We give a construction of a class of magnetic Laplacian operators on finite directed graphs. We study some general combinatorial and algebraic properties of operators in this class before applying the Harrell-Stubbe Averaged Variational Principle to derive several sharp bounds on sums of eigenvalues of such operators. In particular, among other inequalities, we show that if is a directed graph on vertices arising from orienting a connected subgraph of -regular loopless graph on vertices, then if is any magnetic Laplacian on , of which the standard combinatorial Laplacian is a special case, and are the eigenvalues of then for we have
Eigenvalue Sums of Magnetic Laplacians]Eigenvalue Sums of Combinatorial Magnetic Laplacians on Finite Graphs
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160 USA
magnetic graph Laplacian; graph Laplacian; eigenvalue inequalities; eigenvalue sums; adjacency matrix; graph spectrum; half-band
We will assume to be a finite directed graph without repeated (directed) edges or self edges (loopless), which we realize as a pair of maps of finite sets, such that if then and if with then The maps are called the source and target, respectively. The set is called the set of directed edges, and the set is called the set of vertices.
If is an edge, we think of the vertex as the “source” of the edge and the vertex as the “target” of the edge If and with we adopt the notations and for
In this paper we shall primarily study a class of operators defined as follows. Given a map such that if then ; if , for let
Such operators are referred to in the literature as discrete magnetic Laplacians, magnetic combinatorial Laplacians, or discrete magnetic Schödinger operators . The usual (combinatorial) graph Laplacian corresponds to the choice for all The signless Laplacian corresponds to the choice for all Note that since it is irrelevant for the choice whether for
The operators may be connected to electromagnetism, justifying the term “magnetic graph Laplacian.” However, our focus in this paper is primarily mathematical. We study the combinatorial and spectral properties of these operators in the case of finite graphs without symmetry. Moreover, we do not insist that the graph be planar or that the values arise from some “flux” in a physical model. Our study yields a number of inequalites on the sum of eigenvalues of magnetic Laplacian operators on graphs that apply, in particular, to the classical combinatorial graph Laplacian.
The outline of this paper is as follows. First, in the following section we define magnetic graph Laplacians. We study some algebraic and combinatorial properties of these operators. Next, in section 3 we recall the averaged variational principle from . Finally, in section 4 we apply the variational principle to derive bounds for eigenvalue sums of graph magnetic Laplacians.
2 Magnetic Graph Laplacians
We adopt the following notation. If is a finite set, let be the set of complex valued functions on For let be the indicator function at that is is for and for Then the form a basis for As is isomorphic to where denotes the cardinality of it is a Hilbert space with inner product induced by our choice of standard basis As a convention we take all inner products to be conjugate linear in the first argument. Also, if is an operator on a finite dimensional Hilbert space, by we mean its adjoint, that is the operator whose matrix representation in the standard basis is the conjugate transpose of the matrix representing in the standard basis. For a vertex, let be the degree of For any complex valued function, we may consider the corresponding multiplication operator , defined by its action on basis vectors, for Then the adjoint of is the multiplication operator, again defined by its values on the basis of , By the above notation we denote it by Let . Then, as above, we may also consider as a multiplication operator Observe that induce operators by extending linearly from the action on basis vectors and for The algebraic point of view we take is similar to the one in .
Let such that if with then . Define a quadratic form by Then let and define the (combinatorial) magnetic graph Laplacian
Then Note, by its factorization as a square, is a positive, self-adjoint operator.
2.1 Properties of Magnetic graph Laplacians
Let Then by expanding the product we see
Note that is the standard graph Laplacian and that
It can be seen from either the definition in terms of or from the quadratic form that both the standard Laplacian and are independent of orientation, as interchanging the roles of and for any given edge leaves them invariant. Note, however, that in general is highly dependent on the orientation.
Since if and only if the lowest eigenvalue and since the set is compact, as the space is finite dimensional; we have if and only if there exists an with This occurs, by the form of given above, if and only if for all This suggests the following result.
Suppose is a positive integer. An undirected graph is bipartite if and only if for some orientation. An undirected graph is tripartite if and only if for some orientation.
Without loss of generality we may assume the graph is connected since we may consider its components separately. Let a positive integer. Let Suppose for some orientation that for some By re-scaling if necessary and possibly multiplying by a global phase, since we may assume that for some Then since the graph is connected we have that takes on at most the values for . Let be the set of vertices where takes on values for even and the set of vertices where takes on values for odd. Since for any edge vertices in can only be connected to vertices in and vertices in can only be connected to vertices in . So is a bipartition. Conversely, suppose is a bipartition of an undirected graph. Then define Define an orientation by having always take values in , values in . Then for any we have Hence
As for the second assertion, let and suppose for some orientation that for some Again by re-scaling if necessary and possibly multiplying by a global phase, since we may assume that for some Then since the graph is connected we have that takes on at most the values So define a tripartition with for the set of vertices where takes on the value . Conversely, suppose is a tripartition of an undirected graph. Define Then define an orientation by the following rules. For any edge between an vertex and a vertex, take to be the vertex, the vertex. For any edge between a vertex and an vertex, set to be the vertex, to be the vertex. For any edge between a vertex and an vertex, set to be the vertex, to be the vertex. Then, by construction, for any directed we have Hence and the proof is complete. ∎
The above proposition hints at the computational difference between determining whether a graph is 2-colorable or 3-colorable. Indeed, using the proposition to determine whether a graph is 2-colorable requires only computing one determinant since is independent of orientation. However, for a graph with edges, using the proposition to determine whether it is -colorable requires checking at most determinants, one for each orientation.
Following , we call a unitary operator a gauge transformation if it is multiplication operator with respect to the basis of vertices. So for where with
is unitarily equivalent under a gauge transformation to the standard Laplacian if and only if
We may assume the underlying graph is connected since has a decomposition as a direct sum of corresponding operators on each connected component.
Since one direction is clear. For the other, suppose Let , normalized with the supremum norm such that Since is normalized and the graph is connected, by the equation
the vanishing of ensures that . Define by for and extending linearly. Since is a gauge transformation.
Define sesquilinear forms by
Let We have
Then if the result is If and non-adjacent, then If is an edge, substituting for and for we have . Similarly an edge of the form results in But also
which is deg if , if and non-adjacent, and any edge of the form or results in Hence Therefore which implies is unitarily equivalent to under a gauge transformation. ∎
If is an oriented edge, say , let be the reverse edge. Then extend by Then if is a closed (unoriented) walk, the flux is defined by Hence, if is an oriented edge, then it contributes to the flux; and if is an edge, then contributes to the flux.
Note a directed graph has an underlying undirected graph with edge relation if or is a directed edge. By a walk in a graph we mean a finite list of vertices such that for A closed walk is a walk with the initial and final vertex coinciding.
The proof of the following proposition, in the case of may be found in . In order to derive this version from the one presented there, note that gauge transformations are diagonal in the standard basis for and thus commute with
Two magnetic Laplacians are unitarily equivalent under a gauge transformation if and only if and induce the same fluxes through closed walks.
3 Averaged Variational Principle
In this section we develop a tool (see ) for its origin) that will allow estimates on sums of eigenvalues of finite Laplacian operators.
If is a self-adjoint matrix, we denote its eigenvalues by and a corresponding orthonormal basis of eigenvectors by If is any dimensional subspace of and an orthonormal basis for then we define
Tr( is independent of the basis chosen. Indeed, let be the projection onto . Then So
using the spectral decomposition of Since the left hand side of the above string of equalities is independent of basis, the result holds.
We begin by stating the following classical result .
With notation as above, for we have
In particular if is any collection of orthonormal vectors, we have
(Harrell-Stubbe) Let be a self adjoint matrix with eigenvalues and corresponding normalized eigenvectors Suppose is a (positive) measure space and is measurable with . Then if for any we have
Note that provided that we have that
4 Inequalities for Sums of Eigenvalues of Magnetic Laplacians
We now apply the averaged variational principle, Theorem 1, to For the remainder of this section we further assume that is connected, and for any , if then In other words, we assume arises from orienting a connected, loopless, undirected graph without repeated edges. If has vertices, let denote the eigenvalues of
Let be the complete, loopless, undirected graph on the vertices of . Orient with some orientation such that the orientation of its restriction to is the orientation on . Suppose is a -regular directed subgraph of with a directed subgraph of This, in particular, implies that is connected on vertices. Note this is always possible by taking and with . However, for example, for may be taken to be or Let be the graph complement of in with the induced orientation. If is an oriented edge in we shall denote . We call two vertices adjacent and write if there is some oriented edge between them, either or If the graph is not clear from the context, we write if we wish to restrict the relation to Then let and We shall denote pairs in by . Let Define on by for an edge in , for an edge in , and for all Let Then, extend and to all of by setting them equal to outside of edges of
Define by where
Hence for any
We wish to calculate Note that for an edge in , we have For fixed and for any vertex exactly one of the three following possibilities occurs: , is adjacent to in , or is adjacent to in . Hence, since edges and their opposites occur in pairs in both and , and since is -regular and is regular, we have
Let Now we calculate There are four cases. If is an oriented edge in then
If is an oriented edge in then
If and neither nor is an oriented edge in then
Lastly, for any ,
For a finite set, let denote the cardinality of Then we have
We may achieve great simplifications of the above inequality if we take to contain only edges or reverse edges of or also, if needed, “loops” of the form .
Before continuing, we note the following. The quantity is known in graph theory literature as the first Zagreb index of (see ), where is the degree of in . Then note that
Indeed, for each appears once in exactly terms in the sum.
For what follows, let Then (2) simplifies to
Since we will be wishing to minimize this quantity, we define such that
However, since can take on any number greater than or equal to if then the optimal choice is Hence, we have proven the following theorem.
Suppose is a directed graph arising from orienting a connected, loopless, undirected graph without repeated edges. Let be the degree of a regular subgraph of containing as a subgraph. Then if is an integer with we have
Note that if is the degree matrix, and Hence we may rewrite the above inequality as follows. For as in the previous theorem, we have
We may increase the bound on by admitting a combination of reverse edges of and loops to Then the cosine terms cancel in pairs for reverse edges and loops add terms proportional to the degree.
In  the half-filled band, corresponding to the case that is studied. As a corollary we provide an inequality for the half-filled band in the case of a regular graph. Let . Then Note in this case and Note further that any is a sum of magnetic Laplacians corresponding to individual edges, each being a positive operator. It follows that eigenvalue sums for a subgraph are bounded above by corresponding sums for the graph. Therefore we have the following result for the half-band.
Suppose is a directed graph on vertices arising from orienting a connected, undirected subgraph of a -regular undirected loopless graph on vertices without repeated edges. Then for we have
Note the above bounds hold for all choices of In particular they hold for the standard combinatorial Laplacian. This connects to the ”flux phase” problem investigated for the case of planar graphs in , that is to find the choice of that maximizes the sum of the half-band eigenvalues. For different classes of graphs, the optimal choice of may vary, leading to the possibility for improvements to the above bounds in such cases for particular choices of
We give two simple examples. Let with some orientation. The condition is so the only non-trivial choice for is . The above inequality reduces to This is sharp since taking constant equal to on any orientation yields a spectrum of
Consider the cycle Then the spectrum of the standard Laplacian is Hence the inequality is sharp at for this example, as the sum of the first half of the spectrum is where
I would like to thank Evans Harrell for helpful discussion of the ideas in this paper and comments on a draft. Also, I would like to thank the anonymous reviewer for comments that led to substantial improvements to an earlier version of this paper.
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