Sparsity Counts in Group-Labeled Graphs and Rigidity

Sparsity Counts in Group-labeled Graphs and Rigidity

Rintaro Ikeshita1 Shin-ichi Tanigawa2

1University of Tokyo 2CWI & Kyoto University

July 15, 2015

1 / 23 I Examples

I k = ` = 1: forest

I k = 1, ` = 0: pseudoforest

I k = `: Decomposability into edge-disjoint k forests (Nash-Williams)

I ` ≤ k: Decomposability into edge-disjoint k − ` pseudoforests and ` forests

I general k, `: Rigidity of graphs and scene analysis

(k, `)-sparsity

def I A ﬁnite undirected graph G = (V , E) is (k, `)-sparse ⇔

|F | ≤ k|V (F )| − `

for every F ⊆ E with k|V (F )| − ` ≥ 1.

2 / 23 (k, `)-sparsity

def I A ﬁnite undirected graph G = (V , E) is (k, `)-sparse ⇔

|F | ≤ k|V (F )| − `

for every F ⊆ E with k|V (F )| − ` ≥ 1.

I Examples

I k = ` = 1: forest

I k = 1, ` = 0: pseudoforest

I k = `: Decomposability into edge-disjoint k forests (Nash-Williams)

I ` ≤ k: Decomposability into edge-disjoint k − ` pseudoforests and ` forests

I general k, `: Rigidity of graphs and scene analysis

2 / 23 I Examples

I k = ` = 1: graphic matroid

I k = 1, ` = 0: bicircular matroid

I ` ≤ k: union of k − ` copies of bicircular matroid and ` copies of graphic matroid

I k = 2, ` = 3: generic 2-rigidity matroid (Laman70)

Count Matroids

I Suppose ` ≤ 2k − 1. Then Mk,`(G) = (E, Ik,`) forms a matroid, called the (k, `)-count matroid, where

Ik,` = {I ⊆ E : I is (k, `)-sparse}.

3 / 23 Count Matroids

I Suppose ` ≤ 2k − 1. Then Mk,`(G) = (E, Ik,`) forms a matroid, called the (k, `)-count matroid, where

Ik,` = {I ⊆ E : I is (k, `)-sparse}.

I Examples

I k = ` = 1: graphic matroid

I k = 1, ` = 0: bicircular matroid

I ` ≤ k: union of k − ` copies of bicircular matroid and ` copies of graphic matroid

I k = 2, ` = 3: generic 2-rigidity matroid (Laman70)

3 / 23 Group-labeled Graphs

I A group-labeled graph (Γ-labeled graph) (G, ψ) is a directed ﬁnite graph whose edges are labeled invertibly from a group Γ.

I G: the underlying directed graph

I ψ : E(G) → Γ

+ + a b + a2b + a + a2 + Γ = {−, +} 1Γ 2 2 Γ = {1Γ, a, a , ab, a b} − 1Γ

4 / 23 I I ⊆ E is independent iﬀ ( 0 if F is balanced |F | ≤ |V (F )| − 1 + (F ⊆ I ) 1 otherwise

I What would be (k, `)-sparsity in group-labeled graphs??

Frame Matroids

I In the frame matroid (or, Dowling geometry, bias matroid) of (G, ψ), I ⊆ E is independent iﬀ each connected component of I contains no cycle or just one cycle, which is unbalanced if exists.

− − − −

− + unbalanced balanced

I Examples I graphic matroid

I bicircular matroid

I signed graphic matroid

I 1-dimensional locally completability matroid!!

5 / 23 I What would be (k, `)-sparsity in group-labeled graphs??

Frame Matroids

I In the frame matroid (or, Dowling geometry, bias matroid) of (G, ψ), I ⊆ E is independent iﬀ each connected component of I contains no cycle or just one cycle, which is unbalanced if exists.

− − − −

− + unbalanced balanced

I Examples I graphic matroid

I bicircular matroid

I signed graphic matroid

I 1-dimensional locally completability matroid!! I I ⊆ E is independent iﬀ ( 0 if F is balanced |F | ≤ |V (F )| − 1 + (F ⊆ I ) 1 otherwise

5 / 23 Frame Matroids

I In the frame matroid (or, Dowling geometry, bias matroid) of (G, ψ), I ⊆ E is independent iﬀ each connected component of I contains no cycle or just one cycle, which is unbalanced if exists.

− − − −

− + unbalanced balanced

I Examples I graphic matroid

I bicircular matroid

I signed graphic matroid

I 1-dimensional locally completability matroid!! I I ⊆ E is independent iﬀ ( 0 if F is balanced |F | ≤ |V (F )| − 1 + (F ⊆ I ) 1 otherwise

I What would be (k, `)-sparsity in group-labeled graphs??

5 / 23 Motivation: Rigidity of Symmetric Frameworks

I Characterization of the inﬁnitesimal rigidity of symmetric bar-joint frameworks

I A framework (G, p) is Γ-symmetric if

I Γ acts on G through θ :Γ → Aut(G) and

I p(θ(γ)v) = γp(v) for any γ ∈ Γ and v ∈ V

I Extension of classical theorem — count conditions on the underlying quotient group-labeled graphs (Malestein-Theran2010,2012, Ross2011, ...)

6 / 23 I R1(G, p) = M1,1(G) for any 1-d realization (G, p)

I Theorem(Laman 1970) G can be realized as an inﬁnitesimally rigid framework in R2 iﬀ G contains a spanning subgraph G 0 with 2|V (G)| − 3 edges satisfying the following count condition:

|F | ≤ 2|V (F )| − 3 ∅ 6= ∀F ⊆ E(G 0)

I R2(G, p) = M2,3(G) for any generic 2-d realization (G, p)

Characterization of Rigid Graphs

1 I G can be realized as an infinitesimally rigid framework in R iff G is connected iff G contains a spanning subgraph G 0 with |V (G)| − 1 edges satisfying the following count condition:

|F | ≤ |V (F )| − 1 ∅ 6= ∀F ⊆ E(G 0)

7 / 23 I Theorem(Laman 1970) G can be realized as an inﬁnitesimally rigid framework in R2 iﬀ G contains a spanning subgraph G 0 with 2|V (G)| − 3 edges satisfying the following count condition:

|F | ≤ 2|V (F )| − 3 ∅ 6= ∀F ⊆ E(G 0)

I R2(G, p) = M2,3(G) for any generic 2-d realization (G, p)

Characterization of Rigid Graphs

1 I G can be realized as an infinitesimally rigid framework in R iff G is connected iff G contains a spanning subgraph G 0 with |V (G)| − 1 edges satisfying the following count condition:

|F | ≤ |V (F )| − 1 ∅ 6= ∀F ⊆ E(G 0)

I R1(G, p) = M1,1(G) for any 1-d realization (G, p)

7 / 23 Characterization of Rigid Graphs

1 I G can be realized as an infinitesimally rigid framework in R iff G is connected iff G contains a spanning subgraph G 0 with |V (G)| − 1 edges satisfying the following count condition:

|F | ≤ |V (F )| − 1 ∅ 6= ∀F ⊆ E(G 0)

I R1(G, p) = M1,1(G) for any 1-d realization (G, p)

I Theorem(Laman 1970) G can be realized as an inﬁnitesimally rigid framework in R2 iﬀ G contains a spanning subgraph G 0 with 2|V (G)| − 3 edges satisfying the following count condition:

|F | ≤ 2|V (F )| − 3 ∅ 6= ∀F ⊆ E(G 0)

I R2(G, p) = M2,3(G) for any generic 2-d realization (G, p)

7 / 23 One Approach (Schulze-T13)

I Compute the block-diagonalization of rigidity matrix R(G, p)   R0(G, p) 0 >  ..  T R(G, p)S :=  .  0 Rr (G, p)

I Its ”sparsity pattern” is carried over (??):

I In each block, each row and each column are associated with an edge orbit and a vertex orbit, respectively. d I The zero-nonzero pattern of each block is the same as I (G/Γ) ⊗ R , where I (G/Γ) denotes the incidence matrix of G/Γ.

I Characterize the rank of each block in terms of combinatorial conditions of the underlying quotient graph G/Γ

8 / 23 1-dimensional case

I (G, p): Z2-symmetric framework with Z2 = {−, +} 1 I p : V → R with p((−)v) = −p(v) for each v ∈ V I R (G, p) 0 T >R(G, p)S := 0 0 R1(G, p)

I R1(G, p) = I (G/Z2) I The row-independence of R0(G, p) cannot be described by G/Z2...

9 / 23 Quotient Group-labled Graphs

I Quotient group labeled graph (G/Γ, ψ)

I G/Γ: a directed quotient graph

I ψ : E(G/Γ) → Γ encoding the covering map 1 C32

C33 3 2 C31 C 2 C id 3 C22 3 C C1 C23 C3 id 2 (G/Γ, ψ) C2 C21 G

10 / 23 I ⇔ independent in the frame matroid of (G/Z2, ψ). I ⇔ ∀F ⊆ E/Z2, ( 0 if F is balanced |F | ≤ |V (F )| − 1 + 1 if F is unbalanced

1-dimensional case

I (G, p): Z2-symmetric framework 1 I p : V → R with p((−)v) = −p(v) for each v ∈ V I R (G, p) 0 T >R(G, p)S := 0 0 R1(G, p)

I R1(G, p) = I (G/Z2) I R0(G, p) = I (G/Z2, ψ) I where I (G/Z2, ψ) is row independent iﬀ each connected component contains no cycle or just one cycle, which is unbalanced if exists

11 / 23 I ⇔ ∀F ⊆ E/Z2, ( 0 if F is balanced |F | ≤ |V (F )| − 1 + 1 if F is unbalanced

1-dimensional case

I (G, p): Z2-symmetric framework 1 I p : V → R with p((−)v) = −p(v) for each v ∈ V I R (G, p) 0 T >R(G, p)S := 0 0 R1(G, p)

I R1(G, p) = I (G/Z2) I R0(G, p) = I (G/Z2, ψ) I where I (G/Z2, ψ) is row independent iﬀ each connected component contains no cycle or just one cycle, which is unbalanced if exists I ⇔ independent in the frame matroid of (G/Z2, ψ).

11 / 23 1-dimensional case

I (G, p): Z2-symmetric framework 1 I p : V → R with p((−)v) = −p(v) for each v ∈ V I R (G, p) 0 T >R(G, p)S := 0 0 R1(G, p)

11 / 23 2-dimensional reﬂection symmetry (Schulze-T 13)

I (G, p): 2-dimensional framework with reﬂection symmetry 2 I p : V → R is ”regular” under the reﬂection symmetry I R (G, p) 0 T >R(G, p)S := 0 0 R1(G, p)

I Ri (G, p) is row-independent iﬀ ∀F ⊆ E/Γ, ( 0 if F is balanced |F | ≤ 2|V (F )| − 3 + 2 − i if F is unbalanced

I A similar count works for grid-like frameworks (Kitson)

− + + + + + + −

12 / 23 I Examples of counts in rigidity with symmetry:  0 hF i is trivial  v |F | ≤ 2|V (F )|−3+ 2 hF iv is nontrivial and cyclic (F ⊆ E/Γ) 3 otherwise Given an integer i,  0 hF iv is trivial  1 hF i ' |F | ≤ 2|V (F )|−3+ v Z2 (F ⊆ E/Γ) 2 hF iv ' Zk for some k ∈ S(n, i)  3 otherwise where S(n, i) = {k : n ≡ 0 (mod k) and i ∈ {−1, 0, 1} (mod k)}\{2}.

Count Conditions on Group Labeled Graphs

I Notation:

I (G/Γ, ψ): Γ-labeled graph

I ψ(W ): the total gain though a closed walk W

I hF iv := hψ(W ) : a closed walk W ⊆ F ⊆ E/Γ starting at vi I F is balanced iﬀ hF iv is trivial

13 / 23 Given an integer i,  0 hF iv is trivial  1 hF i ' |F | ≤ 2|V (F )|−3+ v Z2 (F ⊆ E/Γ) 2 hF iv ' Zk for some k ∈ S(n, i)  3 otherwise where S(n, i) = {k : n ≡ 0 (mod k) and i ∈ {−1, 0, 1} (mod k)}\{2}.

Count Conditions on Group Labeled Graphs

I Notation:

I (G/Γ, ψ): Γ-labeled graph

I ψ(W ): the total gain though a closed walk W

13 / 23 Count Conditions on Group Labeled Graphs

I Notation:

I (G/Γ, ψ): Γ-labeled graph

I ψ(W ): the total gain though a closed walk W

I hF iv := hψ(W ) : a closed walk W ⊆ F ⊆ E/Γ starting at vi I F is balanced iﬀ hF iv is trivial I Examples of counts in rigidity with symmetry:  0 hF i is trivial  v |F | ≤ 2|V (F )|−3+ 2 hF iv is nontrivial and cyclic (F ⊆ E/Γ) 3 otherwise Given an integer i,  0 hF iv is trivial  1 hF i ' |F | ≤ 2|V (F )|−3+ v Z2 (F ⊆ E/Γ) 2 hF iv ' Zk for some k ∈ S(n, i)  3 otherwise where S(n, i) = {k : n ≡ 0 (mod k) and i ∈ {−1, 0, 1} (mod k)}\{2}.

13 / 23 Γ I α : 2 → R is polymatroidal if I α(∅) = 0;

I α(X ) ≤ α(Y ) for any X ⊆ Y ⊆ Γ;

I α(X ) + α(Y ) ≥ α(X ∪ Y ) + α(X ∩ Y ) for any X , Y ⊆ Γ;

I α(X ) = α(hX i) for any nonempty X ⊆ Γ; −1 I α(X ) = α(γX γ ) for any nonempty X ⊆ Γ and γ ∈ Γ.

Γ I Theorem(T12) Let α : 2 → {0, 1,..., k} be a polymatroidal function, and (G, ψ) a Γ-labaled graph. Let Iα := {I ⊆ E(G): I satisﬁes the count (?) ∀F ∈ C(I ), v ∈ V (F )}:

|F | ≤ k|V (F )| − k + α(hF iv ) ··· (?)

Then, (E(G), Iα) is a matroid.

I Several applications to rigidity and scene analysis (T12)

Sparsity Count for Group-labeled Graphs

I When a count condition of the form

|F | ≤ k|V (F )| − ` + α(hF iv ) ∀F ∈ C(G) and ∀v ∈ V (F )

for some α : 2Γ → Z induces a matroid?

14 / 23 Γ I Theorem(T12) Let α : 2 → {0, 1,..., k} be a polymatroidal function, and (G, ψ) a Γ-labaled graph. Let Iα := {I ⊆ E(G): I satisﬁes the count (?) ∀F ∈ C(I ), v ∈ V (F )}:

|F | ≤ k|V (F )| − k + α(hF iv ) ··· (?)

Then, (E(G), Iα) is a matroid.

I Several applications to rigidity and scene analysis (T12)

Sparsity Count for Group-labeled Graphs

I When a count condition of the form

|F | ≤ k|V (F )| − ` + α(hF iv ) ∀F ∈ C(G) and ∀v ∈ V (F )

for some α : 2Γ → Z induces a matroid? Γ I α : 2 → R is polymatroidal if I α(∅) = 0;

I α(X ) ≤ α(Y ) for any X ⊆ Y ⊆ Γ;

I α(X ) + α(Y ) ≥ α(X ∪ Y ) + α(X ∩ Y ) for any X , Y ⊆ Γ;

I α(X ) = α(hX i) for any nonempty X ⊆ Γ; −1 I α(X ) = α(γX γ ) for any nonempty X ⊆ Γ and γ ∈ Γ.

14 / 23 Sparsity Count for Group-labeled Graphs

I When a count condition of the form

|F | ≤ k|V (F )| − ` + α(hF iv ) ∀F ∈ C(G) and ∀v ∈ V (F )

for some α : 2Γ → Z induces a matroid? Γ I α : 2 → R is polymatroidal if I α(∅) = 0;

I α(X ) ≤ α(Y ) for any X ⊆ Y ⊆ Γ;

I α(X ) + α(Y ) ≥ α(X ∪ Y ) + α(X ∩ Y ) for any X , Y ⊆ Γ;

I α(X ) = α(hX i) for any nonempty X ⊆ Γ; −1 I α(X ) = α(γX γ ) for any nonempty X ⊆ Γ and γ ∈ Γ.

Γ I Theorem(T12) Let α : 2 → {0, 1,..., k} be a polymatroidal function, and (G, ψ) a Γ-labaled graph. Let Iα := {I ⊆ E(G): I satisﬁes the count (?) ∀F ∈ C(I ), v ∈ V (F )}:

|F | ≤ k|V (F )| − k + α(hF iv ) ··· (?)

Then, (E(G), Iα) is a matroid.

I Several applications to rigidity and scene analysis (T12)

14 / 23 k < `

I Example that does not induce a matroid (Csaba Kir´aly): ( 0 F is balanced |F | ≤ 2|V (F )| − 3 + (F ∈ C(G)) 3 otherwise

v3 v3 v3

g g g g g g g g

v4 v4 v4

v1 v2 v1 v2 v1 v2

15 / 23 Γ I Theorem(Ikeshita-T) Let α : 2 → {0, 1,..., k + 1} be a polymatroidal function, and (G, ψ) a Γ-labaled graph. Let Iα := {I ⊆ E(G): I satisﬁes the count (?) ∀F ∈ C(I ), ∀v ∈ V (F )}: ( min{α(hF i ), k} if F is near-balanced |F | ≤ k|V (F )| − (k + 1) + v α(hF iv ) otherwise

Suppose also that α(Z2) ≤ k. Then, (E(G), Iα) is a matroid.

I Near-balanced sets were ﬁrst observed as ”obstacles” in the analysis of frameworks with cyclic synmmetry (Schulze-T13)

Near-balancedness and Main Theorem

I A Γ-labeled graph (G, ψ) (resp. an edge set) is near-balanced if there is a balanced split (i.e., the inverse of two vertices identiﬁcation)

v1 v1 v1

g g g g

v2 v3 v2 v3 v2 v3 v3

g g g g g g

non-near-balanced near-balanced the split

16 / 23 I Near-balanced sets were ﬁrst observed as ”obstacles” in the analysis of frameworks with cyclic synmmetry (Schulze-T13)

Near-balancedness and Main Theorem

I A Γ-labeled graph (G, ψ) (resp. an edge set) is near-balanced if there is a balanced split (i.e., the inverse of two vertices identiﬁcation)

v1 v1 v1

g g g g

v2 v3 v2 v3 v2 v3 v3

g g g g g g

non-near-balanced near-balanced the split Γ I Theorem(Ikeshita-T) Let α : 2 → {0, 1,..., k + 1} be a polymatroidal function, and (G, ψ) a Γ-labaled graph. Let Iα := {I ⊆ E(G): I satisﬁes the count (?) ∀F ∈ C(I ), ∀v ∈ V (F )}: ( min{α(hF i ), k} if F is near-balanced |F | ≤ k|V (F )| − (k + 1) + v α(hF iv ) otherwise

Suppose also that α(Z2) ≤ k. Then, (E(G), Iα) is a matroid.

16 / 23 Near-balancedness and Main Theorem

I A Γ-labeled graph (G, ψ) (resp. an edge set) is near-balanced if there is a balanced split (i.e., the inverse of two vertices identiﬁcation)

v1 v1 v1

g g g g

v2 v3 v2 v3 v2 v3 v3

g g g g g g

Suppose also that α(Z2) ≤ k. Then, (E(G), Iα) is a matroid.

I Near-balanced sets were ﬁrst observed as ”obstacles” in the analysis of frameworks with cyclic synmmetry (Schulze-T13) 16 / 23 Example 1

I Suppose that Γ does not have an element of order two. Then the count  0 F is balanced  |F | ≤ 2|V (F )| − 3 + 2 F is near-balanced (F ∈ C(G)) 3 otherwise

induces a matroid. 2 I May not be true if ∃g with g = 1Γ

v1 v1 v1

g g g g g

v2 v3 v2 v3 v2 v3 g g g g g g

17 / 23 Example 2

I In general, the count  0 F is balanced  |F | ≤ 2|V (F )|−3+ 2 F is near-balanced or hF iv ' Z2 (F ∈ C(G)) 3 otherwise

induces a matroid.

18 / 23 Example 3

I The count  0 hF i is trivial  v |F | ≤ 2|V (F )|−3+ 2 hF iv is nontrivial and cyclic (F ∈ C(G)) 3 otherwise

induces a matroid if Γ is dihedral (Jord´an-Kaszanitzky-T12) Γ I Lemma α : 2 → Z deﬁned by  0 hX i is trivial  α(X ) = 2 hX i is nontrivial and cyclic (X ⊆ Γ) 3 otherwise

is polymatroidal iﬀ for each x ∈ Γ there is a unique maximal cyclic subgroup containing x.

I Characterization of the (symmetry-forced) rigidity of bar-joint frameworks with dihedral symmetry in the plane (JKT12)

19 / 23 Example 4

I Given integers i, n with 0 ≤ i ≤ n, α : 2Zn → {0,..., 3} deﬁned by 0 hX i is trivial  1 hX i ' α(X ) = Z2 (X ⊆ Γ) 2 hX i ' Zk for some k ∈ S(n, i)  3 otherwise is polymatroidal, where S(n, i) = {k : n ≡ 0 (mod k) and i ∈ {−1, 0, 1} (mod k)}\{2}. I For a Zn-labeled graph, the following count induces a matroid:  0 hF iv is trivial  1 hF i '  v Z2 |F | ≤ 2|V (F )|−3+ 2 hF iv ' Zk for some k ∈ S(n, i) (F ∈ C(G)).   or F is near-balanced  3 otherwise

I Characterization of the inﬁnitesimal rigidity of bar-joint frameworks with cyclic symmetry in the plane (Ikeshita-T)

I The matroid of the i-th block in the block-diagonalization of the rigidity matrix is isomorphic to this count matroid. 20 / 23 I General strategy for proving rigidity

I Show that the base graphs (# vertices ≤2) are inf. rigid

I Show that each operation preserves the corank

Inductive Construction

I Recall our count for k = 2, ` = 3: ( min{α(hF i ), 2} if F is near-balanced |F | ≤ 2|V (F )| − 3 + v α(hF iv ) otherwise

where α : 2Γ → {0, 1, 2, 3} is a polymatroidal function

I Theorem(Ikeshita-T) A ”tight graph” with respect to the above count can be constructed from a tight graph with at most two vertices by a sequence of i-extension (i ∈ {0, 1, 2, }), vertex-splitting, and tight-replacement keeping the sparsity.

21 / 23 Inductive Construction

I Recall our count for k = 2, ` = 3: ( min{α(hF i ), 2} if F is near-balanced |F | ≤ 2|V (F )| − 3 + v α(hF iv ) otherwise

where α : 2Γ → {0, 1, 2, 3} is a polymatroidal function

I General strategy for proving rigidity

I Show that the base graphs (# vertices ≤2) are inf. rigid

I Show that each operation preserves the corank

21 / 23 Inﬁnitesimal Rigidity with Cyclic Symmetry

I Theorem(Ikeshita-T) The count characterizes the inﬁnitesimal rigidity of frameworks in the plane with cyclic symmetry Zn (under Zn-genericity) if n is odd with n ≤ 1000 or n is even with n ≤ 6.

I Each operation preserves the corank of the rigidity matrix.

I The base case is checked with the aid of a computer up to 1000.

I The count fails in the single-vertex graph for even n with n ≥ 8

I Q. Characterize the inﬁnitesimal rigidity of 2-d Zn-symmetric frameworks whose underlying graphs are Caylay graphs of Zn

22 / 23 I Even for d = 2 and Γ ' Zn the problem is challenging... I Cay(Zn, S) is connected iff S generates Zn. I Cay(Zn, S) is connected and bipartite iff S generates Zn and S consists of odd numbers I If Cay(Zn, S) is bipartite, then the corresponding framework is not infinitesimally rigid

I For n = 24 and S = {−1, 1, −5, 5}, the framework is 4-regular connected and bipartite but has two degree of freedom. (Why?)

Thank you for your attention!!

Q. Characterize the inﬁnitesimal rigidity of ”Caylay frameworks”.

I Γ ⊂ Eucl(d): a group −1 I S ⊆ Γ with S = S I G = Cay(Γ, S):

I V (G) = Γ

I E(G) = {ij : j ∈ iS}

I A Caylay framework (G, p) of Γ:

I G = Cay(Γ, S)

I p(gv) = gp(v) for every v ∈ V (G) and g ∈ Γ

23 / 23 Thank you for your attention!!

Q. Characterize the inﬁnitesimal rigidity of ”Caylay frameworks”.

I Γ ⊂ Eucl(d): a group −1 I S ⊆ Γ with S = S I G = Cay(Γ, S):

I V (G) = Γ

I E(G) = {ij : j ∈ iS}

I A Caylay framework (G, p) of Γ:

I G = Cay(Γ, S)

I p(gv) = gp(v) for every v ∈ V (G) and g ∈ Γ I Even for d = 2 and Γ ' Zn the problem is challenging... I Cay(Zn, S) is connected iff S generates Zn. I Cay(Zn, S) is connected and bipartite iff S generates Zn and S consists of odd numbers I If Cay(Zn, S) is bipartite, then the corresponding framework is not infinitesimally rigid

I For n = 24 and S = {−1, 1, −5, 5}, the framework is 4-regular connected and bipartite but has two degree of freedom. (Why?)

23 / 23 Q. Characterize the inﬁnitesimal rigidity of ”Caylay frameworks”.

I Γ ⊂ Eucl(d): a group −1 I S ⊆ Γ with S = S I G = Cay(Γ, S):

I V (G) = Γ

I E(G) = {ij : j ∈ iS}

I A Caylay framework (G, p) of Γ:

I G = Cay(Γ, S)

I For n = 24 and S = {−1, 1, −5, 5}, the framework is 4-regular connected and bipartite but has two degree of freedom. (Why?)

Thank you for your attention!!

23 / 23