What are primes in graphs and how many of them have a given length?

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1 What are pries in graphs an how any of the have a given length? Aurey Terras Math. Club Oct. 30, 2008 A graph is a bunch of vertices connecte by eges. The siplest exaple for the talk is the tetraheron K 4. A prie in a graph is a close path in the graph iniizing the nuber of eges traverse. This eans no backtrack, no tails. Go aroun only once. Orientation counts. Starting point oesn t count. The length of a path is the nuber of eges in the path. The talk concerns the prie nuber theore in this context. Degree of a vertex is nuber of eges incient to it. Graph is regular if each vertex has sae egree. K 4. is 3-regular. We assue graphs connecte, no egree vertices, an graph is not a cycle.

2 Exaples of pries in K 4 e 3 e2 e 5 e 4 e Here are 2 of the in K 4 : [C] =[e e 2 e 3 ] ={e e 2 e 3, e 2 e 3 e, e 3 e e 2 } The [] eans that it oes not atter where the path starts. [D]=[e 4 e 5 e 3 ] [E]=[e e 2 e 3 e 4 e 5 e 3 ] ν(c) =length C= # eges in C ν(c)=3, ν(d)=3, ν(e)=6 E=CD another prie [C n D], n=2,3,4, infinitely any pries assuing the graph is not a cycle or a cycle with hair. Ihara Zeta Function Definition ζ ( u, X ) = ( C ) ν ( u ) [ C ] prie u coplex nuber u sall enough Ihara s Theore (Bass, Hashioto, etc.) A = ajacency atrix of X= V x V atrix of 0s an s with i,j entry iff vertex i ajacent to vertex j Q = iagonal atrix; jth iagonal entry = egree jth vertex -; r = E - V + ζ 2 r 2 (, ux) = ( u) et( I Au+ Qu) 2

3 Labeling Eges of Graphs Orient the eges; i.e., put arrows on the. Label the as follows. Here the inverse ege has opposite orientation. e,e 2,,e, e + =(e ) -,,e 2 =(e ) - e e 7 Note that t these irecte eges are our alphabet neee to express paths in the graph. The Ege Matrix W Define W to be the 2 E 2 E atrix with i j entry if ege i fees into ege j, (en vertex of i is start vertex of j) provie that j the inverse of i, otherwise the i j entry is 0. i j Theore. ζ(u,x) - =et(i-wu). Corollary. The poles of Ihara zeta are the reciprocals of the eigenvalues of W. The pole R of zeta is the closest to 0 in absolute value. R=/Perron-Frobenius eigenvalue of W; i.e., the largest eigenvalue which has to be positive real. See Horn & Johnson, Matrix Analysis, Chapter 8. 3

4 Exaple. W for the Tetraheron Label the eges The inverse of ege j is ege j+6. e 3 e 5 e W = e 4 e 6 e e 7 There are eleentary proofs of the 2 eterinant forulas for zeta, even for X irregular. See y book on y website eu/~aterras/newbook.pf If you are willing to believe these forulas, we can give a rather easy proof of the graph theory prie nuber theore. But first state the prie nuber th an give soe exaples. 4

5 The Prie Nuber Theore π X () = # {pries [C] in X of length } Δ = greatest coon ivisor of lengths of pries in X R = raius of largest circle of convergence of ζ(u,x) If Δ ivies, then π X () Δ R - /, as. R=/q, if graph is q+- regular The proof involves forulas like the following, efining N = # {close paths of length where we count starting point an orientation} log ζ ( u, X) u = Nu u = The proof is siilar to one in Rosen, Nuber Theory in Function Fiels, p Exaples K 4 an K 4 -ege 4 g ζ ζ ( uk, ) 4 = ( u ) ( u )( 2 u )( + u+ 2 u ) ( u, K e) 4 = ( u )( u)( + u )( + u + 2 u )( u 2 u ) 5

6 N for the exaples Exaple. The Tetraheron K 4. N =# close paths of length log ζ ( x, G) x = Nx x x /x log ζ (x,k 4 ) = 24x x x x x x 9 + π(3)=8 ( ) (orientation counts) π (4)=6 π (5)=0 we will show that: 6 6 = N = π( ) = π() + 2 π(2) + 3 π(3) + 6 π(6) Δ = g.c.. lengths of pries = π (6) = 24 Exaple 2. The Tetraheron inus an ege. x /x log ζ (x,k 4 -e) =2x 3 + 8x x x 7 + 8x x 9 + π(3)=4 π (4)=2 π (5)=0 π(6)=2 Δ = g.c.. lengths of pries = Poles of Zeta for K 4 are {,,,-,-,½,r +,r +,r +,r -,r -,r - } where r ± =(-± -7)/4 an r =/ 2 R=½=Pole closest to 0 The prie nuber th π() Δ R - /, as. becoes K 4 π() 2 /, as. Poles of zeta for K 4 -e are {,,-,i,-i,r +,r -,α,β,β } R = α real root of cubic.6573 β coplex root of cubic The prie nuber th becoes for /α.5 K 4 -e π().5 /, as. 6

7 Proof of Prie Nuber Theore Start with the efinition of zeta as a prouct over pries. Take log an a erivative. Use Taylor series. u u X u u u u ν ( C) log ζ (, ) = log( ) [ C] prie = u u = ν ( C) u u j ( C ) ( u ν ) ζ ( ux, ) = ν( C) j ν( C) j [ C] j [ C] j prie j ν ( C ) u log ζ ( u, X ) = u u = j C prie path N u [ C ] prie Taylor Series for log(-x) #[C]=ν(C) non-pries are powers of pries This copletes the proof of the first forula () u log ζ ( u, X) = Nu u Next we note another forula for the zeta function coing fro the original efinition. ( C ) ( u ν ) ζ ( ux, ) = [ C ] prie Recall π(n) = # pries [C] with ν(c)=n n ( u) n π ( n) ζ ( ux, ) =. 7

8 n ( u) n π ( n) ζ ( ux, ) =. n nπ ( n) u u log ζ ( u, X) = = ( ) n π u u n u If you cobine this with forula, which was u log ζ ( u, X) = Nu u Taylor Series for (-x) - you get our 2n forula saying N is a su over the positive ivisors of : (2) N = π ( ) (3) N = π ( ) This is a ath 04 (nuber theory) -type forula an there is a way to invert it using the Mobius function efine by:, if n = μ( n) = 0, if n not square free r ( ), n = p p, r with piistinct pries π ( ) = N μ To coplete the proof, we nee to use one of our 2 eterinant forulas for zeta. ζ(u,x) - =et(i-wu). 8

9 Fact fro linear algebra - Schur Decoposition of a Matrix (Math 02) There is an orthogonal atrix Q (i.e., QQ t =I) an T=upper triangular with eigenvalues of W along the iagonal such that W=QTQ -. So we see that T * * * 0 2 * * = 0 0 * Det I uw Det I uqtq ( ) = ( ) = Det( I ut ) = u i= ( ) i ζ(u,x) - =et(i-wu) = ( u) eigenvalue of W u log ζ ( u, X) = u u log( u) u u u It follows that (4) = N eigenvalue of W = eigenvalue of W = eigenvalue of W ( u) 9

10 N = eigenvalue of W The ain ters in this su coe fro the largest eigenvalues of W in absolute value. There is a theore in linear algebra that you on t learn in a st course calle the Perron-Frobenius theore. (Horn & Johnson, Matrix Analysis, Chapter 8). It applies to our W atrices assuing we are looking at connecte graphs (no egree vertices) & not cycles. The easiest case is that Δ=g.c.. lengths of pries =. Then there is only eigenvalue of W of largest absolute value. It is positive an is calle the Perron-Frobenius eigenvalue. Moreover it is /R, R=closest pole of zeta to 0. So we fin that if Δ=, N R -, as. If Δ=, N R -, as. To figure out what happens to π(), use π ( ) = N μ This allows you to prove the prie nuber theore when Δ= p X () R - /, as. For the general case, see y book 0

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