Some Trigonometric Identities Involving Fibonacci and Lucas Numbers

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1 Joural of Iteger Sequeces, Vol. 12 (2009), Article Some Trigoometric Idetities Ivolvig Fiboacci ad Lucas Numbers Kh. Bibak ad M. H. Shirdareh Haghighi Departmet of Mathematics Shiraz Uiversity Shiraz Ira khmath@gmail.com shirdareh@susc.ac.ir Abstract I this paper, usig the umber of spaig trees i some classes of graphs, we prove the idetities: F = 2 (1 cos kπ 3kπ cos ), 2, (1 + 4 si 2 kπ ) = L 2 2 = F 2+2 F 2 2 2, 1, where F ad L deote the Fiboacci ad Lucas umbers, respectively. Also, we give a ew proof for the idetity: F = (1 + 4 si 2 kπ ) = (1 + 4 cos 2 kπ ), 4. 1 Itroductio Let F ad L deote the Fiboacci ad Lucas umbers respectively. That is, F +2 = F +1 + F, for 1 with F 1 = F 2 = 1, ad L +2 = L +1 + L, for 1 with L 1 = 1 ad L 2 = 3. 1

2 I this paper, we derive the idetities: F = 2 (1 cos kπ 3kπ cos ), 2, (1) (1 + 4 si 2 kπ ) = L 2 2 = F 2+2 F 2 2 2, 1. (2) To prove idetity (1), we apply the umber of spaig trees i a special class of graphs kow as circulat graphs. Idetity (2) is derived from the umber of spaig trees i a wheel. Applyig the same techique to a graph kow as fa gives us a ew proof for the followig idetity: F = (1 + 4 si 2 kπ ) = (1 + 4 cos 2 kπ ), 4, (3) appeared i [6] ad its correspodig refereces. Also, applyig this techique to the path P ad the cycle C gives us a ew proof for the well-kow idetities: si kπ 2 = si kπ = 2 Techiques ad Proofs 2, 2, (4) 2, 2. (5) For a graph G, a spaig tree i G is a tree which has the same vertex set as G. The umber of spaig trees i a graph (etwork) G, deoted by t(g), is a importat ivariat of the graph (etwork). It is also a importat measure of reliability of a etwork. I the sequel, we assume our graphs are loopless but multiple edges are allowed. A famous ad classic result o the study of t(g) is the followig theorem, kow as the Matrix-tree Theorem. The Laplacia matrix of a graph G is defied as L(G) = D(G) A(G), where D(G) ad A(G) are the degree matrix ad the adjacecy matrix of G, respectively. Sice this theorem is first proved by Kirchhoff [7], L(G) is also kow as the Kirchhoff matrix of the graph G. Theorem 1. For every coected graph G, t(g) is equal to ay cofactor of L(G). The umber of spaig trees of a coected graph G ca be expressed i terms of the eigevalues of L(G). Sice by defiitio, L(G) is a real symmetric matrix, it therefore has o-egative real eigevalues, where is the umber of vertices of G. Aderso ad Morley 2

3 [1, Theorem 1] proved that the multiplicity of 0 as a eigevalue of L(G) equals the umber of compoets of G. Therefore, the Laplacia matrix of a coected graph G has 0 as a eigevalue with multiplicity oe. Theorem 2. ([5]) Suppose G is a coected graph with vertices. Let λ 1,...,λ be the eigevalues of L(G), with λ = 0. The t(g) = 1 λ 1 λ. As the first example, we prove idetity (4). Proof of idetity (4). Cosider the path P. It is kow that the eigevalues of the Laplacia matrix of P are 2 2 cos kπ (0 k 1) (see, e.g., [4]). O the other had, we kow that t(p ) = 1, therefore by usig Theorem (2) we obtai (4). Now, we state some more defiitios ad theorems. Defiitio 3. A matrix C = (c ij ) is called a circulat matrix if its etries satisfy c ij = c 1, j i+1, where subscripts are reduced modulo ad lie i the set {1, 2,...,}. Defiitio 4. Let 1 s 1 < s 2 < < s k <, where ad s 2 i (1 i k) are positive itegers. A udirected circulat graph C (s 1,s 2,...,s k ) is a 2k-regular graph with vertex set V = {0, 1,..., 1} ad edge set E = {{i,i + s j ( mod )} i = 0, 1,..., 1, j = 1, 2,...,k}. The Laplacia matrix of C (s 1,s 2,...,s k ) is clearly a circulat matrix. By a direct usig of Theorem 4.8 of [12], we obtai the followig lemma: Lemma 5. The ozero eigevalues of L(C (s 1,s 2,...,s k )) are where ω = e 2πi. 2k ω s 1j ω s kj ω s 1j ω s kj, 1 j 1, With combiig Theorem 2 ad the lemma above, we obtai the followig corollary: Corollary 6. The umber of spaig trees i G = C (s 1,s 2,...,s k ) is equal to: t(g) = 1 ( k (2 2 cos 2js iπ )). j=1 i=1 Proof of idetity (1). Cosider the square cycle C (1, 2). We ca use Corollary 6 to obtai the umber of spaig trees of C (1, 2). O the other had, Kleitma ad Golde [8] proved that t(c (1, 2)) = F. 2 Now, with a little additioal algebraic maipulatio, idetity (1) follows. Proof of idetity (5). Look at the cycle C (1) = C. We kow that t(c ) =, therefore by applyig Corollary 6 to it, (5) follows. Defiitio 7. The joi W = C K1 of a cycle C ad a sigle vertex is referred to as a wheel with spokes. Similarly, the joi F = P K1 of a path P ad a sigle vertex is called a fa. 3

4 Sedlacek [11] ad later Myers [10] showed that t(w ) = L 2 2 = F 2+2 F 2 2 2, 1. Also, Bibak ad Shirdareh Haghighi [2, 3] proved that t(f ) = F 2, 1. Now, we fid the umber of spaig trees i W ad F by applyig Theorem 2. We first eed to determie the eigevalues of L(W ) ad L(F ). Theorem 8. ([9]) Let G 1 ad G 2 be simple graphs o disjoit sets of r ad s vertices, respectively. If S(G 1 ) = (µ 1,...,µ r ) ad S(G 2 ) = (ν 1,...,ν s ) are the eigevalues of L(G 1 ) ad L(G 2 ) arraged i oicreasig order, the the eigevalues of L(G 1 G2 ) are = r+s; µ 1 + s,...,µ r 1 + s; ν 1 + r,...,ν s 1 + r; ad 0. Sice the eigevalues of L(C ) are 2 2 cos 2kπ (0 k 1) (by Lemma 5), ad the eigevalues of L(P ) are 2 2 cos kπ (0 k 1), therefore, by Theorem 8 we ca determie the eigevalues of L(W ) ad L(F ). Theorem 9. The eigevalues of L(W ) are + 1, 0 ad si 2 kπ (1 k 1), ad the eigevalues of L(F ) are + 1, 0 ad si 2 kπ (1 k 1) (or + 1, 0 ad cos 2 kπ (1 k 1) ). 2 Proofs of the idetities (2) ad (3). By Theorems 2 ad 9, the umber of spaig trees of W ad F are, respectively, t(w ) = (1 + 4 si 2 kπ ), 1, t(f ) = (1 + 4 si 2 kπ 2 ) = (1 + 4 cos 2 kπ ), 2. 2 O the other had, as we already referred, t(w ) = L 2 2 = F 2+2 F 2 2 2, 1 ad t(f ) = F 2, 1. Therefore, we obtai (2) ad (3). Refereces [1] W. N. Aderso ad T. D. Morley, Eigevalues of the Laplacia of a graph, Liear Multiliear Algebra 18 (1985), [2] Kh. Bibak ad M. H. Shirdareh Haghighi, Recursive relatios for the umber of spaig trees, Appl. Math. Sci. 3 (2009), [3] Kh. Bibak ad M. H. Shirdareh Haghighi, The umber of spaig trees i some classes of graphs, Rocky Moutai J. Math., to appear. [4] A. E. Brouwer, A. M. Cohe ad A. Neumaier, Distace-Regular Graphs, Spriger- Verlag, [5] D. Cvetkovič, M. Doob ad H. Sachs, Spectra of Graphs: Theory ad Applicatios, third ed., Joha Ambrosius Barth,

5 [6] N. Garier ad O. Ramaré, Fiboacci umbers ad trigoometric idetities, Fiboacci Quart. 46 (2008), 1 7. [7] G. Kirchhoff, Über die Auflösug der gleichuge auf, welche ma bei der utersuchug der lieare verteilug galvaischer Ströme geführt wird, A. Phy. Chem. 72 (1847), [8] D. J. Kleitma ad B. Golde, Coutig trees i a certai class of graphs, Amer. Math. Mothly 82 (1975), [9] R. Merris, Laplacia graph eigevectors, Liear Algebra Appl. 278 (1998), [10] B. R. Myers, Number of spaig trees i a wheel, IEEE Tras. Circuit Theory 18 (1971), [11] J. Sedlacek, O the skeletos of a graph or digraph, I Proc. Calgary Iteratioal Coferece o Combiatorial Structures ad their Applicatios, Gordo ad Breach, 1970, pp [12] F. Zhag, Matrix Theory: Basic Results ad Techiques, Spriger-Verlag, Mathematics Subject Classificatio: Primary 11B39, Secodary 05C05, 15A18. Keywords: Fiboacci umbers, Lucas umbers, spaig tree, trigoometric idetity. (Cocered with sequeces A ad A ) Received November ; revised versio received November Published i Joural of Iteger Sequeces, November Retur to Joural of Iteger Sequeces home page. 5