Some Applications of Generalized Action Graphs and Monodromy Graphs
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Some applications of generalized action graphs and monodromy graphs Tomaˇz Pisanski (University of Primorska and University of Ljubljana) 2015 International Conference on Graph Theory UP FAMNIT, Koper, May 26–28, 2015 Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Thanks Thanks to my former students Alen Orbani´c, Mar´ıa del R´ıo-Francos and their and my co-authors, in particular to Jan Karab´aˇs and many others whose work I am freely using in this talk. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Snub Cube - Chirality in the sense of Conway Snube cube is vertex-transitive (uniform) chiral polyhedron. It comes in two oriented forms. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Motivation We would like to have a single combinatorial mechanism that would enable us to deal with such phenomena. Our goal is to unify several existing structures in such a way that the power of each individual structure is preserved. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Example: The Cube Example The cube may be considered as an oriented map: R, r, as a map: r0, r1, r2 as a polyhedron .... Question What is the dual of a cube and how can we describe it? Question How can we tell that the cube is a regular map and amphichiral oriented map? Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Plan We define generalized action graphs as semi-directed graphs in which the edge set is partitioned into directed 2-factors (forming an action digraph) and undirected 1-factors (forming a monodromy graph) and use them to describe several combinatorial structures, such as maps and oriented maps. The quotient of the action graph with respect to its automorphism group (or some of its subgroup) is called the symmetry type graph and is very useful in connection with map symmetries and orientation preserving symmetries. Several usual cases of regular, edge-transitive, vertex-transitive, chiral, etc. maps and oriented maps are revisited. Our symmetry type graphs are closely related to Delaney-Dress symbols and orbifolds. The theory is very general and applies to a variety of discrete structures such as hypermaps, abstract polytopes and maniplexes. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Pregraphs and Graphs A pre-graph Γ is a quadruple (V, D, i, r) where V and D are disjoint non-empty sets V being the set of vertices, D the set of darts, i : D → V is a mapping that assigns to each dart its initial vertex and r : D → D is an involution assigning each dart its reverse a b c d e f g h i r b a c e d g f i h dart. i 3 2 2 2 1 2 1 1 1 Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Pregraphs and Graphs Let Γ = (V, D, i, r) be a pre-graph. The orbits of r are called edges of Γ. The edges corresponding to fixed points of r are called pending edges or semi-edges, while other edges are called proper edges. A pre-graph Γ = (V, D, i, r) is graph, if r is fixed-point free. a b d e f g h i r b a e d g f i h i 3 2 2 1 2 1 1 1 Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Digraphs A digraph Γ is a quadruple (V, A, j, t) where V and A are disjoint non-empty sets V being the set of vertices, A the set of arcs, j : A → V is a mapping that assigns to each dart its initial vertex and a d g h t : A → V is a mapping that j 3 1 2 1 assigns to each dart its terminal t 2 2 1 1 vertex. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Partially Directed Graphs A partially directed graph Γ is a hextuple (V, D, A, i, r, j, t) where (V, D, i, r) is a pre-graph and (V, A, j, t) is a digraph. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Morphisms, Isomorphisms, Automorphisms A morphism ϑ between two partially directed graphs (V, D, A, i, r, j, t) and (V 0,D0,A0, i0, r0, j0, t0) satisfies the following, for any d ∈ D and a ∈ A: i0(ϑ(d)) = ϑ(i(d)) r0(ϑ(d)) = ϑ(r(d)) j0(ϑ(a)) = ϑ(j(a)) t0(ϑ(a)) = ϑ(t(a)) Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Action graphs We generalize and adapt the notion of action graph, first introduced by A. Malniˇc in 2002 as follows. Let Φ be a finite non-empty set of flags or vertices. Let R = [R1,R2,...,Rm], where m ≥ 0 be a collection of permutations ∀i ∈ I : Ri ∈ Sym(V ),I = {1, 2, . , m}, called (the set of) rotations. Let % = [r1, r2, . , rn] be a collection of involutions on Φ, i.e. 2 ∀j ∈ J : rj ∈ Sym(V ), rj = Id,J = {1, 2, . , m}, called reflections. The structure Γ = (Φ; R; %, I, J) is called an action graph on the vertex set Φ of type (I,J) with signature (m, n), m = |I|, n = |J|. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Action graphs In practice we omit the type or replace it by signature when it is not needed. The subgroup Mon Γ = hR; %i ≤ Sym(Φ) is called the monodromy group of Γ. The elements of Mon Γ act from right on the elements of Φ, as it is defined by the convention. The action graph Γ is connected if Mon Γ acts transitively on Φ. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Action graphs The action graph Γ = (Φ; R, ρ) can be viewed as a partially directed regular graph on the vertex set Φ with edges coloured by the corresponding generators of Mon Γ. The incidence relation in Γ reads as u ∼ v ⇐⇒ v = u.g, where g is a generator of Mon Γ. The valence of Γ with signature (m, n) is 2m + n. A fixed point of Ri is exhibited as a loop, while a fixed point of rj is considered as a semiedge. We will always represent a pair of the opposite arcs arising from the action of reflections by the undirected edge; hence Γ is partially directed. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Example - Cayley graphs as action graphs Example A (directed) Cayley graph with m + n generators, out of which there are n involutions, can be regarded as an action graph with signature (m + k, n − k), 0 ≤ k ≤ n. Note that an involution may play different roles. It may be counted as permutation or as involution. This explains different signatures. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Example - Abstract polytopes as action graphs Example An abstract polytope of rank n is an action graph with signature (0, n). The vertices are flags and the n involutions represent exchange maps on the flags of the polytope: Γ = (Φ; ∅, {r0, r1, . , rn−1}), rirj = rjri, |i − j| > 2. For Γ to represent an abstract polytopes two further technical conditions have to be met: the diamond condition and strong connectivity. Instead of abstract polytopes one could study more general maniplexes (without diamond condition and strong connectivity). Maniplexes were introduced by Steve Wilson in 2012. See also crystallizations by Ferri and Gagliardi, introduced in the 70s, not to forget Lins and his work in the 90s. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Maniplex vs. Abstract Polytope Proposition Every abstract polytope is a maniplex. Proposition Dodecahedral Space There exist a maniplex that is not an abstract polytope, e.g. the Poincar´e v = 5, e = 10, f = 6,F = 1 Homology Sphere, There are 120 flags! Triple (vertex-edge-face) gives rise to two flags! Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Dodecahedral Space is a Maniplex Dodecahedral Space alias Poincar´e Homology 3-Sphere. The 120 flags are vertices of the dual of the barycentric subdivision of the ordinary dodecahedron on the sphere. The flag graph of the dodecahedron is a trivalent 0-,1-,2- edge-colored spanning subgraph of the flag graph of the dodecahedral space. 3-edges connect corresponding flags in antipodal pentagonal faces. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Question about maniplexes. Question Is there an easy way to tell from action graph whether a given maniplex is a polytope? E.g. if an i-edge has both end vertices in the same J-face and i∈ / J, the maniplex is not a polytope.