NUMBER OF SPANNING TREES OF NEW JOIN GRAPHS

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1 Available olie at Algebra Letters, 03, 03:4 ISSN 0-0 NUMBER OF SPANNING REES OF NEW JOIN GRAPHS S. N. DAOUD, Departmet of Applied Mathematics, Faclty of Applied Sciece, aibah Uiversity, Al-Madiah, K.S.A. Departmet of Mathematics, Faclty of Sciece, El-Mifiya Uiversity, Shebee El-Kom, Egypt Abstract. I mathematics, oe always tries to get ew strctres from give oes. his also applies to the realm of graphs, where oe ca geerate may ew graphs from a give set of graphs. I this paper we derive simple formlas of the complexity, mber of spaig trees, of some ew oi graphs, sig liear algebra, Chebyshev polyomials ad matrix aalysis techiqes. Keywords: Nmber of spaig trees;chebyshev polyomials; Joi graphs. Mathematics Sbect Classificatio: 0C0, 0C0.. Itrodctio I this workwe deal with simple ad fiite directed graphs G ( V, E ), where V is the vertex setad E is the edgeset. For a graph G, a spaig tree i G is a tree which has the same vertex set as G. he mber of spaig trees i G, also called, the complexity of the graph, deoted by ( G ), is a well-stdied qatity (for log time). A classical reslt of Kirchhoff, [9] ca be sed to determie the mber of spaig trees for G ( V, E ).Let V { v, v,..., v }, the the Kirchhoff matrix H defied as characteristic matrix H D A,where D is the diagoal matrix of the degrees of G ad A is the adacecy matrix of G, H [ a i ] defied as follows: (i) ai v i ad v are adacet ad i, (ii) a i eqals the degree of vertex v i if i, ad (iii) a 0 otherwise. All of co-factors of H are eqal to ( G ). here are i Received April3, 03

2 S. N. DAOUD other methods for calclatig ( G ). Let... p deote the eigvalesof H matrix of a p poit graph. he it is easily show that p 0 k. Frthermore, p Kelmas ad Chelokov [8] show that, ( G ).he formla for the k p mber of spaig trees i a d-reglar graph G p ( G) ( d k ) p k ca be expressed as where 0 d,,,..., p are the eigevales of the correspodig adacecy matrix of the graph. However, for a few special families of graphs there exists simple formlas that make it mch easier to calclate ad determie the mber of correspodig spaig trees especially whe these mbers are very large. Oe of the first sch reslt is de to Cayley[4] who showed that complete graph o vertices, ( ) K, K has spaig trees that he showed q p. Aother reslt, ( K ) p q, p, q, where K, is the pq, complete bipartite graph with bipartite sets cotaiig p ad q vertices, respectively. It is well kow, as i e.g., [,]. Aother reslt is de to Sedlacek [3] who derived a formla for the wheel o vertices, W, he showed that 3 3 ( W ) ( ) ( ), for 3. Sedlacek [4] also later derived a formla for the mber of spaig trees i a Mobis ladder, M ( M ) [( 3) ( 3) ] for.aother class of graphs for which a explicit formla has bee derived is based o a prism.boesch, et.al. [,3]. Daod,[6-6] later derived formlas for the mber of spaig trees for may graphs. he Lcas mbers are the seqece { L } defied by the liear recrrece eqatio pq, L L L With L0, L ad L 3. he vales of L for,,3,... are (),3,4,7,,8,9,47,76,3,...,[,7 ]. he Fiboacci mbers are the seqece { F } defied by the liear recrrece eqatio: F F F ()

3 NUMBER OF SPANNING REES OF NEW JOIN GRAPHS 3 With F0 0 ad F F. he vales of F for,,3,... are,,,3,,8,3,,...,[]. Now, we ca itrodce the followig lemma: Lemma. ( G ) det( I D A ) where A, D are the adacecy ad degree matrices of G, the complemet of G, respectively, ad I is the it matrix. he advatageof thisformla is to express ( G ) directly as a determiat rather tha i terms of cofactors as i Kirchhoff theorem or eigevales as i Kelmas ad Chelokov formla.. Chebyshev Polyomial I this sectio we itrodce some relatios cocerig Chebyshev polyomials of the first ad secod kid which we se it i or comptatios. We begi from their defiitios,yapig, et. al. []. Let A be matrix sch that: x 0 0 A x, where all other elemets are zeros. Frther we recall that the Chebyshev polyomials of the first kid are defied by: ( x ) cos( arccos x ) (3) he Chebyshev polyomials of the secod kid are defied by d si( arccos x ) U ( x ) ( x ) dx si(arccos x ) (4) It is easily verified that U ( x ) xu ( x ) U ( x ) 0 () It ca the be show from this recrsio that by expadig det A ( x ) oe gets U ( x ) det( A ( x )), Frthermore by sig stadard methods for solvig the recrsio (3), oe obtais the explicit formla U x x x x x x ( ) [( ) ( ) ], (6) (7)

4 4 S. N. DAOUD Where the idetity is tre for all complex x (except at x where the fctio ca be take as the limit). he defiitio of U easily yields its zeros ad it ca therefore be verified that U ( x ) ( x cos ) (8) Oe frther otes that U ( x ) ( ) U ( x ) (9) hese two reslts yield aother formla for U ( x ), U ( x ) 4 ( x cos ) (0) Fially, simple maiplatio of the above formla yields the followig, which also will be extremely sefl to s latter: U x ( ) ( x cos ) () 4 Frthermore oe ca show that ad U ( x ) [ ] [ (x )] ( x ) ( x ), () ( x ) [( x x ) ( x x ) ]. (3) Aother iterestig fact follows by comparig (7) with the will kow closed form formla for the Fiboacci mbers F. F [( ) ( ) ] (4) amely, U ( ) F [( ) ( ) ] () Also we ca prove that : ( ) L ( ) ( ) (6)

5 NUMBER OF SPANNING REES OF NEW JOIN GRAPHS Lemma. Let B ( x ) be matrix sch that: x 0 0 x 0 B 0 x 0 0 x he Proof: x det( B ( x )) ( x ) U ( ). Straightforward idctio sig properties of determiats ad above metioed defiitio. Lemma. Let C be matrix, 3, x sch that: x 0 0 x C x 0 0 x he x det( C ( x )) ( x ) U ( ) Proof: Straightforward idctio sig the properties of determiats ad lemma.. Lemma.3 Let D be matrix, 3, x 4 sch that : x D x ( x 3) x he: det( D( x )) [ ( ) ] x 3 Proof: Straightforward idctio sig properties of determiats ad above metioed defiitios of Chebyshev polyomial of the first ad secod kid.

6 6 S. N. DAOUD Lemma.4 Let E be matrix, x sch that: he x E x det( E ) ( x )( x ) Proof From the defiitio of the circlat determiats, we have: x 3 det( E ( x )) det ( x... ) Lemma.,[0] x 3 ( x... ) ( x... ), ( x ) ( x ). m Let A, B, C F F m F m m ad D F assme that AD,, are osiglar matrices. he: A B C D m m det ( ) det( A BD C ) det D ( ) det A det( D CA B ) Formla i lemma. give some sort of symmetry i some matrices which facilitates or calclatio of determiats. 3. Complexity of some ew oi Graphs Defiitio 3. he oi graph P N with 3 vertices, is formed from a simple path P with vertices,,,..., ad ll graph N with vertices, v, v,..., v sch that v N adacet with ad i P ad v N adacet with ad i P ad so o. See Fig. (3).

7 NUMBER OF SPANNING REES OF NEW JOIN GRAPHS 7 v v 4 v 3 v v Fig(3) : P0 N heorem 3. For, ( P N ) L. Proof: Applyig lemma., we have: ( P N ) det(3 I D A ) (3 ) A( ) B ( ) det det (3 ) (3 ) B C ( ) ( ) detc det( A BC B ) (3 ) det ( ) det (3 ) Straightforward idctio sig relatios betwee Lcas mbers ad properties of Chebyshev polyomials of first kid ad secod kid, yields:

8 8 S. N. DAOUD (3 ) 3 3 ( P N ) [ ( )][ ( )] ( ) (3 ) 3 3 [( ) ( ) ] [(3 ) (3 ) ] L. Defiitio 3.3 he oi graph P P with 3 vertices, is formed from a simple path P with vertices,,,..., ad path P with vertices, v, v,..., v sch that v P adacet with ad i P ad v Padacet with ad o. See Fig. (). v P i ad so v 4 v 3 v v Fig.(): P0 N heorem 3.4 Let For, ( P P ) L. Proof: Applyig lemma., we have: ( P P) det(3 I D A ) (3 ) (3 ) A( ) B ( ) det det (3 ) ( ) ( ) 4 0 B C (3 ) detc det( A BC B ) Straightforward idctio sig properties of determiats ad relatios of Chebyshev polyomials of first kid ad secod kid, yields:

9 NUMBER OF SPANNING REES OF NEW JOIN GRAPHS 9 3 ( ) U ( ) ( P P ) [( ) U ()][ ] ( ) U ( ) (3 ) 3 (3 ) ( ) U () [(3 ) (3 ) ][( ) ( ) ]. Defiitio 3. he oi graphc N o 3 vertices, is formed from a simple path C with vertices,,,..., ad ll N with vertices, v, v,..., v sch that v N adacet with ad i C ad v N adacet with ad i C o. See Fig. (3). ad so 6 v 3 3 v v 4 Fig(3) : C6 N 3 heorem 3.6: for, ( C N ) F. Proof: Applyig lemma., we have: ( C N ) det(3 I D A ) (3 ) A( ) B ( ) det det (3 ) (3 ) B C ( ) ( ) detc det( A BC B ) (3 )

10 0 S. N. DAOUD Straightforward idctio sig relatios betwee Fiboacci mbers ad properties of Chebyshev polyomials of first kid ad secod kid, yields: ( C N ) [(3 ) (3 ) ] F. Defiitio 3.7 he oi graphc P with 3 vertices, is formed from a simple path C with vertices,,,..., ad path P with vertices, v, v,..., v sch that v P adacet with ad i C ad v Padacet with ad i C o. See Fig. (). ad so 6 v 3 3 v v 4 Fig.(4): C6 P3 heorem3.8 For, ( C P) [(3 ) (3 ) ][( ) ( ) ]. 0 Proof: Applyig lemma 3., we have: ( C P) det(3 I D A ) (3 ) (3 ) A( ) B ( ) det det (3 ) ( ) ( ) 4 0 B C

11 NUMBER OF SPANNING REES OF NEW JOIN GRAPHS (3 ) detc det( A BC B ) Straightforward idctio sig properties of determiats ad relatios of Chebyshev polyomials of first kid ad secod kid, yields: ( C P ) [(3 ) (3 ) ][( ) ( ) ];. 0 4.Coclsio he mber of spaig trees ( G ) i graphs (etworks) is a importat ivariat. he evalatio of this mber is ot oly iterestig from a mathematical (comptatioal) perspective, bt also, it is a importat measre of reliability of a etwork ad desigig electrical circits. Some comptatioally hard problems sch as the travellig salesma problem ca be solved approximately by sig spaig trees. De to the high depedece of the etwork desig ad reliability o the graph theory we itrodced the above importat theorems ad lemmas ad their proofs.. Ackowledgemets he athor is deeply idebted ad thakfl to the deaship of the scietific research for his helpfl ad for distict team of employees at aibah iversity, Al- Madiah Al-Mawarah, K.S.A. for their cotios helps ad the ecoragemet s to ialize this work. his research work was spported by a grat No. (30/434) from the deaship of the scietific research at aibah iversity, Al-Madiah Al- Mawwarah, K.S.A. REFERENCES [] Adrews G., Nmber theory,w.b. Saders Compay, Lodo, (97). [] Boesch, F.. ad Bogdaowicz, Z. R. : he mber of spaig trees i a Prism, Iter. J. Compt. Math.,, 9-43 (987). [3] Boesch, F.. ad Prodiger, H., Spaig tree Formlas ad Chebyshev Polyomials, J. of Graphs ad Combiatorics, 9-00 (986). [4] Cayley, G. A.,A heorm o rees, Qart. J. Math. 3, (889). [] Clark, L., O the emeratio of mltipartite spaig trees of the complete graph, Bll. Of the ICA 38, 0 60 (003). [6] Daod S. N., Some Applicatios of spaig trees i Complete ad Complete bipartite graph : America Joral of Applied Sci. Pb.,9(4),84-9, (0). [7] Daod S. N., Complexity of Some Special amed Graphs ad Chebyshev polyomials Iteratioal oral applied Mathematical ad Statistics,Ceser Pb. Vol. 3, No.,77-84, (03).

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