Discriminants, symmetrized graph monomials, and sums of squares
Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels of the original graph. In the 20-th century this construction was studied by several authors. We pose the question for which graphs this polynomial is a non-negative resp. a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on discriminant of the derivative of a univariate polynomial, and an interesting example of P. and A. Lax of a graph with edges whose symmetrized graph monomial is non-negative but not a sum of squares. We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations.
In what follows by a graph we will always mean a (directed or undirected) graph with (possibly) multiple edges but no loops. The classical construction of J. J. Sylvester and J. Petersen [7, 8] associates to an arbitrary directed loopless graph a symmetric polynomial as follows:
Let be a directed graph, with vertices and adjacency matrix Define first its graph monomial as follows
where is the number of edges joining with
The symmetrized graph monomial of is defined as
Notice that if the original is undirected one can still define up to a sign by choosing an arbitrary orientation of its edges. Symmetrized graph monomials are closely related to -invariants and covariants and were introduced in 1870’s in attempt to find new tools in the invariant theory. Namely, to obtain an -coinvariant from a given we have to perform two standard operations. First we express the symmetric polynomial in variables in terms of the elementary symmetric functions and obtain the (inhomogeneous) polynomial . Secondly, we perform the standard homogenization of a polynomial of a given degree
The following fundamental proposition apparently goes back to A. Cayley, see Theorem 2.4 of .
(i) If is a -regular graph with vertices then is either an -invariant of degree in variables or it is identically zero.
(ii) Conversely, if is an -invariant of degree and order then there exist -regular graphs with vertices and integers such that
Recall that a graph is called -regular if every its vertex has valency . Notice that if is an arbitrary graph then it is natural to interpret its polynomial as the -coinvariant.)
The natural question about the kernel of the map sending to (or to ) was already discussed by J. Petersen who claimed that he has found a necessary and sufficient condition when belongs to the kernel, see . This claim turned out to be false. (An interesting correspondence between J. J. Sylvester, D. Hilbert and F. Klein related to this topic can be found in .) The kernel of this map seems to be related to several open problems such as Alon-Tarsi  and the Rota basis conjecture . (We want to thank Professor A. Abdesselam for this valuable information, see .)
In the present paper we are interested in examples of graphs whose symmetrized graph monomial are non-negative resp. sum of squares. Our interest in this matter has two sources.
The first one is a recent conjecture by F. Sottile and E. Mukhin formulated on the AIM meeting ’Algebraic systems with only real solutions’ in October 2010.
The discriminant of the derivative of a polynomial of degree is the sum of squares of polynomials in the differences of the roots of
Based on our calculations and computer experiments we propose the following extension of the latter conjecture. We call an arbitrary graph with all edges of even multiplicity a square graph. Notice that the symmetrized graph monomial of a square graph is obviously is a sum of squares.
For any non-negative integer the discriminant of the th derivative of a polynomial of degree is a finite positive linear combination of the symmetrized graph monomials where all underlying graphs are square graphs with vertices. On other words, lies in the convex cone spanned by the symmetrized graph monomials of the square graphs with vertices and edges.
Observe that and is, therefore, even. We use the following agreement in our figures below. If a shown graph has fewer vertices than then we always assume that it is appended by the required number of isolated vertices. The following examples support the above conjectures.
If then is proportional to where is the complete graph on vertices with all edges of multiplicity .
For , the discriminant equals
In other words, where the graph is given in Fig. 1 (appended with isolated vertices).
For we conjecture that the discriminant equals
where the graphs and are given in Fig. 2. (This claim is verified for )
The discriminant is given by
where are given in Fig. 3.
where are given in Fig. 4. (Note that this representation as sum of graphs is by no means unique.)
The second motivation of the present study is an interesting example of a graph whose symmetrized graph monomial is non-negative but not a sum of squares. Namely, the main result of  shows that for the graph given in Fig. 5 has this property.
Our main computer-aided results regarding the case with 4 resp. 6-edged graphs are given below. Notice that there exist graphs with edges and graphs with edges. We say that two graphs are equivalent if their symmetrized graph monomials are non-vanishing identically and proportional.
(i) graphs with edges have identically vanishing symmetrized graph monomial. (ii) The remaining graphs are divided into 4 equivalence classes presented in Fig 6. (iii) The first two classes contain square graphs and, thus, their symmetrized monomials are non-negative. (iv) The third graph is non-negative (as a positive linear combination of the Lax graph and a square graph). Since it effectively depends only on three variables, it is SOS, see . (v) The last graph is the Lax graph which is thus the only non-negative graph with edges not being a SOS.
(i) graphs with edges have identically vanishing symmetrized graph monomial. (ii) The remaining graphs are divided into 27 equivalence classes. (iii) of these classes can be expressed as non-negative linear combinations of square graphs, i.e. lie in the convex cone spanned by the square graphs. (iv) Of the remaining 15 classes, symmetrized graph monomial of of them change sign. (v) Of the remaining 8 classes (which are presented on Fig. 7) the first 5 are sums of squares, given as matrix representations in the Appendix below. (Notice however that these symmetrized graph monomials do not lie in the convex cone spanned by the square graphs.) (vi) The last 3 classes contain all non-negative graphs with edges, which are not SOS and, therefore, give new examples of graphs a’la Lax.
It is classically known that for any given number of vertices and edges , the linear span of the symmetrized graph monomials coming from all graphs with vertices and edges coincides with the linear space of all symmetric translation-invariant polynomials of degree in variables.
Notice that translation invariant symmetric polynomials appeared also in the early 1970’s in the study of integrable -body problems in mathematical physics (apparently) see the famous paper of F. Calogero . A few much more recent publications related to the ring of such polynomials in connection with the investigation of multi-particle interactions and the quantum Hall effect were printed since then, see e.g. , . In particular, the ring structure and the dimensions of the homogeneous components of this ring were calculated. In particular, it was shown in § IV of  and  that the ring of translation invariant symmetric polynomials (with integer coefficients) in is isomorphic as a graded ring to the polynomial ring where stands for the -th elementary symmetric function in . From this fact one can easily show that the dimension of its -th homogeneous component equals the number of distinct partitions of where each part is strictly bigger than and the number of parts is at most . Several natural linear bases were also suggested for each such homogeneous component, see (29) in  and . It seems that the authors of the latter papers were unaware of the mathematical developments in this field related to graphs.
Acknowledgements. We are sincerely grateful to Professors A. Abdesselam, F. Sottile, B. Sturmfels for discussions and important references and to Doctor P. Rostalski for his assistance with computer-aided proof of the fact that certain symmetrized graph monomials are SOS.
2. Some generalities about symmetrized graph monomials
Let be a directed graph with edges and vertices Let be an integer partition of A partition-coloring of with is an assignment of colors to the edges and vertices of satisfying the following:
For each color we paint the vertex and edges connected to with the color
Each edge of is painted with exacly one color.
Each vertex is painted at most once.
An edge is odd-colored if it has color and is directed to a vertex with the same color. The coloring is said to be negative if there is an odd number of odd-colored edges in and positive otherwise.
Given a polynomial and a multi-index we use the notation to denote the coefficient in front of in
Note that we may view as a partition of the sum of the indices.
Let be a directed graph with edges and vertices Then is given by the number of positive partition-colorings of with minus the number of negative partition-colorings.
See [10, Lemma 2.3]. ∎
2.1. Bases for
It is known that the dimension of with is given by the number of integer partitions of where each part is at least of size 2. Such integer partition will be called a 2-partition.
To each 2-partition , we associate the following graph : For each we have a connected component of consisting of a center vertex, connected to other vertices, with the edges directed away from the center vertex. Since is an integer partition of it follows that has exacly edges. This type of graph will be called a partition graph.
Let be a polynomial. Then
This is standard, by a straightforward combinatorical argument. ∎
If is non-negative, then is non-negative.
If is a sum of squares, then is a sum of squares.
2.2. Partition graphs
We will use the notation that every symmetric polynomial associated with a graph on edges is symmetrized over variables. Corollary 3 says that if a relation holds for symmetrizations in variables, it will also hold for variables, and therefore, each relation derived in this section also holds for variables.
Let be a partition graph with edges, and let be a 2-partition.
We will try to color the graph with
Since we may only color the center vertices of Hence, all edges in each component of must have the same color as the center vertex. It is clear that such coloring is impossible if If we see that each coloring has positive sign, since only center vertices are colored and all connected edges are directed outwards.
The only difference between two colorings must be the assignment of the colors to the center vertices. Hence, components with the same size can permute colors, which yields
number of ways to color with the partition ∎
It follows that all partition graphs yield linearly independent polynomials, since each partition graph unique contributes with the monomial The number of partition graphs on edges equals the dimension of and must therefore span the entire vector space.
2.3. Square graphs
We will use the notation to denote a partition where are the odd parts in decreasing order, and are the even parts in decreasing order. Parts are allowed to be equal to 0, so that can be used as multi-index over variables.
Now we define a second type of graphs that we associate with 2-paritions of even integers:
Let , be a 2-partition of Since this is a partition of an even integer, must be even.
For each even we have a connected component of consisting of a center vertex, connected to other vertices, with the edges directed away from the center vertex, and with multiplicity 2.
For each pair of odd parts, we have a connected component consisting of two center vertices and such that is connected to other vertices for with edges of multiplicity 2, and the center vertices are connected with a double edge. This type of component will be called a glued component.
Thus, each edge in has multiplicity 2, and the number of edges, counting multiplicity, is This type of graph will be called a square graph. Note that is a sum of squares.
Let be a square graph where Then
Similarly to Lemma 6, it is clear that a coloring of with colors require that each center vertex is painted.
The center vertex of a component with only two vertices is not uniquely determined, so we have choices of centers.
It is clear that each glued component contributes with exacly one odd edge for every coloring, and the sign is therefore the same for each coloring. The number of glued components are precisely
Lastly, we may permute the colors corresponding to center vertices with the same degree. These observations together yields the formula
Define a total order on 2-partitions as follows:
Let and be 2-partitions. We say that if for and one of the following holds:
is odd and is even.
This generalizes to
Let be a square graph. Then we may write
It suffices to show that there is no partition-coloring of with if since this implies
We now have three cases to consider:
Case 1: for and where and are either both odd or both even.
We must paint a center vertex and connected edges, since
There is no vacant center vertex in with degree at least all such centers have already been painted with the colors Hence a coloring is impossible in this case.
Case 2: for , is odd and is even. This condition impies that
Every component of has an even number of edges, and only vertices with degree at least three can be colored with an odd color. Therefore, glued components must be colored with exacly zero or two odd colors, and non-glued component must have an even number of edges of each present color. This implies that a coloring is only possible if but this is not true in the considered case.
Hence, there is no coloring of with the colors given by and therefore, the coefficient in front of is 0 in implying ∎
The polynomials obtained from the square graphs with edges is a basis for if is even.
3. Final remarks
Some obvious challenges related to this project are as follows.
1.ï¿½ Prove Conjectures 1 and 2.
2.ï¿½ Describe the boundary of the convex cone spanned by all square graphs with a given number of (double) edges and vertices.
3. Find more examples of graphs a’la Lax.
Here we give a SOS presentation of the symmetrized graph monomial for the first classes given in Fig. 7. Each row yield the same polynomial, up to a constant. The symmetrized graph monomial from row is a constant multiple of the polynomial where is the coefficient vector and is the corresponding symmetric positive semi-definite matrix, given below. This certifies that the first 5 classes of graphs are sum of squares.
By using Matlab together with Yalmip, one may verify that the last three classes cannot be expressed as sums of squares. It is relatively straightforward to verify that the polynomials indeed are non-negative, using methods similarly to .
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