path between 0 and. ISBN: ---- ISSN: 0-0 (Pint); ISSN: 0-0 (Online) WCE 0

4 Poceedings of the Wold Congess on Engineeing 0 Vol I WCE 0, July -, 0, London, U.K. Fig Rabin numbe of G 1 1 P = (0, n +, n +,..., ) of length n ( ) + if n ( ) x(mod ), whee x {,,..., 1}. By Theoem 1, the length of othe paths ae less than o equal to n ( ) +. Also it is easy to see that d (0, ) = n ( ) + d (i, j) fo all vetices i, j taken modulo n in G. Remak. If =, then n ρ (G) = whee G be the ciculant gaph G(n; ±{1,,..., }). Theoem. If G is an undiected ciculant gaph, then n ( + ) ρ +1 (G(n; ±{1,,..., })) = fo all = 1,,..., n/. Poof. Since G is vetex-tansitive, let us assume that 1,,..., be the faulty nodes. Let G = G\{1,,..., }. Then G is -connected. We assume that u = 0 and v =. Since the neighbouhood of 0 is the set N(0) = {n 1,..., n }, each path in the containe C (0, ) contains exactly one membe fom N(0). Conside a path P in C (0, ) passing though n 1. In ode to compute ρ +1 (G) we choose P to be a shotest path between 0 and. of length P = (0, n 1, n 1 +,..., ). By Theoem 1, the length of othe. It is also easy n (+) paths ae less than o equal to to see that d (0, ) = n (+) n (+) d (i, j) fo all vetices i, j taken modulo n in G. See Fig.. We now poceed to detemine the w-rabin numbe fo an undiected ciculant gaph. Theoem. If G is an undiected ciculant gaph, then n m 1 (G(n; ±{1,,..., m})) = m fo all m = 1,,..., n. Poof. By definition of Rabin numbe, we choose m points (say u, u 1, u,..., u m 1 ) in G. It is enough to conside the case when these points ae consecutive points in C n labeled i, i 1,..., i m 1 modulo n. We claim that, thee ae m 1 vetex-disjoint paths fom i to i k, 1 k m 1 of length atmost n m +1. We in fact, constuct m 1 vetex-disjoint paths fom i to i k, 1 k m 1 as follows. The edge (i, i 1 ) is a path of length 1 fom i to i 1. Futhe (i, i n m+l1, i 1 ), 1 l 1 m 1 and (i, i n+l, i 1 ), l m ae m numbe of paths in G of length fom i to i 1. The edge (i, i ) is a path of length 1 fom i to i. Futhe (i, i 1, i ), (i, i n+l1, i ), l 1 m and (i, i n m+l, i ), l m 1 ae m numbe of paths in G of length fom i to i. Also (i, i + n, i + n,..., i +, i ) is a path of length n m. Similaly, the m 1 vetex-disjoint paths fom i to i k, k m 1 ae of length less than o equal to n m +1. Theoem. If G is an undiected ciculant gaph G(n; ±{1,,..., m}), then { n m m (G) = m +, if n 1(mod m) n m +, othewise fo all m = 1,,..., n. Poof. This theoem is an easy consequence of Theoem. See Fig.. IV. CONCLUSION In this pape we have found the w-wide diamete, w-fault diamete, w-fault wide diamete and w-rabin numbe fo a ciculant undiected gaph G. It would be inteesting to study eliability measues in inteconnection netwok such as hypecube netwok, buttefly netwok, mesh netwok and so on. ACKNOWLEDGMENT We thank Pof. Paul Manuel, Depatment of Infomation Science, Kuwait Univesity, Kuwait fo his fuitful discussions. We also thank the unknown efeee fo his useful suggestions on evising the pape. REFERENCES [1] J.C. Bemond, F. Comellas and D.F. Hsu, Distibuted loop compute netwoks, A suvey Jounal of Paallel and Distibuted Computing, vol., no. 1, pp. -,. [] R. Beivide, E. Heada, J.L. Balc aza and A. Auabaena, Optimal Distance Netwoks of Low Degee fo Paallel Computes, IEEE Tansactions on Computes, vol. 0, no., pp. 1-, 1. [] F.T. Boesch and J. Wang, Reliable ciculant netwoks with minimum tansmission delay, IEEE Tansactions on Cicuit and Systems, vol., no. 1, pp. -1,. [] F. Cao, D. Du, D.F. Hsu and S. Teng, Fault Toleance Popeties of Pyamid Netwoks, IEEE Tansactions on Computes, vol., no. 1, pp. -,. [] W.Y.C. Chen, V. Fabe and E. Knill, Resticted Routing and Wide Diamete of the Cycle Pefix Netwok, DIMACS Seies in Discete Mathematics and Theoetial Compute Science, vol. 1, pp. 1-,. [] D.F. Hsu, On Containe Width and Length in Gaphs, Goups and Netwoks, IEICE Tans. Fundamentals of Electonics, Comm., and Compute Sciences, vol. E-A, pp. -0,. [] K.E. Jia and A.Q. Zhang, On Wide diamete of Cayley gaphs, Jounal of Inteconnection Netwoks, vol., no., pp. -1, 00. [] M. Kalin, New binay coding esults by ciculants, IEEE Tansactions on Infomation Theoy, vol. 1, no. 1, pp. 1-,. [] M.S. Kishnamoothy and B. Kishnamuthy, Fault diamete of inteconnection netwoks, Comput. Math. Appl., vol. 1, no. -, pp. -,. [] S.C. Liaw and G.J. Chang, Wide Diametes of Buttefly Netwoks, Taiwanese Jounal of Mathematics, vol., no. 1, pp. -,. [] S.C. Liaw and G.J. Chang, Rabin numbe of Buttefly Netwoks, Discete Math., vol., no. 1-, pp. -,. ISBN: ---- ISSN: 0-0 (Pint); ISSN: 0-0 (Online) WCE 0

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