Homotopy in Distance-Regular Graphs Ideals of Designs Spherical Codes Association Schemes and Duality Polynomial ideals associated to combinatorial objects William J. Martin Department of Mathematical Sciences and Department of Computer Science Worcester Polytechnic Institute JCCA 2018, Sendai May 21, 2018 William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Spherical Codes Association Schemes and Duality Thanks To you for inviting me, to the organizers for such a nice conference, to Dr. Shoichi Tsuchiya for so much personal help, to my friends here at Tohoku for making math so interesting! Date Masamune William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality Distance-Regular Graphs (DRGs) A graph Γ = ( X , R ) of diameter d is distance-regular (DRG) if there exist constants b 0 , b 1 , . . . , b d − 1 ; c 1 , c 2 , . . . , c d such that, whenever x and y are vertices at distance i , there are exactly ◮ c i neighbors of y at distance i − 1 from x , and ◮ b i neighbors of y at distance i + 1 from x . k − c i − b i c i b i • • y x Γ i − 1 ( x ) Γ i ( x ) Γ i +1 ( x ) William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality Distance-Regular Graphs (DRGs) Examples: ◮ all five Platonic solids ◮ regular graphs with just three eigenvalues (“strongly regular”) ◮ n -cubes and Hamming graphs ◮ incidence graphs of symmetric designs ◮ Moore graphs and generalized polygons ◮ . . . many other connections! William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality Coxeter Graph The Coxeter graph is a cubic distance-regular graph (DRG) of diameter 4 on 28 vertices having girth 7. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality Distance-Regular Graphs (DRGs) A graph Γ = ( X , R ) of diameter d is distance-regular (DRG) if there exist constants b 0 , b 1 , . . . , b d − 1 ; c 1 , c 2 , . . . , c d such that, whenever x and y are vertices at distance i , there are exactly ◮ c i neighbors of y at distance i − 1 from x , and ◮ b i neighbors of y at distance i + 1 from x . k − c i − b i c i b i • • y x Γ i − 1 ( x ) Γ i ( x ) Γ i +1 ( x ) William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality Distance Distribution in the Coxeter Graph 0 0 1 1 3 1 2 1 2 1 1 2 0 1 2 3 4 The Coxeter graph is distance-regular: b 0 = 3, b 1 = b 2 = 2, b 3 = 1; c 1 = c 1 = c 3 = 1, c 4 = 2. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x Starting at vertex x , we build a closed walk representing an element of our homotopy group. 11 edges total. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality A Closed Walk in the Coxeter Graph x We say this walk (of length 11) has essential length 7. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality An Excursion into Homotopy The following idea appears in the thesis work of Heather Lewis ( Discrete Math. (2000)) under the supervision of Paul Terwilliger. v s x w t u Consider equivalence classes of closed walks in Γ starting and ending at basepoint x . William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality Discrete Homotopy on a Graph v v s s x x w w t t u u Closed walk xtwx is in the same equivalence class as xtwswx . In general, walk q ′ = q 1 pp − 1 q 2 is equivalent to walk q = q 1 q 2 : q ′ ∼ q William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality Discrete Homotopy on a Graph v v v s s s x x x w w w r t t t u u u These three walks all have “essential length” 3. William J. Martin Ideals in Combinatorics
Homotopy in Distance-Regular Graphs Ideals of Designs Distance-Regular Graphs Spherical Codes Homotopy of a graph: trivial? Association Schemes and Duality Discrete Homotopy on a Graph v v v s s s x x x w w w t t t u u u Our group operation is concatenation of walks. Of course, the concatenation of these two walks is represented by another cycle: xtwx ⋆ xwsvx = xtwxwsvx ∼ xtwsvx William J. Martin Ideals in Combinatorics
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