rigidity in the euclidean plane - college of sciencesrigidity in the euclidean plane xiaofeng gu...
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![Page 1: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/1.jpg)
Rigidity in the Euclidean plane
Xiaofeng Gu(University of West Georgia)
31st Cumberland Conferenceon Combinatorics, Graph Theory and Computing
UCF
May 19, 2019
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Introduction Some Results
Background
Rigidity, arising in discrete geometry, is the property of astructure that does not flex.A d-dimensional framework is a pair (G, p), where G(V,E)is a graph and p is a map from V to Rd. Roughly speaking,it is a straight line realization of G in Rd.Two frameworks (G, p) and (G, q) are equivalent if||p(u)− p(v)|| = ||q(u)− q(v)|| holds for every edge uv ∈ E,where || · || denotes the Euclidean norm in Rd.Two frameworks (G, p) and (G, q) are congruent if||p(u)− p(v)|| = ||q(u)− q(v)|| holds for every pair u, v ∈ V .The framework (G, p) is rigid if there exists an ε > 0 suchthat if (G, p) is equivalent to (G, q) and ||p(u)− q(u)|| < εfor every u ∈ V , then (G, p) is congruent to (G, q).
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Introduction Some Results
Background (cont...)
A generic realization of G is rigid in Rd if and only if everygeneric realization of G is rigid in Rd.Hence the generic rigidity can be considered as a propertyof the underlying graph.A graph is rigid in Rd if every/some generic realization of Gis rigid in Rd.Laman provides a combinatorial characterization of rigidgraphs in R2.
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Introduction Some Results
Rigid graphs
Let G = (V,E) be a graph.
A graph G is sparse if |E(X)| ≤ 2|X| − 3 for all X ⊆ V (G)with |X| ≥ 2.
If in addition |E(G)| = 2|V (G)| − 3, then G is minimallyrigid.A minimally rigid graph is also called a Laman graph.A graph G is rigid if G contains a spanning minimally rigidsubgraph.Every rigid graph with at least 3 vertices is 2-connected.Combinatorial rigidity theory.
![Page 5: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/5.jpg)
Introduction Some Results
Rigid graphs
Let G = (V,E) be a graph.
A graph G is sparse if |E(X)| ≤ 2|X| − 3 for all X ⊆ V (G)with |X| ≥ 2.If in addition |E(G)| = 2|V (G)| − 3, then G is minimallyrigid.
A minimally rigid graph is also called a Laman graph.A graph G is rigid if G contains a spanning minimally rigidsubgraph.Every rigid graph with at least 3 vertices is 2-connected.Combinatorial rigidity theory.
![Page 6: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/6.jpg)
Introduction Some Results
Rigid graphs
Let G = (V,E) be a graph.
A graph G is sparse if |E(X)| ≤ 2|X| − 3 for all X ⊆ V (G)with |X| ≥ 2.If in addition |E(G)| = 2|V (G)| − 3, then G is minimallyrigid.A minimally rigid graph is also called a Laman graph.
A graph G is rigid if G contains a spanning minimally rigidsubgraph.Every rigid graph with at least 3 vertices is 2-connected.Combinatorial rigidity theory.
![Page 7: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/7.jpg)
Introduction Some Results
Rigid graphs
Let G = (V,E) be a graph.
A graph G is sparse if |E(X)| ≤ 2|X| − 3 for all X ⊆ V (G)with |X| ≥ 2.If in addition |E(G)| = 2|V (G)| − 3, then G is minimallyrigid.A minimally rigid graph is also called a Laman graph.A graph G is rigid if G contains a spanning minimally rigidsubgraph.
Every rigid graph with at least 3 vertices is 2-connected.Combinatorial rigidity theory.
![Page 8: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/8.jpg)
Introduction Some Results
Rigid graphs
Let G = (V,E) be a graph.
A graph G is sparse if |E(X)| ≤ 2|X| − 3 for all X ⊆ V (G)with |X| ≥ 2.If in addition |E(G)| = 2|V (G)| − 3, then G is minimallyrigid.A minimally rigid graph is also called a Laman graph.A graph G is rigid if G contains a spanning minimally rigidsubgraph.Every rigid graph with at least 3 vertices is 2-connected.
Combinatorial rigidity theory.
![Page 9: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/9.jpg)
Introduction Some Results
Rigid graphs
Let G = (V,E) be a graph.
A graph G is sparse if |E(X)| ≤ 2|X| − 3 for all X ⊆ V (G)with |X| ≥ 2.If in addition |E(G)| = 2|V (G)| − 3, then G is minimallyrigid.A minimally rigid graph is also called a Laman graph.A graph G is rigid if G contains a spanning minimally rigidsubgraph.Every rigid graph with at least 3 vertices is 2-connected.Combinatorial rigidity theory.
![Page 10: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/10.jpg)
Introduction Some Results
Examples of minimally rigid graphs
Recall: A graph G is minimally rigid if |E(G)| = 2|V (G)| − 3 and|E(X)| ≤ 2|X| − 3 for all X ⊆ V (G) with |X| ≥ 2.
|V (G)| = 2 |V (G)| = 3
v
|V (G)| = 4
v
|V (G)| = 5
Extension operations:1. Add a new vertex v and two edges.2. Subdivide an edge by a new vertex v and add a new edge.
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Introduction Some Results
Non-rigid example
Constructed by Lovasz and Yemini (1982):
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Introduction Some Results
An alternative definition
(Lovasz and Yemini, 1982)A graph G is rigid if
∑X∈G(2|X| − 3) ≥ 2|V | − 3 for every
collection G of sets of V (G) such that {E(X), X ∈ G}partitions E(G), where E(X) is the edge set of thesubgraph of G induced by X.
In the example, choose a collection G in this way: thevertex set of each K5 together with vertex set of each ofother single edges.∑
X∈G(2|X| − 3) = 8(2 · 5− 3) + 20(2 · 2− 3) = 76
However, 2|V | − 3 = 2 · 40− 3 = 77.
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Introduction Some Results
An alternative definition
(Lovasz and Yemini, 1982)A graph G is rigid if
∑X∈G(2|X| − 3) ≥ 2|V | − 3 for every
collection G of sets of V (G) such that {E(X), X ∈ G}partitions E(G), where E(X) is the edge set of thesubgraph of G induced by X.In the example, choose a collection G in this way: thevertex set of each K5 together with vertex set of each ofother single edges.
∑X∈G(2|X| − 3) = 8(2 · 5− 3) + 20(2 · 2− 3) = 76
However, 2|V | − 3 = 2 · 40− 3 = 77.
![Page 14: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/14.jpg)
Introduction Some Results
An alternative definition
(Lovasz and Yemini, 1982)A graph G is rigid if
∑X∈G(2|X| − 3) ≥ 2|V | − 3 for every
collection G of sets of V (G) such that {E(X), X ∈ G}partitions E(G), where E(X) is the edge set of thesubgraph of G induced by X.In the example, choose a collection G in this way: thevertex set of each K5 together with vertex set of each ofother single edges.∑
X∈G(2|X| − 3) = 8(2 · 5− 3) + 20(2 · 2− 3) = 76
However, 2|V | − 3 = 2 · 40− 3 = 77.
![Page 15: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/15.jpg)
Introduction Some Results
An alternative definition
(Lovasz and Yemini, 1982)A graph G is rigid if
∑X∈G(2|X| − 3) ≥ 2|V | − 3 for every
collection G of sets of V (G) such that {E(X), X ∈ G}partitions E(G), where E(X) is the edge set of thesubgraph of G induced by X.In the example, choose a collection G in this way: thevertex set of each K5 together with vertex set of each ofother single edges.∑
X∈G(2|X| − 3) = 8(2 · 5− 3) + 20(2 · 2− 3) = 76
However, 2|V | − 3 = 2 · 40− 3 = 77.
![Page 16: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/16.jpg)
Introduction Some Results
Known results on rigid graphs
Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.
There exist 5-connected non-rigid graphs.Theorem (Jackson, Servatius and Servatius, 2007)Every 4-connected essentially 6-connected graph is rigid.Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.
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Introduction Some Results
Known results on rigid graphs
Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.There exist 5-connected non-rigid graphs.
Theorem (Jackson, Servatius and Servatius, 2007)Every 4-connected essentially 6-connected graph is rigid.Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.
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Introduction Some Results
Known results on rigid graphs
Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.There exist 5-connected non-rigid graphs.Theorem (Jackson, Servatius and Servatius, 2007)Every 4-connected essentially 6-connected graph is rigid.
Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.
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Introduction Some Results
Known results on rigid graphs
Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.There exist 5-connected non-rigid graphs.Theorem (Jackson, Servatius and Servatius, 2007)Every 4-connected essentially 6-connected graph is rigid.Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.
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Introduction Some Results
Edge-disjoint spanning rigid subgraphs
Theorem (Jordan, 2005)Every 6k-connected graph contains k edge-disjointspanning rigid subgraphs.
Theorem (Cheriyan, Durand de Gevigney, Szigeti, 2014) Asimple graph G contains edge-disjoint k spanning rigidsubgraphs if G− Z is (6k − 2k|Z|)-edge-connected forevery Z ⊂ V (G).Kriesell conjectured that there exists a (smallest) integerf(p) such that every f(p)-connected graph G a spanningtree T such that G− E(T ) is p-connected.
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Introduction Some Results
Edge-disjoint spanning rigid subgraphs
Theorem (Jordan, 2005)Every 6k-connected graph contains k edge-disjointspanning rigid subgraphs.Theorem (Cheriyan, Durand de Gevigney, Szigeti, 2014) Asimple graph G contains edge-disjoint k spanning rigidsubgraphs if G− Z is (6k − 2k|Z|)-edge-connected forevery Z ⊂ V (G).
Kriesell conjectured that there exists a (smallest) integerf(p) such that every f(p)-connected graph G a spanningtree T such that G− E(T ) is p-connected.
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Introduction Some Results
Edge-disjoint spanning rigid subgraphs
Theorem (Jordan, 2005)Every 6k-connected graph contains k edge-disjointspanning rigid subgraphs.Theorem (Cheriyan, Durand de Gevigney, Szigeti, 2014) Asimple graph G contains edge-disjoint k spanning rigidsubgraphs if G− Z is (6k − 2k|Z|)-edge-connected forevery Z ⊂ V (G).Kriesell conjectured that there exists a (smallest) integerf(p) such that every f(p)-connected graph G a spanningtree T such that G− E(T ) is p-connected.
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Introduction Some Results
Spanning Trees Packing Theorem
Theorem (Nash-Williams and Tutte, 1961, independently)A graph G has k edge-disjoint spanning trees if and only iffor any partition π of V (G), eG(π) ≥ k(|π| − 1).
CorollaryEvery 2k-edge-connected graph contains a packing of kspanning trees.
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Introduction Some Results
Spanning Trees Packing Theorem
Theorem (Nash-Williams and Tutte, 1961, independently)A graph G has k edge-disjoint spanning trees if and only iffor any partition π of V (G), eG(π) ≥ k(|π| − 1).
CorollaryEvery 2k-edge-connected graph contains a packing of kspanning trees.
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Introduction Some Results
Partition condition for edge-disjoint spanning rigidsubgraphs
Let Z ⊂ V (G) and π be a partition of V (G− Z) with n0 trivialparts v1, v2, · · · , vn0 . We define nZ(π) to be
∑1≤i≤n0
|Zi| whereZi is the set of vertices in Z that are adjacent to vi for1 ≤ i ≤ n0. If Z = ∅, then nZ(π) = 0.Theorem (G. 2018)A simple graph G contains k edge-disjoint spanning rigidsubgraphs if for any partition π of V (G− Z) with n0 trivial parts,eG−Z(π) ≥ k(3− |Z|)n′0 + 2kn0 − 3k − nZ(π) for everyZ ⊂ V (G).
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Introduction Some Results
More sufficient conditions
Recall: Theorem (Jackson, Servatius and Servatius, 2007)Every 4-connected essentially 6-connected graph is rigid.
Theorem (G. 2018)Every 4k-connected and essentially 6k-connected graph Gcontains edge-disjoint k spanning rigid subgraphs.Recall: Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.Theorem (G. 2018)A simple graph G contains edge-disjoint k spanning rigidsubgraphs if G is 4k-edge-connected, and G− Z isessentially (6k − 2k|Z|)-edge-connected for everyZ ⊂ V (G).
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Introduction Some Results
More sufficient conditions
Recall: Theorem (Jackson, Servatius and Servatius, 2007)Every 4-connected essentially 6-connected graph is rigid.Theorem (G. 2018)Every 4k-connected and essentially 6k-connected graph Gcontains edge-disjoint k spanning rigid subgraphs.
Recall: Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.Theorem (G. 2018)A simple graph G contains edge-disjoint k spanning rigidsubgraphs if G is 4k-edge-connected, and G− Z isessentially (6k − 2k|Z|)-edge-connected for everyZ ⊂ V (G).
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Introduction Some Results
More sufficient conditions
Recall: Theorem (Jackson, Servatius and Servatius, 2007)Every 4-connected essentially 6-connected graph is rigid.Theorem (G. 2018)Every 4k-connected and essentially 6k-connected graph Gcontains edge-disjoint k spanning rigid subgraphs.Recall: Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.
Theorem (G. 2018)A simple graph G contains edge-disjoint k spanning rigidsubgraphs if G is 4k-edge-connected, and G− Z isessentially (6k − 2k|Z|)-edge-connected for everyZ ⊂ V (G).
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Introduction Some Results
More sufficient conditions
Recall: Theorem (Jackson, Servatius and Servatius, 2007)Every 4-connected essentially 6-connected graph is rigid.Theorem (G. 2018)Every 4k-connected and essentially 6k-connected graph Gcontains edge-disjoint k spanning rigid subgraphs.Recall: Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.Theorem (G. 2018)A simple graph G contains edge-disjoint k spanning rigidsubgraphs if G is 4k-edge-connected, and G− Z isessentially (6k − 2k|Z|)-edge-connected for everyZ ⊂ V (G).
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Introduction Some Results
Try other sufficient conditions
Recall: Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.
Algebraic connectivity µ2(G) is the second smallesteigenvalue of the Laplacian matrix of a graph G.It is well known that the vertex connectivity is boundedbelow by the algebraic connectivity (Fiedler 1973).CorollaryIf µ2(G) ≥ 6, then G is rigid.
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Introduction Some Results
Try other sufficient conditions
Recall: Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.Algebraic connectivity µ2(G) is the second smallesteigenvalue of the Laplacian matrix of a graph G.
It is well known that the vertex connectivity is boundedbelow by the algebraic connectivity (Fiedler 1973).CorollaryIf µ2(G) ≥ 6, then G is rigid.
![Page 32: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/32.jpg)
Introduction Some Results
Try other sufficient conditions
Recall: Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.Algebraic connectivity µ2(G) is the second smallesteigenvalue of the Laplacian matrix of a graph G.It is well known that the vertex connectivity is boundedbelow by the algebraic connectivity (Fiedler 1973).
CorollaryIf µ2(G) ≥ 6, then G is rigid.
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Introduction Some Results
Try other sufficient conditions
Recall: Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.Algebraic connectivity µ2(G) is the second smallesteigenvalue of the Laplacian matrix of a graph G.It is well known that the vertex connectivity is boundedbelow by the algebraic connectivity (Fiedler 1973).CorollaryIf µ2(G) ≥ 6, then G is rigid.
![Page 34: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/34.jpg)
Introduction Some Results
New result on rigid graphs
Theorem (Cioaba and G. 2019+)Let G be a graph with minimum degree δ ≥ 6k. If
µ2(G) > 2 +2k − 1
δ − 1,
then G has at least k edge-disjoint spanning rigidsubgraphs. In particular, if µ2(G) > 2 + 1
δ−1 , then G is rigid.
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Introduction Some Results
Another application
Theorem (Fiedler 1973)If µ2(G) ≥ p, then G is p-connected.
Question What spectral conditions can guaranteeedge-disjoint spanning p-connected subgraphs?Cioaba and Wong started the problem for p = 1 and poseda conjecture.Theorem (Liu, Hong, G., Lai 2014)Let G be a simple graph with δ ≥ 2k. If µ2(G) > 2k−1
δ+1 , thenG has at least k edge-disjoint spanning trees.Theorem (Cioaba and G. 2019+)Let G be a simple graph with δ ≥ 6k. If µ2(G) > 2 + 2k−1
δ−1 ,then G has at least k edge-disjoint spanning 2-connectedsubgraphs.
![Page 36: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/36.jpg)
Introduction Some Results
Another application
Theorem (Fiedler 1973)If µ2(G) ≥ p, then G is p-connected.Question What spectral conditions can guaranteeedge-disjoint spanning p-connected subgraphs?
Cioaba and Wong started the problem for p = 1 and poseda conjecture.Theorem (Liu, Hong, G., Lai 2014)Let G be a simple graph with δ ≥ 2k. If µ2(G) > 2k−1
δ+1 , thenG has at least k edge-disjoint spanning trees.Theorem (Cioaba and G. 2019+)Let G be a simple graph with δ ≥ 6k. If µ2(G) > 2 + 2k−1
δ−1 ,then G has at least k edge-disjoint spanning 2-connectedsubgraphs.
![Page 37: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/37.jpg)
Introduction Some Results
Another application
Theorem (Fiedler 1973)If µ2(G) ≥ p, then G is p-connected.Question What spectral conditions can guaranteeedge-disjoint spanning p-connected subgraphs?Cioaba and Wong started the problem for p = 1 and poseda conjecture.
Theorem (Liu, Hong, G., Lai 2014)Let G be a simple graph with δ ≥ 2k. If µ2(G) > 2k−1
δ+1 , thenG has at least k edge-disjoint spanning trees.Theorem (Cioaba and G. 2019+)Let G be a simple graph with δ ≥ 6k. If µ2(G) > 2 + 2k−1
δ−1 ,then G has at least k edge-disjoint spanning 2-connectedsubgraphs.
![Page 38: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/38.jpg)
Introduction Some Results
Another application
Theorem (Fiedler 1973)If µ2(G) ≥ p, then G is p-connected.Question What spectral conditions can guaranteeedge-disjoint spanning p-connected subgraphs?Cioaba and Wong started the problem for p = 1 and poseda conjecture.Theorem (Liu, Hong, G., Lai 2014)Let G be a simple graph with δ ≥ 2k. If µ2(G) > 2k−1
δ+1 , thenG has at least k edge-disjoint spanning trees.
Theorem (Cioaba and G. 2019+)Let G be a simple graph with δ ≥ 6k. If µ2(G) > 2 + 2k−1
δ−1 ,then G has at least k edge-disjoint spanning 2-connectedsubgraphs.
![Page 39: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/39.jpg)
Introduction Some Results
Another application
Theorem (Fiedler 1973)If µ2(G) ≥ p, then G is p-connected.Question What spectral conditions can guaranteeedge-disjoint spanning p-connected subgraphs?Cioaba and Wong started the problem for p = 1 and poseda conjecture.Theorem (Liu, Hong, G., Lai 2014)Let G be a simple graph with δ ≥ 2k. If µ2(G) > 2k−1
δ+1 , thenG has at least k edge-disjoint spanning trees.Theorem (Cioaba and G. 2019+)Let G be a simple graph with δ ≥ 6k. If µ2(G) > 2 + 2k−1
δ−1 ,then G has at least k edge-disjoint spanning 2-connectedsubgraphs.
![Page 40: Rigidity in the Euclidean plane - College of SciencesRigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory](https://reader034.vdocuments.us/reader034/viewer/2022042403/5f1575eb5b1fb905296481bb/html5/thumbnails/40.jpg)
Introduction Some Results
Thank You