On Characteristic Polynomial of Directed Divisor Graphs

Transcription

1 Iter. J. Fuzzy Mathematical Archive Vol. 4, No., 04, 47-5 ISSN: (P), (olie) Published o April 04 Iteratioal Joural of V. Maimozhi a ad V. Kaladevi b a Departmet of mathematics, Chidambarampillai College for Wome Maachaallur, Trichy, Tamiladu, Idia b Departmet of mathematics, Seethalakshmi Ramaswami College, Trichy Tamiladu, Idia Received February 04; accepted March 04 Abstract. Let N be the set of all positive itegers ad let S be the fiite subset of N. A divisor graph G(S) is a graph whose vertex set is labeled from the set V=S ad ay two vertices, u,v of v are adjacet if either u divides v or v divides u. The divisor graph is a directed divisor graph if all the edges are directed from low to high levels. I this paper the characteristic polyomials of directed divisor graphs cycle, path & Bistar graphs are obtaied from the correspodig skew adjacecy matrix. Keywords: Divisor graph, directed divisor graph, skew adjacecy matrix. AMS Mathematics Subject Classificatio (00): 05C6. Itroductio Let DD be a directed divisor graph of order with vertex set V(DD) = {v,v, v } ad arc set Γ(DD) V(DD) x V(DD). Throughout this paper the graph DD has o loops ad multiple arcs. The skew adjacecy matrix of DD is the x matrix. if ( vi, v j ) Γ( DD) ( DD) = Sij where sij = if ( v j, vi ) Γ( DD) 0 otherwise Because of the assumptios o DD, S(DD) is ideed a skew-symmetric matrix. Hece the eigevalue { λ, λ,... λ } of S(DD) are all purely imagiary umbers, ad the λ, λ,... λ of its eigevalues. Cosequetly, the eergy of S(DD), which is defied as the sum of the absolute values of its eigevalues. The eergy or the skew eergy of S(DD) is deoted by sigular values of S(DD) coicide with the absolute values { } E s (DD) ad is defied by λ. i i= Defiitio.. Let N be the set of all positive itegers ad S be the fiite subset of N. A divisor graph G(S) is a graph whose vertex set is labeled from the set V=S ad ay two vertices u, v of V are adjacet if either u divides v (or) v divides u. The divisor graph is a directed divisor graph if all the edges are directed from low to high. 47

2 V. Maimozhi ad V. Kaladevi Note: Thecharacteristic polyomial of ay graph is deoted by φ ( DD; λ ) = λ + Cλ + Cλ C. Defiitio.. The path is a tree with odes of vertex degree ad the other - odes of vertex degree. Defiitio.: The bistar graph, is the graph obtaied from K by joiig m pedat edges to oe ed ad pedat edges to the other ed of K.. Characteristic polyomial of directed cycle graph Theorem..The characteristic polyomial of skew adjacecy matrix of directed divisor cycle graph ( is eve) is ( 3) 4 ( 4)( 5) 6 φ ( C; λ ) = λ + λ + λ + λ ( 5)( 6) ( 7) 8 + λ C 4! if 0(mod 4) where C λ = if (mod 4). Proof: Let V( )={v,v,.v } be the vertex set of. Let E( )={v i v i+, v v i ; } be the edge set of, Whe is eve. Let v ad v of V(C ) be the start ad ed vertices respectively. The cycle graph C becomes the divisor graph with vertex labelig as below. v =p, v = i pi for i, vi = pi pi+ for i. The divisor cycle graph C becomes a directed cycle graph if the directio of edges is from low to high levels. For example, the directed divisor cycle graph C 6 is give i the Fig. Figure : 48

3 φ c ; λ = λ + 4λ 4 4 Also ( ) φ 6 ( c ; λ 6 ) = λ + 6λ + 9λ + 4 φ 8 4 ( ; λ ) = λ + 8 λ + 0 λ + 6 λ c 8 ( 3) ( 4)( 5) φ λ λ λ λ λ ( 5)( 6)( 7) 8 + λ c 4!, if 0(mod 4) where c λ =, if (mod 4). 4 6 ( c ; ) = Characteristic polyomial of directed path graph ad directed Bistar Theorem 3.. The characteristic polyomial of skew adjacecy matrix of directed divisor path graph P is φ ( P ; λ) = λ + λ + ( )( ) λ + ( )( 3)( 4) λ c. + if & is eve where c = if 3 & is odd. Proof: Let V(P ) ={v,v, v + } be the vertex set of P. Let E(P ) = {v i v i+, < i < }be the edge set of P. The path graph P becomes the divisor graph with vertex labelig as below. v i = Pi, i, vi+ = PP i i+ i v =. PP i The divisor path graph P becomes a directed path graph if the directio of edges is from low to high labels. For example, directed path graph P 5 is give i the Fig.. Figure : 49

4 V. Maimozhi ad V. Kaladevi The skew adjacecy matrix S(DD) is Also, 3 φ( P ; λ) = λ + λ 4 φ( P3 ; λ) = λ + 3λ ( P4 ; ) = ( P5 ; ) = λ +, if ad is eve where c =, if 3 ad is odd ( P5 ; ) = + + ( ) ( ) + ( )( 3)( 4) c Theorem 3..The characteristic polyomial of skew adjacecy matrix of directed divisor bistat B, is ( B ) ( ) 4 φ, ; λ = λ + λ + λ. Proof: Let V(B, ) = {v,v,..v, v,v,,v } be the vertex set of B,. E (B, ) ={ v i v i+,v i v (i+), < i < } be the edge set of B,. The bistar graph B, becomes the divisor graph with vertex labelig as below. The skew adjacecy matrix S(DD) is V = p V i = p p i+ for < i <- V i+ = p p i+ for < i <-. V = p p V (i) =p p p i+4 for < i< - V (i+) = p p p i+5 for < i < - The skew adjacecy matrix s (DD) is 50

5 Also φ ( B ; λ) = λ + 7λ + 9λ 4, ( B5,5; ) = ( B6,6; ) = ( ) φ ( B, ; λ) = λ + λ + λ. 3. Coclusio I this paper, a ew family of directed graphs are costructed from the divisor graph. I particular the characteristic polyomial of directed divisor graph C, P ad B, are studied. REFERENCES. C.Adiga ad R.Balakrisha, The skew eergy of a digraph, Liear Algebra ad its Applicatios, 43 (00) N.Murugesa ad K.Meeakshi, Skew eergy of digraphs ad relatio betwee graph eergy ad distace eergy, Iteratioal Joural of Combiatioal Graph Theory ad Applicatios, A () (0) Igor Shparliski, O the eergy of some circulat graphs, Liear Algebra ad its Applicatios, 44 (006)