Research Article On Simple Graphs Arising from Exponential Congruences

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1 Appled Mathematcs Volume 2012, Artcle ID , 10 pages do: /2012/ Research Artcle On Smple Graphs Arsng from Exponental Congruences M. Aslam Malk and M. Khald Mahmood Department of Mathematcs, Unversty of the Punjab, Lahore 54590, Pakstan Correspondence should be addressed to M. Khald Mahmood, Receved 19 March 2012; Revsed 17 June 2012; Accepted 3 September 2012 Academc Edtor: Maurzo Porfr Copyrght q 2012 M. A. Malk and M. K. Mahmood. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. We ntroduce and nvestgate a new class of graphs arrved from exponental congruences. For each par of postve ntegers a and b, letg n denote the graph for whch V {0, 1,...,n 1} s the set of vertces and there s an edge between a and b f the congruence a x b mod n s solvable. Let n p k1 1 pk2 2 pkr r be the prme power factorzaton of an nteger n, wherep 1 <p 2 < <p r are dstnct prmes. The number of nontrval self-loops of the graph G n has been determned and shown to be equal to r 1 φ pk 1. It s shown that the graph G n has 2 r components. Further, t s proved that the component Γ p of the smple graph G p 2 s a tree wth root at zero, and f n s a Fermat s prme, then the component Γ φ n of the smple graph G n s complete. 1. Introducton The noton of congruence s ntrnsc n number theory. Modular arthmetc has clutched a vtal contrvance for most of the number theoretc mathematcs. In recent years, studyng graphs through congruences s of charsmatc and an ndependent nterest of number theorsts. It has cemented a novel approach to ntroduce a premum connecton between Number Theory and Graph Theory. On behalf of modular arthmetc, we can manpulate many fascnatng features of graphs. We frst encounter the deas used n 1 5, where dgraphs from congruences were dscussed. The condtons for regularty and semregularty of such dgraphs are presented n 1, 2. The necessary and suffcent condtons for the exstence of solated fxed ponts have been establshed n 3. The structures of symmetrc dgraphs have been studed n 4, 5. In ths paper we dscuss the graph G n arrvng from exponental congruences. We assgn to each par of postve ntegers a and b an edge a, b of the graph G n f the congruence a x b mod n s solvable, where V {0, 1,...,n 1} s the set of vertces of G n and E V V s the set of edges of G n. Then the graph G n

2 2 Appled Mathematcs has a loop at a vertex a f and only f a x a mod n admts a soluton. Snce the congruence relaton s an equvalence relaton, so a x a mod n, a {0, 1,...,n 1} always admts a soluton x 1. Thus we call x 1 a trval soluton and, so far, the loops due to trval soluton are called trval loops. Thus t becomes nterestng to fnd the number of vertces n {0, 1,...,n 1} such that the congruence a x a mod n admts a nontrval soluton. The loops at vertces correspondng to nontrval solutons are called nontrval loops. Snce for each α 2, the congruence a α a mod n, fora 0, 1, so the loops at vertces 0 and 1 wll be consdered as nontrval loops. We denote the number L n for nontrval loops of the graph G n. Letn p k 1 r be the prme power factorzaton of an nteger n, where p 1 <p 2 < <p r are dstnct prmes. The number L n of the graph G n has been determned and shown to be equal to r 1 φ pk 1, where φ s the Euler s ph functon. It has been shown that the graph G n has 2 r components. We label these components as Γ d, where d n or d φ n. The order of each component of the graph G n has been determned. Further, t s proved that the component Γ p of the smple graph G p 2 s always a tree wth root at zero. Also f n s a Fermat s prme, then the component Γ φ n of the smple graph G n s a complete graph. 2. Prelmnares A graph G s smple f t s free from loops and multedges. The vertces a 1,a 2,...,a k 1,a k wll consttute a cycle of length k f each of the followng congruences s solvable: a x 1 a 2 mod n, a x 2 a 3 mod n,. 2.1 a x k a 1 mod n. A graph G s sad to be connected f there s a path from x to y, for each par of vertces x and y. A maxmal connected subgraph s called a component 6. In Fgure 1, the smple graph G 30 has eght components. A graph G s complete f every two dstnct vertces of G are adjacent. Thus G n s complete f a x b mod n s solvable for each dstnct par of vertces n V. The degree of a vertex v n G s the number of edges ncdent wth v. It s denoted by deg v. Ifdeg v r for each vertex v of G n, then G n s called r-regular or regular graph of degree r. The smple graph G 30 has two 3-regular and two 1-regular components. We recall the defnton of Euler s ph functon 7, and some of ts propertes. Defnton 2.1. For n 1, let φ n denote the number of postve ntegers not exceedng n whch are relatvely prme to n. Notethatφ 1 1, because gcd 1, 1 1. Thus we can wrte { 1, f n 1, φ n Number of ntegers less than n and co-prme to n, f n /

3 Appled Mathematcs Fgure 1: The smple graph of G 30. By the defnton of Euler s ph functon, t s clear that f a b, then φ a φ b. Alsoφ p p 1 f and only f p s a prme number. We wll need the followng results of 7, regardng Euler s ph functon, for use n the sequel. Theorem 2.2. If p s a prme and k>0, then ( φ p k) p k 1( p 1 ). 2.3 Let f be an arthmetc functon. One recalls that f s sad to be multplcatve f f mn f m f n, gcd m, n 1. The followng theorem s the generalzaton of the well-known Fermat s Lttle theorem whch states that f a, p 1, then a p 1 mod p. Theorem 2.3 Euler. If n 1 and a, n 1, thena φ n 1 mod n. Defnton 2.4 see 7. Asetofk ntegers s a complete resdue system CRS modulo k f every nteger s congruent to exactly one of the modulo k. If we delete ntegers from CRS, whch are not prme to k, then the remanng ntegers wll consttute the reduced resdue system RRS modulo k.

4 4 Appled Mathematcs Theorem 2.5 countng prncple 8. If a work s dstrbuted over r 1,r 2,...,r t, objects wth r 1 object occurng s 1 ways, r 2 object occurng s 2 ways, and r t object occurng s t ways, then the work can be done n s 1 s 2 s t ways. Theorem 2.6 the ncluson-excluson prncple 9. Let A 1,A 2,...,A n be n fnte sets. Then n A A 1 1 n 1 <j n A A j 1 <j<k n n A A j A k 1 n 1 A Applcatons of Euler s Ph Functon In Graph Theory, a loop or a self-loop s an edge that connects a vertex to tself. A vertex of an undrected graph s called an solated vertex f t s not the endpont of any edge. Ths means that t s not connected wth any other vertex of the graph. Thus n our case, the vertex v s an solated vertex f and only f v x u mod n, v / u s not solvable. Now f we omt the condton of smplcty n our graph, then ths solated vertex wll contrbute twce n the degree as there wll be a self-loop at v as the congruence v x v mod n s solvable. Ths leads to the followng results. Theorem 3.1. Let n 2p 1 p 2 p k be the prme factorzaton of n, wherep 1 <p 2 < <p k are dstnct, odd prmes. Then 0 and p 1 p 2 p k are solated vertces of G n. That s, the vertces 0 and p 1 p 2 p k are not the endponts of any edge n graph G n except loop at these vertces. Proof. The element a, where 0 a n 1, s an solated pont of the graph G n f and only f the solvablty of a x b mod n mples that a b. It s trval that 0 s an solated pont of G n as a x 0 mod 2p 1 p 2 p k s solvable f and only f a 0 mod 2p 1 p 2 p k. Moreover, p 1 < p 2 < < p k are dstnct, odd prmes, so for some nteger β > 1, p 1 p 2 p k β 1 1 s an even nteger. Thus 2p 1 p 2 p k p 1 p 2 p k p 1 p 2 p k β 1 1 and hence p 1 p 2 p k x p 1 p 2 p k mod2p 1 p 2 p k s solvable wth root β. Next, we clam that the congruence p 1 p 2 p k x a mod2p 1 p 2 p k, a / p 1 p 2 p k mod2p 1 p 2 p k s not solvable. To prove our asserton, let a / p 1 p 2 p k mod 2p 1 p 2 p k and suppose there exsts an nteger α 1, such that the congruence, p 1 p 2 p k α a mod2p 1 p 2 p k s balanced. Then there exsts some nteger t such that ( p1 p 2 p k ) α a 2p1 p 2 p k t. 3.1 Ths mples that p 1 p 2 p k p 1 p 2 p k α 1 2t a. But then p 1 p 2 p k a. Leta t 1 p 1 p 2 p k for some nteger t 1.Ift 1 s even, then 2p 1 p 2 p k a and hence by 3.1, 2p 1 p 2 p k p 1 p 2 p k α, a contradcton as p 1 p 2 p k α s an odd nteger. So let t 1 2r 1, for some nteger r. Then a t 1 p 1 p 2 p k 2r 1 p 1 p 2 p k and ths shows that 2p 1 p 2 p k a p 1 p 2 p k whch s a contradcton aganst the fact that a / p 1 p 2 p k mod 2p 1 p 2 p k. Fnally, the congruence p 1 p 2 p k x 0 mod2p 1 p 2 p k s not solvable snce an odd nteger s not dvsble by an even nteger. Hence 0 and p 1 p 2 p k are not adjacent as well. Corollary 3.2. If n s an odd square free nteger, then 0 s the only solated vertex of the graph G n.

5 Appled Mathematcs 5 Theorem 3.3. Let n p k 1 r be the prme power factorzaton of a non-square-free nteger n, wherep 1 <p 2 < <p k are dstnct prmes, k 1 and r 1. Then 0 and kp, wherek 0, 1,..., n/p, and p p 1 p 2 p k, are always adjacent vertces n G n. Proof. We show that the congruence kp x 0 mod n s solvable for all k 0, 1,..., n/p. Let p, q be two dstnct prmes such that n pq 2. Then kpq 2 k 2 p 2 q 2 k 2 p pq 2 0 mod n. Thusx 2 s the soluton of the congruence kpq x 0 mod pq 2. We follow the fashon explaned above. For ths, take n p k 1 r.lett lcm p 1,p 2,...,p k, then there exst ntegers t 1,t 2,...,t k such that t t k, for each 1, 2,...,k.Letm r 1 pk t 1. Then t s easy to see that ( ) t kp k t p t 1k 1 1 p t 2k 2 2 p t rk r r k t mn 0 mod n. 3.2 Ths shows that x t lcm p 1,p 2,...,p k s the soluton of the congruence, kp x 0 mod n. Hence, kp and 0 are the adjacent vertces n G n. The followng results gve a formula for fndng the number of nontrval self-loops of the graph G n. Lemma 3.4. Let p be a prme number. Let n p k, k 2 and let φ be the Euler s ph functon. Then L n φ p k 1. Proof. Let n p k, k 2. A vertex v of the graph G n has a self-loop f and only f v x v mod n s solvable. Thus to fnd the number of self-loops n G n, we need to count the number of vertces v n CRS mod n such that the congruence v x v mod n s solvable. By Euler s Theorem, f a, n 1, then a x 1 mod n s solvable wth x φ n as ts root, whence a x a mod n s solvable wth x φ n 1. Snce p, 2p,...,p k 1 p are the p k 1 ntegers whch are not prme to p k, so there are p k p k 1 φ p k ntegers n CRS mod n for whch the congruence a x a mod n s solvable. Also, for each k 1, the vertex 0 has selfloop n G p k. Thus there are φ p k 1 self-loops. For the case, f a {p, 2p,...,p k 1 p},wthα as the soluton of a x a mod p k. Then a α 1 1 mod p k 1 mples that a α 1 1 mod p. Ths yelds a contradcton aganst the fact that f a {p, 2p,...,p k 1 p}, then a α 1 0 mod p. Hence, there exst no vertex a {p, 2p,...,p k 1 p}, for whch a x a mod n s solvable. Thus φ p k 1 are the only vertces for whch the graph G n has self-loops. The followng theorem provdes us the cardnalty of the set of those vertces of G n whch have nontrval self-loops, where n p k 1 r and r 2. Theorem 3.5. Let n p k 1 r be the prme power factorzaton of an nteger n,wherep 1 <p 2 < <p r are dstnct prmes, k 1 and r 2. Then L n r ( ( φ 1 p k ) ) Proof. We apply nducton on r. By Lemma 3.4, result s true for r 1. Suppose result s true for r 1 dstnct prme factors. That s, f m p k 1 1 r 1, then L m r 1 1 φ pk 1. Let n p k 1 r mp k r r. Now, the congruence a x a mod mp k r r s solvable f and only f the congruences a x a mod m and a x a mod p k r r are solvable. But by nducton, the

6 6 Appled Mathematcs graph G m has r 1 1 φ pk 1 self-loops and, by Lemma 3.4, the graph G p k r r has φ p k r r 1 self-loops. Hence, by Theorem 2.5, the graph G n has r 1 φ pk 1 self-loops. Ths through our result. In Fgure 2, we depct Theorem 3.5 for n 24. Corollary 3.6. Let n p 1 p 2 p r be the prme factorzaton of a square-free nteger n, wherep 1 < p 2 < <p k are dstnct prmes and r 2. Then L n p 1 p 2 p r. 3.4 Proof. It s easy to see that φ p 1 p, for any prme p. Thus by Theorem 3.5, corollary follows. That s, L n p 1 p 2 p r. 3.5 Corollary 3.7. Let m and n be two non-square-free ntegers such that gcd m, n 1. Then, L mn L m L n Components and Ther Characterstcs Recall that a maxmal connected subgraph of a graph G s called a component. For nstance, the vertces v 1,v 2,...,v r from CRS mod n wll consttute a component of the graph G n f for each, 1 r, there exst some j, 1 j r such that v x v j mod n s solvable, for all / j. The followng theorem explores some nterestng characterstcs of components for the smple graph G n. Theorem 4.1 man theorem. Let n p k 1 r be the prme power factorzaton of an nteger n,wherep 1 <p 2 < <p r are dstnct prmes, k 1 and r 1. Then, a G n has 2 r components, b f n p 2, p s a prme number, then Γ p s a tree wth root at 0, c f n s a Fermat s prme, then Γ φ n s complete. Let n p k 1 r.thenp 1,p 2,...,p r,p 1 p 2,...,p 1 p r,...,p 1 p 2 p r are the possble square-free postve dvsors of n. We see that the sets of vertces generated by these dvsors wll consttute components of G n. We label these components by Γ p1, Γ p2,...,γ pr, Γ p1 p 2,...,Γ p1 p 2 p r. Moreover, the set of all those resdues of n whch are prme to n wll provde us another component. Snce ths set contans φ n vertces from V {0, 1, 2,...,n 1}, so, for the sake of convenence, we label ths component by Γ φ n. Before gvng the proof of Theorem 4.1, we prove the followng results. Lemma 4.2. a Let n p k,k >0,wherep s a prme number. Then G n has 2 components, namely, Γ φ n and Γ p. b Let n p k 1 r and k Π s 1 p, 1 s r. Then the order of the component Γ k s V Γ k n n k kp s<j r j s<j<k r n kp j p k 1 r s n kp s 1 p s 2 p r. 4.1

7 Appled Mathematcs Fgure 2: The graph G 24 has 15 nontrval self-loops. Proof. a Recall that an nteger a such that a, p k 1scalled a prmtve root modulo p k only f a s of order p k 1 p 1. Letr be a prmtve root of p k. Then the smallest nteger k s called the ndex of a wth respect to r f r k a mod p k. Moreover, t s well known that the exponental congruence a x b mod p k s solvable f and only f d nd r b, where d nd r a, p k 1 p 1. We dvde the numbers 0, 1, 2,..., p 1 p k nto two sets, one of whch s {0,p,2p,..., p 1 p k 1 } and the other s RRS modulo p k. We clam that these two sets consttute ndependent components of G p k.letv {r 1,r 2,...,r φ p k } where each r, 1 φ p k s prme to p k, the set of vertces of the component Γ φ p k. Then every vertex of Γ φ p k must have some ndex wth respect to prmtve root r. Thus the exponental congruence a x 1 mod p k s solvable snce nd r 1 0 and s dvsble by d, where, d nd r a, p k 1 p 1. Ths means that each vertex of the set Γ φ p k s adjacent wth 1. So t must contrbute as one component of the graph G p k. Moreover, for any nteger m 0, p k mp k and hence mp x 0 mod p k s solvable wth x k as ts root. Thus each vertces of the set {0,p,2p,..., p 1 p k 1 } s adjacent wth 0. Then V Γ p {0,p,2p,..., p 1 p k 1 } wll form another component of the graph G p k. Next we clam that none of the vertex Γ φ p k s adjacent wth any of the vertex Γ p. To prove our asserton, let u and v be two vertces of G p k such that u Γ φ p k and v Γ p. Then there exst ntegers r and s such that u r 1 ( mod p k) (, v s 0 mod p k). 4.2

8 8 Appled Mathematcs Let t lcm r, s, then there exst ntegers r 1 and s 1 such that t rr 1 ss 1. Hence by 4.2, we get u t 1 ( mod p k) (, v t 0 mod p k). 4.3 Now, f u and v are adjacent, then there must be some nteger α such that u α v mod p k. Then by 4.3, weobtan,0 1 mod p k whch s absurd. Ths s through the clam. Hence Γ φ n and Γ p are the only components of G p k. The part b s the drect consequence of Theorem 2.6. Corollary 4.3. For k 1, V Γ p {0}. Two graphs or two components of a graph are sad to be somorphc to each other f there s an somorphsm between ther vertex sets. Thus, f G n and G m are somorphc, then V G n V G m. Moreover f the vertces u and v are adjacent n G n, then f u and f v must be adjacent n G m, where f s an somorphsm between ther sets of vertces. The followng corollary can easly be proved by usng Lemma 4.2 and Corollary 3.7. Corollary 4.4. If n 2p 1 p 2 p r and l, m are prme dvsors of n,thenγ l and Γ m, are nonsomorphc components of G n. Proof. Snce both l and m are prme numbers, so by Lemma 4.2 b, the order of components Γ l and Γ m must be dfferent. Hence both can never be somorphc. For nstance, take n 30, l 3, and m 5. Then by Lemma 4.2 b, 30 V Γ V Γ 5 5 ( ( ) ) , Hence by 4.4, the components Γ 3 and Γ 5 are not somorphc n G 30. However, V Γ 3 V Γ 6 4. Thus the somorphsm can be establshed between the components Γ 3 and Γ 6 n G 30 f they possess the same structure as well. Ths leads to the followng corollary. Corollary 4.5. Let n 2p 1 p 2 p r and d n. IfΓ d and Γ 2d are the components of G n, then Γ d Γ2d. Proof of Man Theorem. a It s well known that a x b mod p k 1 r s solvable f and only f the congruence a x b mod p k s solvable for each 1, 2,...,r. Then the proof s analogous to Theorem 2.5 together wth Lemma 4.2. b It s easy to see that each element of V Γ p {0,p,..., p 1 p} n G p 2 s connected wth 0 snce for each α 2, tp α 0 mod p 2,1 t p 1 s satsfed. Ths shows that the congruence a x 0 mod p 2 s solvable for each a V Γ p n G p 2. To show that Γ p n G p 2 s a tree, we clam that the nonzero vertces n V Γ p are not connected to each other. To prove our asserton, let t 1 p α t 2 p mod p 2 for α 2and1 t 1 <t 2 p 1. Then by Cancelaton Law of congruences, we obtan, t α 1 pα 1 t 2 mod p. Asp t α 1 pα 1 t 2 and p t α 1 pα 1,soby

9 Appled Mathematcs 9 the smple dvsblty rules, p t 2.Butt 2 p 1 <p, thus we arrve at a contradcton. Ths through our clam and fnally the component Γ p n G p 2 s a tree wth root at zero. c It s wellknown that F n, n>4 s always composte. Thus the only Fermat s prmes are 3, 5, 17, 257, and Let us wrte n 2 k 1, k 1, 2, 4, 8, 16. Then φ n 2 k, k 1, 2, 4, 8, 16 snce n s prme. To show that the component Γ φ n s complete, we need to show that the congruence a x b mod n s solvable for each a, b {1, 2,...,n 1}. By 9, every odd prme p has a prmtve root. Snce Fermat s numbers are always odd, so each of the Fermat s prme has a prmtve root. Let t be r. Then the congruence a x b mod n s solvable f and only f the congruence x nd r a nd r b mod 2 k s solvable, where k 1, 2, 4, 8, 16. As nd r a,nd r b {1, 2,...,n 1}, so the later congruence reduced to the lnear congruence αx β mod 2 k, where α nd r a and β nd r b. Then the smple graph G n wll be complete f the congruence αx β mod 2 k, or the congruence βx α mod 2 k, s solvable. To prove our asserton, t s easy to see that ether α, 2 k 1 β, 2 k or α, 2 k 2 t β, 2 k, where 1 t 2 k. For the frst case, the congruence αx β mod 2 k, s solvable and has a unque soluton. To dscuss the rest of the case, we let α, 2 k 2 t 1 and β, 2 k 2 t 2,1 t 1, t 2 2 k.ift 1 t 2, then α, 2 k β, 2 k, so the lnear congruence αx β mod 2 k s solvable and has 2 t 1 solutons. But f t 2 <t 1, we replace α and β and solve the congruence βx α mod 2 k, whch s solvable and has 2 t 2 solutons. Thus n ether case, we see that there s an edge α, β, α, β {1, 2,...,n 1} n component Γ φ n of the smple graph G n when n s a Fermat s prme. The followng corollares are the drect consequences of Theorem 4.1. Corollary 4.6. Let n t 2,wheret s square free nteger. Then Γ t s a tree wth root at zero. Corollary 4.7. If n s Fermat prme, then Γ φ n s regular of degree φ n Conclusons Ths pece of work descrbes the relatonshp of exponental congruences wth graphs. It has been explored that an exponental congruence yelds a set of graphs. Also certan components of the graphs form trees f self-loops are suppressed. The types of graphs and trees were hence yelded form a pattern based on the nature of varables wthn the congruence. The major results formed show that the component Γ φ n ofthesmplegraphg n s complete f n s Fermat s prme and also that the component Γ t 2, where t s a square free nteger, s always a tree. Intutvely, the results formed fnd ther place n varous applcatons of number theory, encrypton, and algorthms. In varous problems of data decrypton usng brute force method t s desrable to fnd the nature of a number and establsh f t s dvsble by some other prme number or not. Such problems can be tackled by formng a graph of ts exponental congruence and determne the pattern formed by ts components. Moreover these exponental congruences can be establshed as a succnct representaton of graphs of a specfc nature for ther applcatons n varous computer algorthms. Acknowledgment The authors are very thankful to the referees for ther valuable comments and advces. Ths has made the paper more nterestng and nformatve.

10 10 Appled Mathematcs References 1 J. Skowronek-Kazów, Some dgraphs arsng from number theory and remarks on the zero-dvsor graph of the rng Z n, Informaton Processng Letters, vol. 108, no. 3, pp , B. Wlson, Power dgraphs modulo n, The Fbonacc Quarterly, vol. 36, no. 3, pp , L. Somer and M. Křížek, On a connecton of number theory wth graph theory, Czechoslovak Mathematcal Journal, vol. 54, no. 2, pp , G. Deng and P. Yuan, Symmetrc dgraphs from powers modulo n, Open Dscrete Mathematcs, vol. 1, no. 3, pp , G. Deng and P. Yuan, On the symmetrc dgraphs from powers modulo n, Dscrete Mathematcs, vol. 312, no. 4, pp , G. Chartrand and L. Lesnak, Graphs and Dgraphs,, Chapman & Hall, London, UK, 3rd edton, A. Adler and J. E. Coury, The Theory of Numbers, Jones and Bartlett Publshers, Boston, Mass, USA, K. H. Rosen, Dscrete Mathematcs and ts Applcaton, WCB/McGraw-HllPress, New York, NY, USA, T. Koshy, Elementry Numbers Theory wth Applcatons, Academc Press, Elsever Inc., New York, NY, USA, 2007.

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