Listening to the cohomology of graphs
We prove that the spectrum of the Kirchhoff Laplacian of a finite simple Barycentric refined graph and the spectrum of the unimodular connection Laplacian of determine each other. Indeed, we note that with is similar to the Hodge Laplacian of which is in one dimensions the direct sum of the Kirchhoff Laplacian and its 1-form analog . In this one-dimensional case, the spectrum of a single choice of one of the three matrices or alone is enough to determine both Betti numbers of as well as the spectrum of the other matrices. It follows from the similarity of and that for a one-dimensional complex which is a Barycentric refinement, the number of connectivity components is the number of eigenvalues of and that the genus is the number of eigenvalues of . It will also lead to much better estimates of spectral radius and algebraic connectivity. For a general abstract finite simplicial complex , we express the Green function values in terms of the stars of and . One can see and as stable and unstable manifolds of a simplex and as heteroclinic intersection numbers or curvatures and the identity as a collection of Gauss-Bonnet formulas. A special case is the previously known . The homoclinic energy by definition satisfies . The matrix which is similar to has a sum of matrix entries which is Wu characteristic . For with dimension we don’t know yet how to recover the Betti numbers from the eigenvalues of the matrix or from . So far, it is only possible to get it from a collection of block matrices, via the Hodge relations . A natural conjecture is that if is a Barycentric refinement of an other complex, then the spectrum of determines the Betti vector . This note only shows this to be true if has dimension .
Key words and phrases:Cohomology, Spectral properties of graphs
1991 Mathematics Subject Classification:11P32, 11R52, 11A41
Any abstract finite simplicial complex defines a graph , where and is the set of pairs of simplices , where either or . As the dimension on is a coloring, the chromatic number of is equal to the clique number . The graph defines so a new complex , the Whitney complex of , which is called the Barycentric refinement of . It consists of all subsets of in which all elements are connected to each other. The connection matrix of is defined as if and else. If , then , with identifying with . The connection matrix of is then an matrix, where .
Given a finite abstract simplicial complex with -vector , its Dirac operator is , where is the exterior derivative from , where is the linear space of antisymmetric functions in variables defined on -dimensional simplices. This space of discrete -forms has dimension , the cardinality of sets in of length . Also the matrices or can then be realized as matrices, where . Both and so depend on the orientation chosen for each simplex. (The choice of orientation is a choice of basis in and has nothing to do with orientability as no compatibility is required). The matrix is the Hodge Laplacian. It decomposes into blocks , the individual form Laplacians. The kernel of has dimension , which is the ’th Betti number. The vector is the Betti vector of .
There are two natural questions: can one read off the Betti vector from the eigenvalues of ? Can one read off the Betti vector from the eigenvalues of ? We will see that in one dimensions, the two questions are related but that there is already a subtlety: there are -isospectral complexes with different but that after a Barycentric refinement, we always can read off from the eigenvalues of . A weaker question is to get the Euler characteristic from the eigenvalues, either in the or case. We have seen that one in general can get from the eigenvalues of as is the number of positive minus the number of negative eigenvalues . In one dimensions, we can get from the eigenvalues of . We don’t know how to get the Euler characteristic from the eigenvalues of in higher dimensions. We know by Hodge only , the nullity of .
For one-dimensional complexes, the single incidence matrix alone determines everything. The matrix is the gradient, its transpose matrix is the divergence. The Hodge Laplacian has a block decomposition for and . The matrix is the Kirchhoff matrix of the graph , where is the diagonal vertex degree matrix and is the adjacency matrix of . The matrix is the Laplacian on -forms. The ’th cohomology group is defined as , the ’st cohomology group is the linear space given by all function on all functions on edges modulo gradients. The nullity of is the number of connectivity components of . The nullity of the matrix is the genus of , which is the number of holes or the number of generators of the fundamental group.
The operators and the operators are all defined on the same -dimensional Hilbert space, where is the number of simplices in . The matrices all depend on the order in which the simplices are arranged. The matrix also depends on the orientation of simplicial complex. We can not only orient the one and higher dimensional simplices by assigning to them a preferred permutation, allowing to define the orientation through the signature, we can also orient the zero dimensional simplices. This changes too. In general, the matrix is always reducible, except if . On the other hand, the connection operator is irreducible if is connected. This means that there is a positive integer such that has only positive entries. Therefore, has a single Perron-Frobenius eigenvalue and this will be inherited by .
When estimating the largest and smallest non-zero eigenvalue of the Laplacian of a one-dimensional complex it does not matter whether we look for or . Both the maximal eigenvalue = spectral radius as well as the second eigenvalue = ground state = algebraic connectivity are of enormous interest. We plan to explore this more elsewhere as in order to appreciate this fully, it requires to compare the improvement with the existing literature on the estimates but a preliminary assessment shows that the formula is quite powerful to estimate both and for Barycentric refinement graphs which, all examples of bipartite irregular graphs if is not circular. (The omission of circular graphs is no issue as we know there all the eigenvalues explicitly). The estimates are often far better than the best established bounds. One of the questions is to estimate how much below the largest eigenvalue has dropped below the obvious upper bound , where is the maximal vertex degree of the graph. We can use that is a non-negative matrix for which much is known .
The simplest spectral estimate of eigenvalues of is the upper bound , where is the maximal vertex degree of the line graph of , the maximal spectral radius of is bound above by which is if is the maximal vertex degree. This is better than estimates ( Theorem 3.5) or ( Theorem 2.3) established for irregular graphs of diameter and vertices. But our new estimate only holds after a Barycentric refinement. But for higher Barycentric refinements we even have as the vertex degree of has then dropped considerably. The relation between and is so useful because is equal o , where is the adjacency matrix of the connection graph of and is the identity matrix. The connection graph of is the graph in which two simplices are connected if they intersect. Any knowledge about the spectrum of adjacency matrices of graphs (like ) translates directly into spectral statements of , whether it is spectral radius, algebraic connectivity or order structures of the eigenvalues. The reason is that the map maps two spectral intervals of piecewise monotonically into the spectral interval of .
Surprisingly little appears to be known about the relation of the Betti vector and the eigenvalues of the Hodge Laplacian, both in the manifold as well as in the case of simplicial complexes. In the graph case, a natural spectral problem is already to relate the spectrum of the adjacency matrix with the topology of the graph. Its spectrum determines the number of edges and the number of triangles of the eigenvalues of . For a -dimensional connected complex for which every unit sphere is a -dimensional circular graph with , we can then read off the Euler characteristic and so the cohomology of a connected oriented surface for which the boundary is a collection of closed circular graphs. For the Kirchhoff matrix of a graph, we get the number of vertices, the trace is twice the number of edges from handshaking.
We also can get the Zagreb index from the spectrum and because is related to the Wu characteristic we also can get the Euler characteristic. (The connection of Wu characteristic with the Zagreb index has been pointed out to us by Tamas Reti.) This analysis however requires the graph to be geometric, like being a nice triangulation of a -dimensional surface with boundary. To illustrate how little is known, one can ask how to read off the orientability of a triangulated surface from the eigenvalues of the Hodge Laplacian , without indication from which -form vector the zero eigenvalues come from. While non-orientability implies a trivial kernel for the -form Laplacian , we don’t yet know how to access non-orientability even in the two dimensional from the spectrum of or from the spectrum of .
2. Relations between spectral data
Let us report first a fact which is probably well known even-so we have not found a reference despite the existence of a rather large literature on spectral graph theory. For books, see [2, 7, 6, 3, 4, 19, 22, 5]. As much of this literature is also about spectra of the adjacency matrix, the here discussed relation between (a shifted adjacency matrix) and (a Laplacian matrix) is useful as this relation is usually only available for regular graphs, where the vertex degree is constant.
In the following, we mean with a graph the -dimensional simplicial complex defined by and not the Whitney complex. For for example, this means , , and not the set of all subsets of . In that case, are determined by the Kirchhoff Laplacian alone:
Lemma 1 (Kirchhoff listens to the genus).
For any graph equipped with the 1-dimensional complex, the eigenvalues of the Kirchhoff matrix determines the Betti numbers and the eigenvalues of as well as the eigenvalues of . Two isospectral graphs are strongly isospectral, meaning -isospectral.
By Euler handshake, . We also have . The number is the number of zero eigenvalues of . Now, by Euler-Poincaré, . From this, one gets
and the right hand side uses only spectral data. ∎
This can be compared with two-dimensional smooth regions in the plane which can be glued together to build a Riemann surface, for which only and matter. The reason is that we essentially deal with a complex one-dimensional curve then and that the double cover ramified over one-dimensional closed curves is linked to the two dimensional region by Riemann-Hurwitz. So, it is no surprise that one can hear the genus of a drum . We are not aware of any other result, both in the simplicial complex, nor in the manifold case, where one can hear larger Betti numbers with from the eigenvalues of any of the Laplacians on a three or higher dimensional manifold without looking at a sequence of form Laplacians which build up the Hodge Laplacian .
In the case of a -dimensional simplicial complex, where are no higher exterior derivatives like the curl have to be considered, the spectrum of determines completely the spectrum of the Hodge operator and so .
Lemma 2 (Hodge listens to the genus).
Let be the Hodge matrix of a -dimensional simplicial complex . The eigenvalues of the matrix alone determine the Betti numbers of .
The matrix has two blocks and . It is a general fact from linear algebra or the Cauchy-Binet formula for the coefficients of the characteristic polynomial  that and are essentially isospectral meaning that their non-zero eigenvalues agree. It is also a very special case of McKean-Singer supersymmetry which in general assures that the non-zero Bosonic and Fermonic spectra agree for the Hodge Laplacian . Now, from , we can access . We can also hear the number of eigenvalues as . The trace is . Therefore, we know both and and so . Knowing and determines and . ∎
Lemma 3 (Connection does not hear the genus).
There exist two one-dimensional simplicial complexes which are -isospectral but which have different .
The following pair of simplicial complexes was given in . The first one, is generated by the sets
The second one is generated by
We know however that in all dimensions, the eigenvalues of determine the Euler characteristic of as we have proven that in general, the Euler characteristic is , where and are the number of positive and negative eigenvalues of the connection Laplacian . In the Barycentric refined case, this will lead to a relation between the eigenvalues and and the Betti numbers. The eigenvectors of and are of course the same.
3. The theorem
Our main result here relates the Hodge operator with the “Hydrogen operator” . The assumption of having chromatic number is not that severe but necessary as the above lemma shows. Every Barycentric refinement of a graph has chromatic number , the color being the dimension function.
Theorem 1 (Hydrogen and Hodge).
Given a graph which is a Barycentric refinement of a one-dimensional complex, then and are similar. In a suitable basis: .
We will prove this in the next section. The etymology of ”Hydrogren” was explained in , where we looked at the functional which is in general of geometric interest as it is . In with Laplacian , the kernel of the inverse is the Newton potential because of Gauss . In quantum mechanics, the Hydrogen Hamiltonian is has at least a formal analogy to .
Here is a simple example. We take leading to a graph with vertices. It is the simplicial complex . First, lets write down the connection Laplacian:
The Dirac matrix is
made of the incidence matrix , (the lower left block) which is the gradient mapping -forms to -forms as well as its adjoint , (the upper right block) which is the divergence mapping -forms to -forms. The Hodge Laplacian has now a diagonal block structure , where is the Helmholtz matrix (a matrix). The -form block is a matrix which is essentially isospectral to . It is an invertible Jacobi matrix in this case, reflecting that :
The Green’s function is
The Hydrogen operator is the sign-less Hodge matrix. It is a non-negative matrix:
The eigenvalues of are , , , , , , , , . The eigenvalues of are , , , , , , , , . The eigenvalues of are , , , , , , , , in accordance to the Zeta function functional equation assuring that and have the same eigenvalues if the complex is one-dimensional .
Using a coordinate change with diagonal matrix having entries for and for , the matrix is the sign-less Hodge Laplacian. Now, . We have implemented the matrices explicitly using a computer algebra system and included the code at the end. There are many puzzles which remain: we have no idea yet for example how to fix a relation between and in the higher dimensional case. We believe that there should be a deformation of which still makes this happen as we have seen examples, where a change of works. For a triangle complex for example, we just have to change the interaction energy between the -dimensional simplex and the others and still get . Maybe, in general, a small tuning suffices to achieve a connection of with a non-negative matrix for which the spectral analysis is easier.
As explained below, one can get intuition from physics. The matrix entries can be seen as manifestations of energy potentials. There are indications in the form of examples which suggest that we can change to still have a Hydrogen formula. This leads to gauge fields. Already the conjugation is a gauge change even so trivial. It corresponds to a gauge field. The hope is that the inclusion of more general gauge fields (which changes the spectrum of ) would allow to save the algebraic relation between and also in higher dimensions.
For a Barycentric refined graph, the sign-less Hodge matrix is a non-negative matrix which satisfies and .
A matrix is called reducible if there is a basis in which it can be written . If no such decomposition is possible, the matrix is called irreducible. For a non-negative matrix, irreducibility is equivalent to the statement that there exists such that is a positive matrix, meaning that all entries are positive. Both and are non-negative matrices in a suitable basis. If the graph is connected and is not zero-dimensional, then is reducible but is irreducible. We can also use that in one dimensions, the spectrum of is the same than the spectrum of .
For a connected Barycentric refined graph, the maximal eigenvalue of the Kirchhoff matrix has multiplicity .
Example. If is a circular graph with even , then are the eigenvalues of and are the eigenvalues of . This example was the first time we have seen the relation . It was essential to find an explicit Zeta function of in the Barycentric limit .
4. The proof of the theorem
In order to prove the result, we need to know the matrix entries of
the Green’s function . We know already the diagonal entries
This means that for a zero-dimensional simplex , we have ,
where is the vertex degree and if is a one-dimensional simplex.
Because , we have on the zero-dimensional sector
and for the one-dimensional sector. This matches the matrix entries
of in the diagonal. In order to prove the result we have to establish:
a) if and .
b) if and or .
c) if and .
d) if .
We can use this data to build each column vector of . Now just compute
the dot product of a row vector of with a column vector of
and compute . If , this is as there is a hit
and then there are terms . For , then the dot product has
only two terms, one being , the other . Here are a bit more details even so
the general case will make this obsolete:
As is a Barycentric refinement, its vertices are 2-colorable. Any coloring with coloring as well as an orientation of the edges defines a basis. The alternating sign change of the basis assures that which is the sign-less Kirchhoff matrix. It is isospectral to . We will show that .
a) First the diagonal: since , where is the Green’s operator, we know all the entries of . In the diagonal we have . As , we have . Note that this works also on the 1-form sector as every edge has exactly two neighbors and therefore for an edge.
b) Now the mixed dimension part: assume where is zero dimensional and is one dimensional. Then we know and . This means that so that has a block structure.
c) Now we look at the case where are both zero dimensional. Then and . This agrees with .
d) Finally look at the case where are both -dimensional. Then if is not empty and else. We can use to get . Also, if do not intersect, then . ∎
The full generalization uses the ”star” of , which is the set of all if . It is a collection of simplices, but not a simplicial complex in general. It defines a graph in the Barycentric refinement . There is a subtlety: while we know that for a simplicial complex , the Euler characteristic of and its Barycentric refinement are the same, this is not true for sets of simplices which are not simplicial complexes. Take which is not a simplicial complex but which has Euler characteristic with . The Barycentric refinement of is now the complete graph which has Euler characteristic . It is the fact that the star is not a simplicial complex which requires us to compute in and does not allow us not escape to its Barycentric refinement, which is a graph.
The following ”Green star formula” is the ultimate answer about the Green function entries.
Proposition 1 (Green Star formula).
(Sketch) We know by Cramer that , where is the matrix with row and column deleted. Now proceed by induction in the same way as for the unimodularity theorem. For proving the formula for a pair , consider an other maximal simplex away from and (which is possible if we don’t deal with a complete graph), then use the multiplicative Poincaré-Hopf formula for the change of the determinant: both sides are multiplied by . See . ∎
The Green star formula gives the inverse in a concrete way. One can also write the matrix multiplication and verify each entry:
Using the notation if intersect: it means for
These are both local Gauss-Bonnet statements similar as in . The Green star formula is equivalent to these two statements about stars in simplicial complexes.
1) We have in particular which means the self-interaction energy of a simplex is the Euler characteristic of its star. 2) If we look at the dual star of , which is the set of all with , then this is a complete simplicial complex with Euler characteristic . We can now write
We see from this that the matrix refers to the inside stable part of the simplices while the inverse matrix refers to the outside unstable part of the simplices.
The connection matrix is conjugated to
The inverse is conjugated via the diagonal matrix to
We see an obvious duality. Mending the two pictures requires to go into the complex. Define the diagonal matrix which has the diagonal entries . Now we can look at
While is isospectral to the Hodge Laplacian , the turned operator is of the form . Now, paired with the energy theorem , we have a relation with the Wu characteristic
which is the total energy of the operator .
For any -dimensional complex which is a Barycentric refinement, we have .
Now this is interesting, as the energy is still a combinatorial invariant, a quantity which does not change if we make a Barycentric refinement. We have actually proven in , see also  that for geometric complexes with boundary, . If we interpret the curvature for as an energy of , we see that it is located on the boundary of a complex and that the total energy is zero in the geometric case. For a closed circular graph for example, the total energy of is zero. In higher dimensions, the gauge fields have to be added differently and it is still unclear whether one can deform to mend the Hydrogen formula. If it is possible, then most likely through a variational mechanism which by wishful thinking should relate to some kind of radiation.
In general, for any complex , the total energy of is . This total energy is zero for geometric graphs without boundary. The energy curvature is supported on the boundary of a geometric space with boundary.
This is not that unfamiliar if we compare a simplicial complex with a space or space time manifold in physics. These manifolds naturally have boundaries as event horizons of singularities. Now, as Hawking famously first pointed out, these boundaries radiate. The analogy is certainly far fetched as what we deal here with relatively basic combinatorial geometry of finite set of sets. Still, it is a mathematical fact that if we define energy of such a geometry as , where is the Euler characteristic and is the Wu characteristic, then due to Dehn-Sommerville, in the interior of Euclidean like parts of space, the energy density (curvature) is zero and all the energy density is at the boundary or located at topological defects of space. History cautions to speculate as the molecular vortex picture debacle reminds. But fundamental questions about the nature of space and time has always motivated mathematics. Here, we deal with remarkable mathematical theorems like the energy theorem which assures that the sum over all interaction energies of simplices in a simplicial complex is the Euler characteristic of the simplicial complex. And this was certainly motivated by physics of the Laplacian in Euclidean space.
Let us explain why for any simplicial complex and any simplex the ”star formula” holds. To see the ”star formula”, we write the unit sphere as the Zykov join of its stable and unstable part and use that the ”genus” is multiplicative . Now, since the stable sphere is the boundary of a simplicial complex, we have . The statement is true because every simplex in is bijectively related to a simplex in . A vertex in corresponds to a simplex in . In some sense, collapsing the simplex in the star to a point and removing that point gives the stable sphere .
The fact that for a bipartite graph, and the Kirchhoff matrix are unitarily equivalent appears in Proposition 2.2 of . The result more generally holds for completely positive graphs, as these are the graphs which have no odd cycles of length larger than . It already does not apply for the Barycentric refinemd triangular graph , where the eigenvalues of are and the eigenvalues of are .
As an application we can compute the eigenvalues of the connection matrices of classes of -dimensional operators like circular graphs. The result also sheds light on the eigenvalue structure. Any integer eigenvalue for different from leads to non-integer eigenvalues of . More applications are likely to follow.
5. Hearing the cohomology
For a -dimensional complex , the Hodge operator decomposes into two blocks . The Betti numbers are then for . The counts the number of connectivity components, the number counts the number of generators of the fundamental group. We in general can not hear the cohomology of a complex, when listening to . The example given is even one-dimensional .
We assume the complex to be a Barycentric refinement of . This implies chromatic number 2 for and that is bipartite. Lets look first at the eigenvalue :
For every connectivity component, we have an eigenvalue of . The eigenvector is a -coloring supported on vertices of which were zero dimensional in . Every connected component has exactly one eigenvalue .
We only have to show that every eigenvector to an eigenvalue is supported on vertices of which were zero dimensional in . This can be done by induction on the number of -dimensional vertices, (vertices in which were -dimensional in ). Lets prove more generally that any eigenvector to the eigenvalue is supported on the -dimensional part of the complex, where it is necessarily a coloring. From the fact that we get . We can now use induction. Lets call a vertex with vertex degree a ”leaf”. An eigenvalue corresponds to an eigenvalue of the adjacency matrix of the connection graph. For every vertex , the average of all values on edges connected to is zero. Assume there is an edge with a leaf attached. Then . So, we can apply induction and remove the leaf. Without any leaf, the original complex must have been a closed loop. Assume that for some . When looking at vertices we see changes sign along the edges of the loop and that the two neighbors have the same sign and add up to zero. ∎
Now, we look at the eigenvalue :
Every homotopically non-trivial closed cycle leads to an eigenvalue . The eigenvector is a coloring of the edges of the cycle and supported on edges. A basis of the eigenspace of corresponds to a generating set of the fundamental group.
Every cycle leads to an eigenvector: just put alternating values on the edges of a cycle and put everywhere else. The fact that every eigenvector can be traced back to a closed path is a consequence of the Hurwicz theorem relating the fundamental group with the first homology group . The Hurwicz homomorphism is explicit for : take a closed path and build from it a function on edges telling how many times an edge has been traversed incorporating the direction. Now apply the heat flow on this function. As the Hodge matrix has only nonnegative eigenvalues, the positive eigenvalue part will die out and the limit will be located on the kernel of , which gives a representative of the cohomology group . It can also be seen from the relation that is similar to as we have already taken care of the eigenvalues of which come from eigenvalues of . and has eigenvectors with the same support than as the conjugation is done by a diagonal matrix. ∎
Can we see from the eigenvalues of , which ones belong to -forms and which one belong to forms? The positive eigenvalues of are the ones from the -forms and the negative eigenvalues of belong to the -forms. For higher dimensional complexes, the eigenvalues of can take both values. In the one dimensional case, the eigenvalues are always non-negative. We can also describe every point of . But in , we have connections between vertices and edges, while in we have connections between vertices and vertices, and edges with edges.
If is the product of two -dimensional complexes which are Barycentric refinements, then the cohomology of is determined by the cohomology of . Assume we know the eigenvalues of , we can from the multiplicities get the eigenvalues of and so the eigenvalues of the Hodge operators and from this the eigenvalues of the Hodge operator of . Can we do that in general for an arbitrary number of products ?
One should probably first focus on the case and try to hear the second cohomology from the spectrum of . We made some experiments with random complexes and counted the number of different eigenvalues of and compared this with . We tried to correlated with spectral data like the fraction , where counts the number of different eigenvalues of the matrix . Also no relation between the factorization of the characteristic polynomial and has been found yet. But there are other correlations still to be tried out, like relations between moments and cohomology or zeta function values with cohomology.
6. Mathematica content
Here is an illustration of the Hydrogen formula . We take a random graph, refine it to get a one-dimensional complex with chromatic number , then build both and the conjugation diagonal matrix .
We illustrate now the Green star formula
The code computes for an arbitrary simplicial complex the Green’s function entires of the inverse matrix of the connection matrix in terms of the stars and . The dimension functional on defines a locally injective function which can be seen as a Morse function. The gradient flow of this functional has stable and unstable manifolds and . Speaking in the language of hyperbolic dynamics, the homoclinic tangle of this ”Morse-Smale” system produces the Green functions. Every simplex is a critical point and the definition of Euler characteristic is a special case of the Morse inequality. Indeed, the component of the -vector of counts the number of critical points having Morse index (The Morse index of a simplex is the dimension of the stable manifold, which is here the plus the dimension of the sphere ).
The index is related to a homoclinic point, the matrix entries of the connection matrix and the matrix entries form the matrix entries of a matrix conjugated to the Green’s function, where is the diagonal matrix with entries. The Wu matrix is conjugated to and its matrix entries add up to the Wu characteristic .
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