CHARACTERISTIC POLYNOMIALS OF SOME WEIGHTED GRAPH BUNDLES AND ITS APPLICATION TO LINKS

. In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weighted K2 (K2)- bundles over a weighted graph F can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs are F As an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link

w(e) if e u,u, e E(F) and # j, w(u,) if i= j, 0 otherwise, Note that if the weight function 2. of F is defined by (e) -1 for e E(F) and  2-(u) deg(u) for u V(F), where deg(u) denotes the degree of u, that is, the number of edges incident to u, then the weighted adjacency matrix A(F) is called the Laplacian matrix of F. We call 2-the Laplacian function of F. The number of spanning trees of a connected graph F is the The characteristic polynomial P(r;)= IAI-A(r)l of the adjacency matrix A(r)is called the characteristic polynomial of the weighted graph r,.A root of P(I',;A) is called an eigenvalue of A runs thro,,gh all non-ze, o eig('nval,ws of .4(Ft)Mor(,ov(',, tl,(' eigenvalues of A(F) used to calc,llate the radius of gyration of a Gaussian molecuh,.For more applicati(ns of the eigenvalues of A(F), the reader is suggested to refer [5].

WEIGHTED GRAPH BUNDLES.
First, we introduce a weighted graph b,m(lle.Every edge of a graph F gives rise to a pail oppositely directed edges.We denote the set of directed edgeb of F bv D(F).By e-' we m,'an the reverse edge to an edge D(F).For any finite group G. a G-voltage ass,gnmen of F i a function 6:D(r)a ,,h that 8(e -) =(, )-for all e D(F).we denote the set of all G- voltage assignments of F by C(F;G).Let .kbe another graph and let C(F;AuI(A)).wh('re Aut(A) is the group of all graph automophislns of A. Now, we construct a graph FxA as follows" v(rx*A)-V(r)xV(A).Two vertices (Ul,L'I)and (u2, v2)are adjacent in either u,u2 D(F) and ,,2 (UlU2)t, or it It and v,,2 E(A).We call F xA the A-bundle over F associated w'th and the natural map p*: F x AF the bundle projectwn.We also call F and A the base and the fibre of Fx*A, respectively.Note that the map p maps vertices to vertices but an image of an edge can be either an edge or a vertex.If A is the complement K,, of the complete graph K, of n vertices, then every A-bundle over F is an n-fold covering graph of F Let F and A, be two weighted graphs and let C'(F;Aut(A)).We define the product of and w wzth respect to ,w x 0, as follows" (1) For each vertex (u,v) of V(Fx*A),(wx*p)(u,v)=w(u)+p(v).
We call the weighted graph (FxeA)% we call it a weighted graph bundle.
the Au-bundle over F, assoczated wth .Briefly, In this section, we give a computation for the characteristic polynomial of a weighted graph bundle F x CA, where A is either complete graph K of two vertices or its complement K2, and study their related topics.Note that Aut(K)= Aut(K)= Z.
For a given graph F with weight function w and for a E C1(F;Z), we define a new weight function w on F as follows: For (2) For v E V(F), ,e'(t,)= w(v).
A subgraph of F is called an elementary conflguratwn if its components are either conplete graph K or K2 or a cycle C,,(m > 3).We denote by E, the set of all elementary configurations of F having k vertices.In [3], the characteristic polynomial of a weighted graph F is giw'n as follows: where S E u I,(s) e e I(S) e C(S) In the above equation, symbols have the following meaning: :(S) is the number of components of S, C(S) the set of all cycles, C,,(m >_ 3), in S, and I,(S)(I(S))is the set of all isolated vertices (edges) in S.Moreover, the product over empty index set is defined to be 1.
For a fixed voltage assignment C(F;Z), we denote by E_ the set of edges of F such that (e)= -1, i.e., Eo_ {e E(r):(e) -1}.Let F(Ee_) be the edge subgraph of r induced by E,_ having weight zero in vertices.If F is a weighted graph, then the weight function of its subgraph S is the restriction of w on S. THEOREM 1.Let Ku be a constant weighted graph, say t(v)= c for v K .Then, for each ( Cl(F; Z:), we have P((r x --)** ; ) P(r.; c)P(r; c).PROOF.Let A(F) be the adjacency matrix of F and let A(F) the adjacency matrix of F. Then we have a(r) A((r\(z_,))) + A(F(E_,)), A(r) A((r\(E_ )))-A(r(E,_ )).
we have our theorem.VI THEOREM 2. Let K (K2,#) be a weighted graph having constant weight on vertices Then, for each b E C(F; Z), we have .)=P(r;--)P(r;A-+c), P((F x *K) %, where c #(v,)= #(v) for the vertices v,,v and c #(e) for the edge e in K.

El
Note that for any CI(F;Aut(A)), the Laplacian function of F xA is the product of Laplacian functions of F and A with respect to .Clearly, the Laplacian function of the K, is the zero function; and the Laplacian function of the K has value and -1 for each of its vertices and its edge, respectively.We shall denote the Laplacian function of a graph by if it makes no confusion.Then Theorem and Theorem 2 give the following corollary.COROLLARY 1.For any C(F; Z), (1) p(r;)P(r,;).
(2) P((rxg),;)= P(rt;)P(rt,;-2).0 Now, we consider another invariant of weighted graphs called the signature.Since A(F,) is symmetric, A(F) can be diagonalized through congruence over R. Let d + denote the number of positive diagonal entries, and d_ the number of negative diagonal entries.
The signature of a weighted graph (F) is defined by a(A(r)) d + -d_ and is denoted by a(F).It is an invariant for weighted 2-isomorphic graphs (see [7]).
From now on, we will consider the weight function on K as zero function and the weight function p on K2 as the map defined by p(v)= 0 for each v V(K) and it(e)= ce for the edge e of K2.Then we can compute the signature of a double covering of F.
We Mso denote the multiplicity of by me(A).By using the above notations d Threm 2, we get the signature of a K-bundle over F. (2) ifc<O, then (tr ct %) trt + MARK.Though the results in this section stated only for a simple graph, it remains true for any graph. 4. APPLICATIONS TO LINKS.
In a signed graph F, an edge e of F is said to be positive if (e) and negative otherwise.
For a signed graph F, we define a new weight function of F by (e)= (e) for any edge e F and (u,) )'= ,, C a,s, where a, is the number of positive edges minus the number of negative edges which have two end vertices u, and u s.Given a knot or link L in , we project it into R so that each crossing point h proper double crossing.The image of L is called a link (or knot) diagram of L, and we do not distinguish betwn a diagram and the image of L.
We may sume without loss of generality that a link diagram of L intersects itself transversely and h only finitely many crossings.The link diagram divides R into finitely many domains, which will be clsified shaded or unshaded.No two shaded or unshaded domains have an edge in common.We now construct a signed plan graph F from L follows: take a point v, from each unshaded domn D,.These points form the set of vertices V(F) of F. If the boundies of D, and D intersect k-times, say, crossing at ct, c,..-,ct, then we form multiple edges eg,e, .,egt on R with common end vertices v, and v, where each edge et pses through a crossing c, for m 1,2,-..,k.To define the weight of edge, first, we define the index (c) to each crossing c of the link diagr in Figure 2. To each edge of F pses through exactly one crossing, say c, of L, the weight (e) will be defined (), (c).(S i 3).The resulting signed planar is called the 9raph of a link with respect to L and is denoted by r( ). he sined planar graph r( aepends not only on g but also on shaain.Conversely, given a signed planar graph 1"0, one can construct uniquely the link diagram L( 0) of a link so that r((r 0//; to. ,,c) = -| FIGURE 4. The index w(c).
Suppose that we are given an oriented link L. The orientation of L induces the orientation of a diagram .We then define the second index w(c), called the twist or writhe at each crossing c as show in Figure 4. We now need the third index r/p(c) at crossing c.Let L be an oriented diagram and p shading on .Let rp(c)= w(c)6,(c){c), where 6 denotes Kronecker's delta.We define r( )= r(c), where the summation runs over all crossing in .T he index r/p( depends not only on the shading p but also on the orientation of .T he following Lemma can be found in ( [7], [4]).
LEMMA 1.The signature a(L) of a link L is a(L) a(P(L ))-r(L ).
El Let and be link diagrams of L1 and L2, respectively.The link L is called a double covering of the link n if F( 2) is a double covering of P( 1) as weighted graphs and it can be extended to a branched covering on R2.Let be a voltage assignment in C(F( );Z2) such that (e)= -1 for some edge e and (e)= otherwise, then F( )x4-is a planar double covering of F( of which the corresponding link is a double covering of L. herefor, one can construct the double covering link diagram L (F,(Z)x4-)of,,, Moreover, we can give an orientation on (P()x -)so that the covering map from L to (P()x 4-) preserves the orientation.We have ( (F()x e--))= 2rp( (see Figure 5).Therefore, by using Lemma and Corollary 2, we get the following theorem.

FIGURE 1 .
FIGURE 1.The graphs C4xeK2 and 3. CHARACTERISTIC POLYNOMIALS.In this section, we give a computation for the characteristic polynomial of a weighted graph a regular matrix of order 2 satisfying M-l[ ] M-[ --00 1 1 " Put X A(F)-A(F(E_ ))+ c 0 Y=A(F(E_I)).