Eigenvalue Sums of Combinatorial Magnetic Laplacians on Finite Graphs
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EIGENVALUE SUMS OF COMBINATORIALMAGNETIC LAPLACIANS ON FINITE GRAPHS J OHN D EVER
Submitted to Operators and MatricesAbstract.
We give a construction of a class of magnetic Laplacian operators on finite directedgraphs. We study some general combinatorial and algebraic properties of operators in this classbefore applying the Harrell-Stubbe Averaged Variational Principle to derive several sharp boundson sums of eigenvalues of such operators. In particular, among other inequalities, we show thatif G is a directed graph on n vertices arising from orienting a connected subgraph of d -regularloopless graph on n vertices, then if ∆ θ is any magnetic Laplacian on G , of which the standardcombinatorial Laplacian is a special case, and λ ≤ λ ≤ ... ≤ λ n − are the eigenvalues of ∆ θ , then for k ≤ n , we have 1 k k − ∑ j = λ j ≤ d − .
1. Introduction
We will assume G to be a finite directed graph without repeated (directed) edgesor self edges (loopless), which we realize as a pair of maps s , t : E → V of finite sets,such that if e ∈ E , then s ( e ) = t ( e ) ; and if e , e ∈ E with e = e , then ( s ( e ) , t ( e )) =( s ( e ) , t ( e )) . The maps s , t are called the source and target, respectively. The set E iscalled the set of directed edges, and the set V is called the set of vertices.If e ∈ E is an edge, we think of the vertex s ( e ) = se ∈ V as the “source” of theedge e and the vertex t ( e ) = te ∈ V as the “target” of the edge e . If e ∈ E and v , w ∈ V with se = v , te = w , we adopt the notations vw and v → w for e . In this paper we shall primarily study a class of operators defined as follows. Givena map θ : E → [ − π , π ] such that if vw , wv ∈ E then θ ( vw ) = − θ ( wv ) ; if f : V → C ,for v ∈ V let ∆ θ ( f )( v ) : = ∑ e ∈ E , te = v ( f ( v ) − e i θ ( e ) f ( se )) + ∑ e ∈ E , se = v ( f ( v ) − e − i θ ( e ) f ( te )) . Such operators are referred to in the literature as discrete magnetic Laplacians, magneticcombinatorial Laplacians, or discrete magnetic Sch¨odinger operators [3]. The usual(combinatorial) graph Laplacian corresponds to the choice θ ( e ) = e ∈ E . The
Mathematics subject classification (2010): 05C50, 05C20,05C15,15B57,15A42.
Keywords and phrases : magnetic graph Laplacian; graph Laplacian; eigenvalue inequalities; eigen-value sums; adjacency matrix; graph spectrum; half-band. θ ( e ) = π for all e ∈ E . Note that since e i π = e − i π = − , it is irrelevant for the choice θ = π whether θ ( uv ) = − θ ( vu ) for uv , vu ∈ E . The operators ∆ θ may be connected to electromagnetism, justifying the term“magnetic graph Laplacian.” However, our focus in this paper is primarily mathemat-ical. We study the combinatorial and spectral properties of these operators in the caseof finite graphs without symmetry. Moreover, we do not insist that the graph be planaror that the θ values arise from some “flux” in a physical model. Our study yields anumber of inequalites on the sum of eigenvalues of magnetic Laplacian operators ongraphs that apply, in particular, to the classical combinatorial graph Laplacian.The outline of this paper is as follows. First, in the following section we definemagnetic graph Laplacians. We study some algebraic and combinatorial properties ofthese operators. Next, in section 3 we recall the averaged variational principle from [2].Finally, in section 4 we apply the variational principle to derive bounds for eigenvaluesums of graph magnetic Laplacians.
2. Magnetic Graph Laplacians
We adopt the following notation. If A is a finite set, let C ( A ) be the set of com-plex valued functions on A . For a ∈ A , let ˆ a be the indicator function at a , that is ˆ a ( x ) is 1 for x = a and 0 for x = a . Then the ( ˆ a ) a ∈ A form a basis for C ( A ) . As C ( A ) isisomorphic to C | A | , where | A | denotes the cardinality of A , it is a Hilbert space withinner product induced by our choice of standard basis ( ˆ a ) a ∈ A . As a convention we takeall inner products to be conjugate linear in the first argument . Also, if T is an operatoron a finite dimensional Hilbert space, by T ∗ we mean its adjoint, that is the operatorwhose matrix representation in the standard basis is the conjugate transpose of the ma-trix representing T in the standard basis. For v a vertex, let d v be the degree of v ∈ V . For f : E → C any complex valued function, we may consider the corresponding mul-tiplication operator ˆ f : C ( E ) → C ( E ) , defined by its action on basis vectors, ˆ e f ( e ) ˆ e for e ∈ E . Then the adjoint of f is the multiplication operator, again defined by itsvalues on the basis of C ( E ) , ˆ e f ( e ) ˆ e . By the above notation we denote it by b f . Let θ : E → [ − π , π ] . Then, as above, we may also consider c e i θ as a multiplication opera-tor C ( E ) → C ( E ) . Observe that s , t induce operators ˆ s , ˆ t : C ( E ) → C ( V ) by extendinglinearly from the action on basis vectors ˆ s ( ˆ e ) : = d s ( e ) and ˆ t ( ˆ e ) : = d t ( e ) for e ∈ E . Thealgebraic point of view we take is similar to the one in [6].Let θ : E → [ − π , π ] such that if v , w ∈ V with vw , wv ∈ E , then θ ( vw ) = − θ ( wv ) .Define a quadratic form Q θ by Q θ ( f ) = ∑ e ∈ E | f ( te ) − e i θ ( e ) f ( se ) | . Then let d θ : = ˆ t − ˆ s d e − i θ and define the (combinatorial) magnetic graph Laplacian ∆ θ : = d θ d ∗ θ = ( ˆ t − ˆ s d e − i θ )( ˆ t ∗ − c e i θ ˆ s ∗ ) . Then Q θ ( f ) = h f , ∆ θ f i . Note, by its factorization as a square, ∆ θ is a positive, self-adjoint operator. 2 .1. Properties of Magnetic graph Laplacians Let D = ˆ s ˆ s ∗ + ˆ t ˆ t ∗ , A θ : = ˆ s d e − i θ ˆ t ∗ + ˆ t c e i θ ˆ s ∗ . Then by expanding the product ( ˆ t − ˆ s d e − i θ )( ˆ t ∗ − c e i θ ˆ s ∗ ) , we see ∆ θ = D − A θ . Note that ∆ = D − A is the standard graph Laplacian and that ∆ π = D + A . It can be seen from either the definition in terms of s , t or from the quadraticform that both the standard Laplacian ∆ and ∆ π are independent of orientation, asinterchanging the roles of se and te for any given edge leaves them invariant. Note,however, that in general ∆ θ is highly dependent on the orientation.Since det ( ∆ θ ) = k f k = Q θ f = , andsince the set k f k = C ( V ) is finite dimensional; we havedet ( ∆ θ ) = f = Q θ ( f ) = . This occurs, by theform of Q θ given above, if and only if f ( te ) = e i θ ( e ) f ( se ) for all e . This suggests thefollowing result.P
ROPOSITION Suppose k is a positive integer. An undirected graph is bipartiteif and only if det ( ∆ π k ) = for some orientation. An undirected graph is tripartite if andonly if det ( ∆ π ) = for some orientation.Proof. Without loss of generality we may assume the graph is connected sincewe may consider its components separately. Let k a positive integer. Let ω = e i π k . Suppose for some orientation that Q π k ( f ) = ∑ e | f ( te ) − ω f ( se ) | = f = . By re-scaling f if necessary and possibly multiplying by a global phase, since f = , we may assume that f ( v ) = v ∈ V . Then since the graph is connected wehave that f takes on at most the values ω j for j ∈ { , , ..., k − } . Let A be the set ofvertices where f takes on values ω j for j even and B the set of vertices where f takeson values ω j for j odd. Since for any edge e , f ( te ) = ω f ( se ) , vertices in A can onlybe connected to vertices in B and vertices in B can only be connected to vertices in A . So A , B is a bipartition. Conversely, suppose A , B is a bipartition of an undirectedgraph. Then define f : = A + ω B . Define an orientation by having s always takevalues in A , t values in B . Then for any e : se → te we have | f ( te ) − ω f ( se ) | = | ω − ω ( ) | = . Hence Q π k ( f ) = . As for the second assertion, let ω : = e i π and suppose for some orientation that Q π ( f ) = ∑ e | f ( te ) − ω f ( se ) | = f = . Again by re-scaling f if nec-essary and possibly multiplying by a global phase, since f = , we may assume that f ( v ) = v . Then since the graph is connected we have that f takes on atmost the values 1 , ω , ω . So define a tripartition A , A , A , with A j , for j = , , , the set of vertices where f takes on the value ω j . Conversely, suppose A , B , C is atripartition of an undirected graph. Define f : = A + ω B + ω C . Then define anorientation by the following rules. For any edge e between an A vertex and a B vertex,take se to be the A vertex, te the B vertex. For any edge e between a B vertex andan C vertex, set se to be the B vertex, te to be the C vertex. For any edge e between3 C vertex and an A vertex, set se to be the C vertex, te to be the A vertex. Then,by construction, for any directed e : se → te , we have | f ( te ) − ω f ( se ) | = . Hence Q π ( f ) = , and the proof is complete.The above proposition hints at the computational difference between determiningwhether a graph is 2-colorable or 3-colorable. Indeed, using the proposition to deter-mine whether a graph is 2-colorable requires only computing one determinant since ∆ π is independent of orientation. However, for a graph with n edges, using the propositionto determine whether it is 3 -colorable requires checking at most 2 n determinants, onefor each orientation.Following [4], we call a unitary operator U : C ( V ) → C ( V ) a gauge transforma-tion if it is multiplication operator with respect to the basis of vertices. So U ˆ v = φ ( v ) ˆ v for v ∈ V where φ : V → C with | φ | = . P ROPOSITION ∆ θ is unitarily equivalent under a gauge transformation to thestandard Laplacian ∆ if and only if det ( ∆ θ ) = . Proof.
We may assume the underlying graph is connected since ∆ θ has a decom-position as a direct sum of corresponding ∆ θ operators on each connected component.Since det ( ∆ ) = , one direction is clear. For the other, suppose det ( ∆ θ ) = . Let φ , normalized with the supremum norm k φ k ∞ = , such that h φ , ∆ θ φ i = . Since φ is normalized and the graph is connected, by the equation Q θ ( φ ) = ∑ e ∈ E | φ ( te ) − e i θ ( e ) φ ( se ) | , the vanishing of h φ , ∆ θ φ i ensures that | φ | = U : C ( V ) → C ( V ) by U ( ˆ v ) : = φ ( v ) ˆ v for v ∈ V and extending linearly. Since | φ | = , U is a gauge transformation.Define sesquilinear forms Q , T by Q ( f , g ) : = h f , ∆ θ Ug i and T ( f , g ) : = h f , U ∆ g i . Let v , w ∈ V . We have Q ( ˆ v , ˆ w ) = h ˆ v , ∆ θ U ˆ w i = h ˆ v , ∆ θ φ ( w ) ˆ w i = φ ( w ) h ˆ v , ∆ θ ˆ w i = φ ( w )( h ˆ v , D ˆ w i − h ˆ v , A θ ˆ w i )= φ ( w )( δ v , w deg ( v ) − h ˆ v , ( ˆ s d e − i θ ˆ t ∗ + ˆ t c e i θ ˆ s ∗ ) ˆ w i ) . Then if v = w , the result is φ ( v ) deg ( v ) . If v = w and v , w non-adjacent, then Q ( ˆ v , ˆ w ) = . If e = vw is an edge, substituting e i θ ( v → w ) φ ( v ) for φ ( w ) and e − i θ ( v → w ) for h ˆ v , A θ ˆ w i , we have Q ( ˆ v , ˆ w ) = − φ ( v ) . Similarly an edge of the form wv results in Q ( ˆ v , ˆ w ) = − φ ( v ) . But also T ( ˆ v , ˆ w ) = h ˆ v , U ∆ ˆ w i = h U ∗ ˆ v , ∆ ˆ w i = h φ ( v ) ˆ v , ∆ ˆ w i = φ ( v ) h ˆ v , ∆ ˆ w i , which is φ ( v ) deg ( v ) if v = w , 0 if v = w and v , w non-adjacent, and any edge of theform vw or wv results in − φ ( v ) . Hence Q = T . Therefore ∆ θ U = U ∆ , which implies ∆ θ is unitarily equivalent to ∆ under a gauge transformation.4f e is an oriented edge, say uv , let e = vu be the reverse edge. Then extend θ by θ ( e ) = − θ ( e ) . Then if v = v , ..., v m = v is a closed (unoriented) walk, the fluxis defined by ∑ m − j = θ ( v j v j + ) (mod 2 π ) . Hence, if v j v j + is an oriented edge, thenit contributes θ ( v j v j + ) to the flux; and if v j + v j is an edge, then v j v j + = v j + v j contributes θ ( v j v j + ) = θ ( v j + v j ) = − θ ( v j + v j ) to the flux.Note a directed graph has an underlying undirected graph with edge relation v ∼ w if vw or wv is a directed edge. By a walk in a graph we mean a finite list of vertices v , v , ..., v n such that v j ∼ v j + for 0 ≤ j < n . A closed walk is a walk with the initialand final vertex coinciding.The proof of the following proposition, in the case of A θ , may be found in [4]. Inorder to derive this version from the one presented there, note that gauge transforma-tions are diagonal in the standard basis for C ( V ) and thus commute with D . P ROPOSITION Two magnetic Laplacians ∆ θ , ∆ θ are unitarily equivalent un-der a gauge transformation if and only if θ and θ induce the same fluxes throughclosed walks.
3. Averaged Variational Principle
In this section we develop a tool (see [2]) for its origin) that will allow estimateson sums of eigenvalues of finite Laplacian operators.If M is a self-adjoint n × n matrix, we denote its eigenvalues by µ ≤ µ ≤ ... ≤ µ n − and a corresponding orthonormal basis of eigenvectors by ( u j ) n − j = . If V is any k dimensional subspace of C n and ( v j ) k − j = an orthonormal basis for V , then we defineTr ( M | V ) : = k − ∑ i = h v i , Mv i i . Tr( M | V ) is independent of the basis chosen. Indeed, let P V be the projection onto V .Then P V = ∑ k − i = v i v ∗ i . So n − ∑ j = µ j k P V u j k = k − ∑ i = n − ∑ j = µ j |h v i , u j i| = k − ∑ i = v ∗ i Mv i , using the spectral decomposition of M . Since the left hand side of the above string ofequalities is independent of basis, the result holds.We begin by stating the following classical result [1].P
ROPOSITION With notation as above, for ≤ k ≤ n , we have k − ∑ j = µ j = inf dim ( V )= k Tr ( M | V ) . In particular if ( v i ) k − i = ⊂ C n is any collection of orthonormal vectors, we have k − ∑ j = µ j ≤ k − ∑ j = h v j , Mv j i . HEOREM (Harrell-Stubbe) Let M be a self adjoint n × n matrix with eigen-values µ ≤ µ ≤ ... ≤ µ n − and corresponding normalized eigenvectors ( u i ) n − i = . Sup-pose ( Z , M , µ ) is a (positive) measure space and φ : Z → C n is measurable with R Z k φ ( z ) k d µ ( z ) < ∞ . Then if Z ∈ M , for any ≤ k ≤ n − , we have µ k Z Z k φ k d µ − k − ∑ j = Z Z |h φ , u j i| d µ ! ≤ Z Z h φ , M φ i d µ − k − ∑ j = Z Z µ j |h φ , u j i| d µ . Note that provided that µ k ∑ k − j = R Z |h φ , u j i| d µ ≤ µ k R Z k φ k d µ , we have that k − ∑ j = Z Z µ j |h φ , u j i| d µ ≤ Z Z h φ , M φ i d µ . (1)We now show that Proposition 4 follows from the the previous Theorem 1, inparticular from (1). Let v , ..., v k − be a collection of orthonormal vectors in C n . Take Z : = { , , ..., n − } and Z : = { , , ..., k − } with the counting measure. Extend the v j to an orthonormal basis for all of C n . Then let φ ( l ) : = v l . Then k − ∑ j = Z Z |h φ , u j i| d µ = k = Z Z k φ k d µ . Hence (1) then states k − ∑ j = µ j n − ∑ l = |h v l , u j i| = k − ∑ j = µ j ≤ k − ∑ j = h v j , Mv j i , and we recover Proposition 4.
4. Inequalities for Sums of Eigenvalues of Magnetic Laplacians
We now apply the averaged variational principle, Theorem 1, to ∆ θ . For the re-mainder of this section we further assume that G is connected, and for any u , v ∈ V ,if uv ∈ E then vu / ∈ E . In other words, we assume G arises from orienting a con-nected, loopless, undirected graph without repeated edges. If G has n vertices, let λ ≤ λ ≤ ... ≤ λ n − denote the eigenvalues of ∆ θ . Let K n be the complete, loopless, undirected graph on the n vertices of G . Orient K n with some orientation such that the orientation of its restriction to G is the orien-tation on G . Suppose H is a d -regular directed subgraph of K n , with G a directedsubgraph of H . This, in particular, implies that H is connected on n vertices. Notethis is always possible by taking H = K n and with d = n − G = C , d may be taken to be 2 , , , or 5 . Let H c be the graph complement of6 in K n with the induced orientation. If e = uv is an oriented edge in K n we shalldenote e : = vu . We call two vertices u , v adjacent and write u ∼ v if there is someoriented edge between them, either u → v or v → u . If the graph G is not clear fromthe context, we write ∼ G if we wish to restrict the relation to G . Then let Z : = V × V and M : = P ( Z ) . We shall denote pairs ( u , v ) in Z by uv . Let a , b ≥ . Define µ on M by µ ( e ) = µ ( e ) = e an edge in H , µ ( e ) = µ ( e ) = a for e an edge in H c ,and µ ( u , u ) = b for all u . Let α : E → [ , π ] . Then, extend α and θ to all of K n bysetting them equal to 0 outside of edges of G . Define φ α , H : Z → C ( V ) ∼ = C n by φ α , H ( uv ) : = b uv , where b uv : = ˆ v − e i α ( uv ) ˆ u , uv ∈ E K n ˆ v + e − i α ( vu ) ˆ u , vu ∈ E K n ˆ u , u = v . Hence for any f , |h f , b uv i| = | f ( u ) | + | f ( v ) | − Re ( e i α ( uv ) f ( u ) f ( v )) , uv ∈ E K n | f ( u ) | + | f ( v ) | + Re ( e − i α ( vu ) f ( u ) f ( v )) , vu ∈ E K n | f ( u ) | , u = v . We wish to calculate ∑ uv ∈ Z µ ( uv ) |h f , b uv i| . Note that for uv an edge in K n , wehave |h f , b uv i| + |h f , b vu i| = | f ( u ) | + | f ( v ) | . For u fixed and for any vertex v , exactly one of the three following possibilities occurs: v = u , v is adjacent to u in H ,or v is adjacent to u in H c . Hence, since edges and their opposites occur in pairs inboth H and H c , and since H is d -regular and H c is n − − d regular, we have ∑ uv ∈ Z µ ( uv ) |h f , b uv i| = ∑ u ∑ v ∼ H u ( | f ( u ) | + | f ( v ) | ) + a ( ∑ u ∑ v ∼ Hc u ( | f ( u ) | + | f ( v ) | ))= ( d + a ( n − − d ) + b ) k f k + d k f k + ( n − − d ) a k f k = ( d + a ( n − − d ) + b ) k f k . Let C ( a , b , d ) : = d + a ( n − − d ) + b . Then ∑ uv ∈ Z µ ( uv ) |h f , b uv i| = C ( a , b , d ) k f k . Let Z ⊂ Z . Now we calculate ∑ uv ∈ Z h b uv , ∆ θ b uv i . There are four cases. If uv isan oriented edge in G , then h b uv , ∆ θ b uv i = | + e i ( α ( uv )+ θ ( uv )) | + d u − + d v − = d u + d v + ( α ( uv ) + θ ( uv )) . If vu is an oriented edge in G , then h b uv , ∆ θ b uv i = | − e − i ( α ( vu )+ θ ( vu )) | + d u − + d v − = d u + d v − ( α ( vu )+ θ ( vu )) . u = v and neither uv nor vu is an oriented edge in G , then h b uv , ∆ θ b uv i = d u + d v . Lastly, for any v , h b vv , ∆ θ b vv i = d v . Hence ∑ uv ∈ Z h b uv , ∆ θ b uv i µ ( uv ) = ∑ { uv ∈ Z | uv ∈ E } d u + d v + ( α ( uv ) + θ ( uv ))+ ∑ { uv ∈ Z | vu ∈ E } d u + d v − ( α ( vu ) + θ ( vu ))+ ∑ { uv ∈ Z | uv , vu / ∈ E , u = v , uv or vu ∈ E H } d u + d v + a ( ∑ { uv ∈ Z | uv , vu / ∈ E , u = v , uv , vu / ∈ E H } d u + d v ) + b ( ∑ { v | vv ∈ Z } d v ) . (2)For A a finite set, let | A | denote the cardinality of A . Then we have ∑ uv ∈ Z k b uv k µ ( uv ) = |{ uv ∈ Z | uv or vu ∈ E H }| + a |{ uv ∈ Z | uv or vu ∈ E H c }| + b |{ u | uu ∈ Z }| . (3)Hence by (1) following Theorem 1, if k is such that 2 kC ( a , b , d ) ≤ ∑ uv ∈ Z k b uv k µ ( uv ) , then k − ∑ j = λ j ≤ ∑ uv ∈ Z h b uv , ∆ θ b uv i µ ( uv ) C ( a , b , d ) . We may achieve great simplifications of the above inequality if we take Z tocontain only edges or reverse edges of E , or also, if needed, “loops” of the form uu .Before continuing, we note the following. The quantity Z G : = ∑ v d v is known ingraph theory literature as the first Zagreb index of G (see [5]), where d v is the degreeof v in G . Then note that ∑ uv ∈ E ( d u + d v ) = Z G . Indeed, for each v , d v appears once in exactly d v terms in the sum.For what follows, let Z = E . Then (2) simplifies to ∑ uv ∈ Z h b uv , ∆ θ b uv i = Z G + ∑ e ∈ E cos ( α ( e ) + θ ( e )) . Since we will be wishing to minimize this quantity, we define α such that α ( e ) + θ ( e ) ≡ π (mod 2 π ) . Then ∑ uv ∈ Z h b uv , ∆ θ b uv i = Z G − | E | . | E | , we have that if 2 kC ( a , b , d ) ≤ | E | , then2 C ( a , b , d ) k − ∑ j = λ j ≤ Z G − | E | . However, since C ( a , b , d ) can take on any number greater than or equal to d , if k ≤ | E | d , then the optimal choice is C ( a , b , d ) = | E | k . Hence, we have proven the followingtheorem.T
HEOREM Suppose G is a directed graph arising from orienting a connected,loopless, undirected graph without repeated edges. Let d be the degree of a regularsubgraph H of K n containing G as a subgraph. Then if k is an integer with k ≤ | E | d , we have k k − ∑ j = λ j ≤ Z G | E | − . Note that if D is the degree matrix, | E | = Tr ( D ) and Z G = Tr ( D ) . Hence wemay rewrite the above inequality as follows. For G as in the previous theorem, we have1 k k − ∑ j = λ j ≤ Tr ( D ) Tr ( D ) − , for k a positive integer with k ≤ d Tr ( D ) . We may increase the bound on k by admitting a combination of reverse edges of G , and loops uu to Z . Then the cosine terms cancel in pairs for reverse edges andloops add terms proportional to the degree.In [4] the half-filled band, corresponding to the case that k = ⌊ n ⌋ , is studied. Asa corollary we provide an inequality for the half-filled band in the case of a d − regulargraph. Let H = G . Then d = d . Note in this case Z G = nd and 2 | E | = nd . Notefurther that any ∆ θ is a sum of magnetic Laplacians corresponding to individual edges,each being a positive operator. It follows that eigenvalue sums for a subgraph arebounded above by corresponding sums for the graph. Therefore we have the followingresult for the half-band.C OROLLARY Suppose G is a directed graph on n vertices arising from orient-ing a connected, undirected subgraph of a d -regular undirected loopless graph on nvertices without repeated edges. Then for k ≤ n , we have k k − ∑ j = λ j ≤ d − . Note the above bounds hold for all choices of θ . In particular they hold for thestandard combinatorial Laplacian. This connects to the ”flux phase” problem investi-gated for the case of planar graphs in [4], that is to find the choice of θ that maximizes9he sum of the half-band eigenvalues. For different classes of graphs, the optimal choiceof θ may vary, leading to the possibility for improvements to the above bounds in suchcases for particular choices of θ . We give two simple examples. Let G = K with some orientation. The conditionis k ≤ , so the only non-trivial choice for k is 1 . The above inequality reduces to λ ≤ . This is sharp since taking θ constant equal to π on any orientation yields aspectrum of 1 , , . Consider the cycle C . Then the spectrum of the standard Laplacian is 0 , , , . Hence the inequality is sharp at k = = n ( d − ) , where n = , d = .
5. Acknowledgments
I would like to thank Evans Harrell for helpful discussion of the ideas in this paperand comments on a draft. Also, I would like to thank the anonymous reviewer forcomments that led to substantial improvements to an earlier version of this paper.
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John Dever,School of MathematicsGeorgia Institute of Technology686 Cherry StreetAtlanta, GA 30332-0160 USAe-mail: [email protected]@math.gatech.edu