Roots of Ehrhart polynomials arising from graphs
Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots of Ehrhart polynomials of polytopes of dimension satisfy , but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order lie in the circle or are negative integers, and (2) a Gorenstein Fano polytope of dimension has the roots of its Ehrhart polynomial in the narrower strip . Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.
Keywords: Ehrhart polynomial, edge polytope, Fano polytope, smooth polytope.
The root distribution of Ehrhart polynomials is one of the current topics on computational commutative algebra. It is well-known that the coefficients of an Ehrhart polynomial reflect combinatorial and geometric properties such as the volume of the polytope in the leading coefficient, gathered information about its faces in the second coefficient, etc. The roots of an Ehrhart polynomial should also reflect properties of a polytope that are hard to elicit just from the coefficients. Among the many papers on the topic, including , , ,  and , Beck et al.  conjecture that:
All roots of Ehrhart polynomials of lattice -polytopes satisfy .
Compared with the norm bound, which is in general , the strip in the conjecture puts a tight restriction on the distribution of roots for any Ehrhart polynomial.
This paper investigates the roots of Ehrhart polynomials of polytopes arising from graphs, namely, edge polytopes and symmetric edge polytopes. The results obtained not merely support Conjecture 0.1, but also reveal some interesting phenomena. Regarding the scope of the paper, note that both kinds of polytopes are “small” in a sense: That is, each edge polytope from a graph without loops is contained in a unit hypercube, and one from a graph with loops, in twice a unit hypercube; whereas each symmetric edge polytope is contained in twice a unit hypercube.
In Section 1, the distribution of roots of Ehrhart polynomials of edge polytopes is computed, and as a special case, that of complete multipartite graphs is studied. We observed from exhaustive computation that all roots have a negative real part and they are in the range of Conjecture 0.1. Moreover, for complete multipartite graphs of order , the roots lie in the circle or are negative integers greater than . And we conjecture its validity beyond the computed range of (Conjecture 1.4).
Simple edge polytopes constructed from graphs with possible loops are studied in Section 2. Roots of the Ehrhart polynomials are determined in some cases. Let be a graph of order with loops and its subgraph of order induced by vertices without a loop attached. Then, Theorem 2.5 proves that the real roots are in the interval , especially all integers in are roots of the polynomial; Theorem 2.6 determines that if , there are real non-integer roots each of which is unique in one of ranges for ; and Theorem 2.7 proves that if , all the integers are roots of the polynomial. We observed that all roots have a negative real part and are in the range of Conjecture 0.1.
The symmetric edge polytopes in Section 3 are Gorenstein Fano polytopes. A unimodular equivalence condition for two symmetric edge polytopes is also described in the language of graphs (Theorem 3.5). The polytopes have Ehrhart polynomials with an interesting root distribution: the roots are distributed symmetrically with respect to the vertical line . We not only observe that all roots are in the range of Conjecture 0.1, but also conjecture that all roots in for Gorenstein Fano polytopes of dimension (Conjecture 3.7).
Before starting the discussion, let us summarize the definitions of edge polytopes, symmetric edge polytopes, etc.
Throughout this paper, graphs are always finite, and so we usually omit the adjective “finite.” Let be a graph having no multiple edges on the vertex set and the edge set . Graphs may have loops in their edge sets unless explicitly excluded; in which case the graphs are called simple graphs. A walk of of length is a sequence of the edges of , where for . If, moreover, holds, then the walk is a closed walk. Such a closed walk is called a cycle of length if for all . In particular, a loop is a cycle of length 1. Another notation, , will be also used for the same cycle with . Two vertices and of are connected if or there exists a walk of such that and . The connectedness is an equivalence relation and the equivalence classes are called the components of . If itself is the only component, then is a connected graph. For further information on graph theory, we refer the reader to e.g. , 
If is an edge of between and , then we define . Here, is the -th unit coordinate vector of . In particular, for a loop at , one has . The edge polytope of is the convex polytope , which is the convex hull of the finite set . The dimension of equals to if the graph is a connected bipartite graph, or , other connected graphs . The edge polytopes of complete multipartite graphs are studied in . Note that if the graph is a complete graph, the edge polytope is also called the second hypersimplex in [30, Section 9].
Similarly, we define for an edge of a simple graph . Then, the symmetric edge polytope of is the convex polytope , which is the convex hull of the finite set . Note that if is the complete graph , the symmetric edge polytope coincides with the root polytope of the lattice defined in .
If is an integral convex polytope, then we define by
We call the Ehrhart polynomial of after Ehrhart, who succeeded in proving that is a polynomial in of degree with . If is the normalized volume of , then the leading coefficient of is .
An Ehrhart polynomial of is related to a sequence of integers called the -vector, , of by
where is the degree of . We call the polynomial in the numerator on the right-hand side of the equation above , the -polynomial of . Note that the -vectors and -polynomials are referred to by other names in the literature: e.g., in , , -vector or -Eulerian numbers are synonyms of -vector, and -polynomial or -Eulerian polynomial, of -polynomial. It follows from the definition that , , etc. It is known that each is nonnegative . If , then for every . Though the roots of the polynomial are the focus of this paper, the -vector is also a very important research subject. For the detailed discussion on Ehrhart polynomials of convex polytopes, we refer the reader to .
1. Edge polytopes of simple graphs
The aim in this section is to confirm Conjecture 0.1 for the Ehrhart polynomials of edge polytopes constructed from connected simple graphs, mainly by computational means.
1.1. Exhaustive Computation for Small Graphs
Let denote the polynomial ring in one variable over the field of complex numbers. Given a polynomial , we write for the set of roots of , i.e.,
We computed the Ehrhart polynomial of each edge polytope for connected simple graphs of orders up to nine; there are connected simple graphs of orders 111 These numbers of such graphs are known; see, e.g., [11, Chapter 4] or A001349 of the On-Line Encyclopedia of Integer Sequences.. Then, we solved each equation in the field of complex numbers. For the readers interested in our method of computation, see the small note in Appendix A.
Let denote , where the union runs over all connected simple graphs of order . Figure 1 plots points of , as a representative of all results. For all connected simple graphs of order –, Conjecture 0.1 holds.
Since an edge polytope is a kind of -polytope, the points in Figure 1 for are similar to those in Figure 6 of . However, the former has many more points, which form three clusters: one on the real axis, and other two being complex conjugates of each other and located nearer to the imaginary axis than the first cluster. The interesting thing is that no roots appear in the right half plane of the figure. The closest points to the imaginary axis are approximately , , and . A polynomial with roots only in the left half plane is called a stable polynomial. This observation raises an open question:
For any and any connected simple graph of order , is always a stable polynomial?
For a few infinite families of graphs, rigorous proofs are known: Proposition 1.2 just below and Examples in the next subsection.
A root of the Ehrhart polynomial of the complete graph satisfies
The Ehrhart polynomial of the complete graph is given in [30, Corollary 9.6]:
In cases where or , the Ehrhart polynomials are binomial coefficients, since the edge polytopes are simplices. Actually, they are:
Thus, there are no roots for , whereas are the roots for .
Hereafter, we assume . It is easy to see that are included in .
We shall first prove that . Let and . Then for a complex number , if and only if , since is . Let us prove for any complex number with a nonnegative real part by mathematical induction on .
holds for any complex number with .
Assume for that is true for any complex number with .
from and , one can deduce
Thus, holds for any complex number with .
Therefore, for any , the inequality holds for any complex number with a nonnegative real part. This implies that the real part of any complex root of is negative.
We shall also prove the other half, that . To this end, it suffices to show that all roots of have negative real parts. Let and be
Then for a complex number , it holds that
Let us prove for any complex number with a nonnegative real part by mathematical induction on .
For , it immediately follows from the inequality between and :
And so we need also as a base case:
Assume for the validity of for any complex number with .
Then, from the fact that
it follows that
Thus, holds for any complex number with .
Therefore, for any , the inequality holds for any complex number with a nonnegative real part. This implies that any complex root of has a negative real part. ∎
1.2. Complete Multipartite Graphs
We computed the roots of the Ehrhart polynomials of complete multipartite graphs as well. Since complete multipartite graphs are a special subclass of connected simple graphs, our interest is mainly on the cases where the general method could not complete the computation, i.e., complete multipartite graphs of orders .
A complete multipartite graph of type , denoted by , is constructed as follows. Let be a disjoint union of vertices with for each and the edge set be . The graph is unique up to isomorphism.
The Ehrhart polynomials for complete multipartite graphs are explicitly given in :
where is a partition of and .
Another simpler formula is newly obtained.
The Ehrhart polynomial of the edge polytope of a complete multipartite graph is
Let denote a complete multipartite graph . We start from the formula (1).
First, it holds that
On the one hand, is the number of combinations with repetitions choosing elements from a set of cardinality . On the other hand,
counts the same number of combinations as the sum of the number of combinations in which the -th smallest number is .
Second, it holds that
Since the outermost summations are the same on both sides, it suffices to show that
The summation of the left-hand side can be transformed as follows:
Finally, substituting these transformed terms into the original formula (1) gives the desired result. ∎
By the new formula above, we computed the roots of Ehrhart polynomials. Let denote , where the union runs over all complete multipartite graphs of order . Figure 2 plots the points of . For all complete multipartite graphs of order –, Conjecture 0.1 holds.
Figure 2, for , shows that the noninteger roots lie in the circle . This fact is not exclusive to alone, but similar conditions hold for all . We conjecture:
For any ,
(1) The leftmost point can only be attained by ; this is shown in Proposition 1.9. Therefore, if we choose , the set of negative integers in the statement can be replaced with the set . However, can be attained by the tree for any ; see Example 1.6 below.
(2) Since can never be a root of an Ehrhart polynomial, Conjecture 1.4 answers Question 1.1 in the affirmative for complete multipartite graphs. Moreover, if Conjecture 1.4 holds, then Conjecture 0.1 holds for those graphs.
(3) The method of Pfeifle  might be useful if the -vector can be determined for edge polytopes of complete multipartite graphs.
The edge polytope of a complete -partite graph for can be obtained as a pyramid from by adjoining a vertex. Therefore, its Ehrhart polynomial is the following:
Each term on the right-hand side is given in Example 1.6 above. By some elementary algebraic manipulations of binomial coefficients, it becomes,
The noninteger root is a real number in the circle of Conjecture 1.4.
Now we prepare the following lemma for proving Proposition 1.9.
For any integer , the polynomial in Proposition 1.3 satisfies:
It is an easy transformation:
Let be a partition of , satisfying . The Ehrhart polynomial of the edge polytope of the complete multipartite graph does not have a root at except when the graph is .
From Proposition 1.3, the Ehrhart polynomial of the edge polytope of is
Since has as one of its roots, it suffices to show that the rest of the expression does not have as one of its roots.
We evaluate at for from to :
by the definition of . If , its sign is since and . In case where , since is zero,
gives the same sign with other values of .
By the conjugate partition of , which is given by , we obtain
where we set, for simplicity, for .
We show that all the coefficients of are nonnegative
for any from to and there is at least one positive
coefficient among them.
The coefficients of are zero for , unless , i.e., when the graph is a complete bipartite graph; the exceptional case will be discussed later. We assume, therefore, for a while. Though equation (2) gives the coefficient of as for , by using Lemma 1.8, we are able to let them be zero and the coefficient of be for . Then all the coefficients of ’s are positive, since the occurrence of integers greater than or equal to in a partition of cannot be greater than .
Each coefficient of in equation (2) is for . By Lemma 1.8, we transfer them to lower terms so as to make the coefficients for be . Then all the coefficients of ’s are nonnegative, since the occurrence of integers greater than or equal to in a partition of cannot be greater than . Moreover, the coefficient is zero for at most one , less than . If and , i.e., in case of , there does not remain a positive coefficient. This exceptional case will be discussed later.
For both (I) and (II), ignoring the exceptional cases, the terms on the right-hand side of equation (2) are all nonnegative when , or nonpositive otherwise, and there is at least one nonzero term. That is, is not a root of
The Ehrhart polynomial is a sum of a polynomial whose roots include and another polynomial whose roots do not include . Therefore, is not a root of .
Finally, we discuss the exceptional cases. The complete bipartite graphs are treated in Example 1.6. In these cases, is not a root of the Ehrhart polynomials. However, is actually a root of the Ehrhart polynomial of the edge polytope constructed from the complete graph , as shown in Proposition 1.2 (1). ∎
2. Edge polytopes of graphs with loops
A convex polytope of dimension is simple if each vertex of belongs to exactly edges of . A simple polytope is smooth if at each vertex of , the primitive edge directions form a lattice basis.
Now, if is an edge of , then cannot be a vertex of if and only if and has a loop at each of the vertices and . Suppose that has a loop at and and that is not an edge of . Then for the graph defined by . Considering this fact, throughout this section, we assume that satisfies the following condition:
If , and if has a loop at each of and , then the edge belongs to .
The graphs (allowing loops) whose edge polytope is simple are completely classified by the following.
([22, Theorem 1.8]) Let denote the set of vertices such that has no loop at and let denote the induced subgraph of on . Then the following conditions are equivalent :
is simple, but not a simplex ;
is smooth, but not a simplex ;
and is one of the following graphs :
is a complete bipartite graph with at least one cycle of length ;
has exactly one loop, is a complete bipartite graph and if has a loop at , then for all ;
has at least two loops, has no edge and if has a loop at , then for all .
From the theory of Gröbner bases, we obtain the Ehrhart polynomial of the edge polytope above. In fact,
([22, Theorem 3.1]) Let be a graph as in Theorem 2.1 (iii). Let denote the set of vertices such that has no loop at and let denote the induced subgraph of on . Then the Ehrhart polynomial of the edge polytope are as follows:
If is the complete bipartite graph on the vertex set with and , then we have
If is the complete bipartite graph on the vertex set with and , then we have
If possesses loops and , then we have
2.1. Roots of Ehrhart polynomials
The consequences of the theorems above support Conjecture 0.1. Recall that denotes the set of roots of given polynomial .
The Ehrhart polynomial for a graph , the induced subgraph of which is a complete bipartite graph , is given in Theorem 2.2 ():