Graphs, Polytopes, Quadrics

Graphs Veroneses

Chris Godsil University of Waterloo

Ferarra, September 2012

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Outline

1 Graphs The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs

2 Veroneses Veronesians on Eigenspaces Designs

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Outline

1 Graphs The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs

2 Veroneses Veronesians on Eigenspaces Designs

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs A Graph

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Eigenvalues

3, √ √ √ 5, 5, 5, 1, 1, 1, 1, 1, 0, 0, 0, 0, −2, −2, −2, −2, √ √ √ − 5, − 5, − 5

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Eigenvectors

−0.3873 0.0 0.0  −0.2887 −0.1706 0.1938     −0.1291 −0.0228 0.3644   −0.1291 0.2391 0.2760    0.1291 0.3042 0.2020   U =  0.2887 0.0826 0.2446      0.1291 −0.1195 0.3450    0.1291 −0.3270 0.1625   −0.1291 −0.3586 0.0690  .  .

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs An Embedding

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Outline

1 Graphs The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs

2 Veroneses Veronesians on Eigenspaces Designs

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics √ UU T represents projection onto the 5-eigenspace. AU = θU

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Properties of U

U T U = I .

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics AU = θU

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Properties of U

U T U = I . √ UU T represents projection onto the 5-eigenspace.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Properties of U

U T U = I . √ UU T represents projection onto the 5-eigenspace. AU = θU

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics The vertices i such that (Uh)i is maximal form a face of the polytope. Each eigenvector determines a parallel pair of faces.

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Eigenvectors and Faces

Any vector Uh is an eigenvector for A.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Each eigenvector determines a parallel pair of faces.

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Eigenvectors and Faces

Any vector Uh is an eigenvector for A.

The vertices i such that (Uh)i is maximal form a face of the polytope.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Eigenvectors and Faces

Any vector Uh is an eigenvector for A.

The vertices i such that (Uh)i is maximal form a face of the polytope. Each eigenvector determines a parallel pair of faces.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics If equality holds and x is the characteristic vector of S, then

|S| x − 1 n is an eigenvector for X with eigenvalue τ.

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Eigenvectors from Ratio-Tight Cocliques

Let X be a k-regular graph on n vertices with least eigenvalue τ. If S is a coclique in X then n |S| ≤ k . 1 − τ

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Eigenvectors from Ratio-Tight Cocliques

Let X be a k-regular graph on n vertices with least eigenvalue τ. If S is a coclique in X then n |S| ≤ k . 1 − τ If equality holds and x is the characteristic vector of S, then

|S| x − 1 n is an eigenvector for X with eigenvalue τ.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Faces from Cocliques

If the size of the coclique S meets the ratio bound, then S and its complement S partition the vertices of the τ-polytope into two parallel faces.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics If you hope to classify the faces, this is not good news!

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Examples in Projective Space

Let W be the incidence matrix of points versus lines in projective space. Then col(W T ) is the sum of the constant functions and the ﬁrst non-trivial eigenspace of the Grassmann graph (on lines). Let S be a subset of the points of projective space, with characteristic vector h. Then the lines ` such that |S ∩ `| is maximal form a face in the θ1-polytope.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Examples in Projective Space

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Outline

1 Graphs The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs

2 Veroneses Veronesians on Eigenspaces Designs

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Covering Radius

Suppose C ⊆ V (X) for some graph X (usually connected and regular, often distance regular). Deﬁne Ci to be the set of vertices in X at distance i from C. (So C0 = C.) Deﬁnition The maximum distance of a vertex from C is its covering radius r.

The partition of V (X) with cells C0,..., Cr is the distance partition relative to C.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Completely-Regular Subsets

Deﬁnition A subset C of V (X) is completely regular if its distance partition is equitable—the number of neighbors in Cj of a vertex in Ci is equal to some constant bi,j . The constants bi−1,i , bi,i , bi,i+1 are the parameters of the partition.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics A ratio-tight coclique in a regular graph is completely regular (with covering radius 1).

If C is completely regular with distance partition C0,..., Cr , then Cr is completely regular. n−t A collection of k−t subsets of {1,..., n} such that any two have at least t points in common is a completely-regular subset in the Johnson graph J(v, k).

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Examples of Completely-Regular Subsets

A graph is distance regular if and only if it is regular and each vertex is a completely regular subset. (The covering radius of a vertex is the diameter of the graph.)

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics If C is completely regular with distance partition C0,..., Cr , then Cr is completely regular. n−t A collection of k−t subsets of {1,..., n} such that any two have at least t points in common is a completely-regular subset in the Johnson graph J(v, k).

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Examples of Completely-Regular Subsets

A graph is distance regular if and only if it is regular and each vertex is a completely regular subset. (The covering radius of a vertex is the diameter of the graph.) A ratio-tight coclique in a regular graph is completely regular (with covering radius 1).

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics n−t A collection of k−t subsets of {1,..., n} such that any two have at least t points in common is a completely-regular subset in the Johnson graph J(v, k).

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Examples of Completely-Regular Subsets

If C is completely regular with distance partition C0,..., Cr , then Cr is completely regular.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Faces of Completely-Regular Subsets

Suppose C is a completely-regular subset of the distance-regular graph X, and let its characteristic vector be x. Let θ0, . . . , θd be the eigenvalues of X, in nonincreasing order. (So θ0 is the valency of X.) Theorem

Let λ be the largest eigenvalue in {θ1, . . . , θd} such that the projection of x onto the λ-eigenspace is not zero. Then C0 and Cr are parallel faces of the λ-polytope.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Corollary n−t A collection of k−t subsets of {1,..., n} such that any two have at least t points in common consists of all k-subsets that contain a speciﬁed set of t points.

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Faces for the Johnson Graph

Theorem

The faces of the θ1-polytope of J(n, k) consist of those k-subsets of {1,..., n} that lie in some subset S of {1,..., n}, and contain some subset T.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Faces for the Johnson Graph

Theorem

The faces of the θ1-polytope of J(n, k) consist of those k-subsets of {1,..., n} that lie in some subset S of {1,..., n}, and contain some subset T.

Corollary n−t A collection of k−t subsets of {1,..., n} such that any two have at least t points in common consists of all k-subsets that contain a speciﬁed set of t points.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Outline

1 Graphs The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs

2 Veroneses Veronesians on Eigenspaces Designs

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Which Distance-Regular Graphs are 1-Skeletons?

Theorem

A distance-regular graph is the 1-skeleton of its θ1-eigenpolytope if and only if it is one of the following: 1 A Johnson graph. 2 A Hamming graph. 3 A halved n-cube. 4 The Schlaeﬂi graph. 5 The Gosset graph. 6 The icosahedron. 7 The dodecahedron.

8 rK2. 9 A cycle.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics The non-trivial eigenspaces of Petersen have dimension 4 and 5.

It follows that the corresponding 1-skeletons must be K10.

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Why Petersen is not a 1-Skeleton

The 1-skeleton of an m-dimensional polytope is m-connected.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics It follows that the corresponding 1-skeletons must be K10.

Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Why Petersen is not a 1-Skeleton

The 1-skeleton of an m-dimensional polytope is m-connected. The non-trivial eigenspaces of Petersen have dimension 4 and 5.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Why Petersen is not a 1-Skeleton

The 1-skeleton of an m-dimensional polytope is m-connected. The non-trivial eigenspaces of Petersen have dimension 4 and 5.

It follows that the corresponding 1-skeletons must be K10.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs Too Many Faces

Lemma

The edges in a spanning regular subgraph of Kn form the vertices of a face in the θ2-polytope of L(Kn); they form a facet if and only if the subgraph is connected and not bipartite.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs Outline

1 Graphs The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs

2 Veroneses Veronesians on Eigenspaces Designs

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs Vanishing Quadratics

Suppose Uh is the shifted characteristic vector of a ratio-tight coclique in a regular graph, and denote its entries by α and β. Then the function (hT x − α)(hT x − β) is a quadratic polynomial that vanishes on the image of each vertex of X.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Example x = a b then 2 2 ver2(x) = 1 a b a ab b

Graphs Veroneses Veronesians on Eigenspaces Designs Veronese Maps

Deﬁnition

The Veronese map verk is a mapping from a vector space of m+k dimension m to a vector space of dimension k such that the entries of verk(x) are all monomials of degree at most k in the entries of x.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs Veronese Maps

Deﬁnition

The Veronese map verk is a mapping from a vector space of m+k dimension m to a vector space of dimension k such that the entries of verk(x) are all monomials of degree at most k in the entries of x.

Example x = a b then 2 2 ver2(x) = 1 a b a ab b

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Lemma There is a quadratic polynomial that vanishes on each row of U if and only if the columns of ver2(U ) are linearly dependent.

Graphs Veroneses Veronesians on Eigenspaces Designs No Quadrics

Deﬁnition

If U is a matrix, then verk(U ) is the matrix we get by applying verk to each row.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs No Quadrics

Deﬁnition

If U is a matrix, then verk(U ) is the matrix we get by applying verk to each row.

Lemma There is a quadratic polynomial that vanishes on each row of U if and only if the columns of ver2(U ) are linearly dependent.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs A Problem

Find graphs where we can use the rank of the Veronese of an eigenspace to rule out the existence of ratio-tight cocliques.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs Strong Arnold

Assume the columns of U are the orthonormal basis for an eigenspace of a graph X, with eigenvalue λ and dimension m. Form the matrix U2 by adjoining to U the vectors U (ei − ej ) for each edge ij. Then the λ-eigenspace satisﬁes the strong Arnold m+1 condition if ver2(U2) has rank 2 .

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs Outline

1 Graphs The Dodedecahedron Eigenpolytopes Completely-Regular Subsets Distance-Regular Graphs

2 Veroneses Veronesians on Eigenspaces Designs

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs Spherical Designs

Deﬁnition d A subset S of the unit sphere in R has strength t if the average over S of any polynomial of degree at most t is equal to its average over the entire sphere. We say S is a t-design if its strength is at least t.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics A connected graph is regular if and only if, for each eigenvalue θ not equal to the spectral radius, the θ-eigenspace is a 1-design.

Graphs Veroneses Veronesians on Eigenspaces Designs 1-Designs

A subset S of the unit sphere is a 1-design if and only if the sum of its elements is zero.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs 1-Designs

A subset S of the unit sphere is a 1-design if and only if the sum of its elements is zero. A connected graph is regular if and only if, for each eigenvalue θ not equal to the spectral radius, the θ-eigenspace is a 1-design.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics A graph X is walk regular if Ar has constant diagonal for all r ≥ 0. A connected graph is walk regular if and only if, for each eigenvalue θ not equal to the spectral radius, the θ-eigenspace is a 2-design.

Graphs Veroneses Veronesians on Eigenspaces Designs 2-Designs

d If x1,..., xn are unit vectors in R , they form a 2-design if P P T and only if xi = 0 and xi xi = (n/d)I .

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics A connected graph is walk regular if and only if, for each eigenvalue θ not equal to the spectral radius, the θ-eigenspace is a 2-design.

Graphs Veroneses Veronesians on Eigenspaces Designs 2-Designs

d If x1,..., xn are unit vectors in R , they form a 2-design if P P T and only if xi = 0 and xi xi = (n/d)I . A graph X is walk regular if Ar has constant diagonal for all r ≥ 0.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs 2-Designs

d If x1,..., xn are unit vectors in R , they form a 2-design if P P T and only if xi = 0 and xi xi = (n/d)I . A graph X is walk regular if Ar has constant diagonal for all r ≥ 0. A connected graph is walk regular if and only if, for each eigenvalue θ not equal to the spectral radius, the θ-eigenspace is a 2-design.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics If the vertices of X form a 3-design, then it follows that (hT x − α)(hT x − β) is orthogonal to all linear functions, in particular to hT x.

Graphs Veroneses Veronesians on Eigenspaces Designs 3-Designs

Suppose we have a subset S of X such that the partition {S, S} is equitable. Assume that the projection of xS on the λ-eigenspace is not zero. Then there is a vector h and scalars α, β such that (hT x − α)(hT x − β) is zero on all vertices of X. Then: For any vector `,

(hT x − α)(hT x − β)`T x

is zero on all vertices.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs 3-Designs

(hT x − α)(hT x − β)`T x

is zero on all vertices. If the vertices of X form a 3-design, then it follows that (hT x − α)(hT x − β) is orthogonal to all linear functions, in particular to hT x.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics (hT x)3 and hT x are odd functions, so their average over the sphere is zero. The average of (hT x)2 over the sphere is not zero, hence α + β = 0.

Corollary Suppose the projection of the characteristic vector of the subset S on the λ-eigenspace is not zero, and that {S, S} is equitable. If the λ-eigenspace is a 3-design, then |S| = |S|. If S is a coclique, X is bipartite.

Graphs Veroneses Veronesians on Eigenspaces Designs A Balanced Result

(hT x −α)(hT x −β)hT x = (hT x)3 −(α+β)(hT x)2 +αβhT x.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics The average of (hT x)2 over the sphere is not zero, hence α + β = 0.

Graphs Veroneses Veronesians on Eigenspaces Designs A Balanced Result

(hT x −α)(hT x −β)hT x = (hT x)3 −(α+β)(hT x)2 +αβhT x. (hT x)3 and hT x are odd functions, so their average over the sphere is zero.

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Corollary Suppose the projection of the characteristic vector of the subset S on the λ-eigenspace is not zero, and that {S, S} is equitable. If the λ-eigenspace is a 3-design, then |S| = |S|. If S is a coclique, X is bipartite.

Graphs Veroneses Veronesians on Eigenspaces Designs A Balanced Result

(hT x −α)(hT x −β)hT x = (hT x)3 −(α+β)(hT x)2 +αβhT x. (hT x)3 and hT x are odd functions, so their average over the sphere is zero. The average of (hT x)2 over the sphere is not zero, hence α + β = 0.

Graphs Veroneses Veronesians on Eigenspaces Designs A Balanced Result

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs A Balanced Result

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics Graphs Veroneses Veronesians on Eigenspaces Designs The End(s)

Chris Godsil University of Waterloo Graphs, Polytopes, Quadrics