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Graphs, Polytopes, Quadrics Graphs Veroneses Graphs, Polytopes, Quadrics 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 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. If you hope to classify the faces, this is not good news! 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 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). If C is completely regular with distance partition
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