graph topology and discrete morse theory - github pagesdiscrete morse theory was developed by robin...
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![Page 1: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/1.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Graph Topology and Discrete Morse Theory
Andrew Rosevear, Andrew TawfeekAdvised by Prof. Ivan Contreras
Amherst College
July 23rd, 2018
![Page 2: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/2.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Outline Part I
OriginsSmooth Morse TheoryDiscrete Morse Theory
Focusing on GraphsDefinitionsGradient Flow
ResultsCharacterizing Critical CellsCharacterizing Gradient FlowWeak Morse Inequalities
Future Directions
![Page 3: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/3.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Outline Part I
OriginsSmooth Morse TheoryDiscrete Morse Theory
Focusing on GraphsDefinitionsGradient Flow
ResultsCharacterizing Critical CellsCharacterizing Gradient FlowWeak Morse Inequalities
Future Directions
![Page 4: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/4.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Outline Part I
OriginsSmooth Morse TheoryDiscrete Morse Theory
Focusing on GraphsDefinitionsGradient Flow
ResultsCharacterizing Critical CellsCharacterizing Gradient FlowWeak Morse Inequalities
Future Directions
![Page 5: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/5.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Outline Part I
OriginsSmooth Morse TheoryDiscrete Morse Theory
Focusing on GraphsDefinitionsGradient Flow
ResultsCharacterizing Critical CellsCharacterizing Gradient FlowWeak Morse Inequalities
Future Directions
![Page 6: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/6.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Smooth Morse TheoryBuilding Intuition
Note that an important result in smooth Morse theory is that given acritical point, we can choose the correct local coordinates so the functiontakes the form of a parabaloid opening upwards/downwards or a saddlepoint.
Lastly, it turns out that there is an important correspondence betweenMorse functions f and gradient-like vector fields for f .
![Page 7: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/7.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Smooth Morse TheoryBuilding Intuition
Note that an important result in smooth Morse theory is that given acritical point, we can choose the correct local coordinates so the functiontakes the form of a parabaloid opening upwards/downwards or a saddlepoint.
Lastly, it turns out that there is an important correspondence betweenMorse functions f and gradient-like vector fields for f .
![Page 8: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/8.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Smooth Morse TheoryBuilding Intuition
Note that an important result in smooth Morse theory is that given acritical point, we can choose the correct local coordinates so the functiontakes the form of a parabaloid opening upwards/downwards or a saddlepoint.
Lastly, it turns out that there is an important correspondence betweenMorse functions f and gradient-like vector fields for f .
![Page 9: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/9.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Shifting View
Discrete Morse theory was developed by Robin Forman around2002, in his published work A Users Guide to Discrete MorseTheory.
Here, he develops an adaption of smooth Morse theory forCW complexes1 that preserves many discrete analogues to theproperties of Morse functions in smooth Morse theory.
We will only focus on the definition with 0-cells (vertices) and1-cells (edges), i.e. graphs.
1CW complexes can be regarded as a generalization of graphs, where notonly can you glue points and edges (S0) together, but higher dimensionalspheres as well.
![Page 10: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/10.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Shifting View
Discrete Morse theory was developed by Robin Forman around2002, in his published work A Users Guide to Discrete MorseTheory. Here, he develops an adaption of smooth Morse theory forCW complexes1 that preserves many discrete analogues to theproperties of Morse functions in smooth Morse theory.
We will only focus on the definition with 0-cells (vertices) and1-cells (edges), i.e. graphs.
1CW complexes can be regarded as a generalization of graphs, where notonly can you glue points and edges (S0) together, but higher dimensionalspheres as well.
![Page 11: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/11.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Shifting View
Discrete Morse theory was developed by Robin Forman around2002, in his published work A Users Guide to Discrete MorseTheory. Here, he develops an adaption of smooth Morse theory forCW complexes1 that preserves many discrete analogues to theproperties of Morse functions in smooth Morse theory.
We will only focus on the definition with 0-cells (vertices) and1-cells (edges), i.e. graphs.
1CW complexes can be regarded as a generalization of graphs, where notonly can you glue points and edges (S0) together, but higher dimensionalspheres as well.
![Page 12: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/12.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Focusing on GraphsDefinitions
For a graph Γ, define an ordering on the cells, Γc = V ∪ E , bydeclaring every vertex lesser than the edge of which it is anendpoint.
DefinitionLet Γ = (V ,E ) be a graph. A discrete Morse function is a functionf : Γc = V ∪ E → R such that for every σ ∈ Γc ,
|{τ ∈ Γc | σ < τ and f (σ) ≥ f (τ)}| ≤ 1; (1)
|{τ ∈ Γc | σ > τ and f (σ) ≤ f (τ)}| ≤ 1. (2)
We say a cell (vertex or edge) is critical if both sets (1) and (2)above are empty. We let c0(f ) denote the number of criticalvertices, and c1(f ) number of critical edges.
![Page 13: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/13.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Focusing on GraphsDefinitions
For a graph Γ, define an ordering on the cells, Γc = V ∪ E , bydeclaring every vertex lesser than the edge of which it is anendpoint.
DefinitionLet Γ = (V ,E ) be a graph. A discrete Morse function is a functionf : Γc = V ∪ E → R such that for every σ ∈ Γc ,
|{τ ∈ Γc | σ < τ and f (σ) ≥ f (τ)}| ≤ 1; (1)
|{τ ∈ Γc | σ > τ and f (σ) ≤ f (τ)}| ≤ 1. (2)
We say a cell (vertex or edge) is critical if both sets (1) and (2)above are empty. We let c0(f ) denote the number of criticalvertices, and c1(f ) number of critical edges.
![Page 14: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/14.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Focusing on GraphsDefinitions
For a graph Γ, define an ordering on the cells, Γc = V ∪ E , bydeclaring every vertex lesser than the edge of which it is anendpoint.
DefinitionLet Γ = (V ,E ) be a graph. A discrete Morse function is a functionf : Γc = V ∪ E → R such that for every σ ∈ Γc ,
|{τ ∈ Γc | σ < τ and f (σ) ≥ f (τ)}| ≤ 1; (1)
|{τ ∈ Γc | σ > τ and f (σ) ≤ f (τ)}| ≤ 1. (2)
We say a cell (vertex or edge) is critical if both sets (1) and (2)above are empty. We let c0(f ) denote the number of criticalvertices, and c1(f ) number of critical edges.
![Page 15: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/15.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Focusing on GraphsDefinitions
![Page 16: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/16.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsGradient Flow
![Page 17: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/17.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsCharacterizing Critical Cells
We call this oriented graph corresponding to a pair (Γ, f ) of agraph and it’s discrete Morse function the gradient flow Γf .
Theorem (A.T.)
1. edge is critical ⇐⇒ undirected in Γf
2. vertex is a critical ⇐⇒ sink in Γf
Another note to have in mind is critical vertices occur at the endof gradient curves.
In fact, we can do even better. We can fully characterize discreteMorse functions with gradient flows.
![Page 18: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/18.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsCharacterizing Critical Cells
We call this oriented graph corresponding to a pair (Γ, f ) of agraph and it’s discrete Morse function the gradient flow Γf .
Theorem (A.T.)
1. edge is critical ⇐⇒ undirected in Γf
2. vertex is a critical ⇐⇒ sink in Γf
Another note to have in mind is critical vertices occur at the endof gradient curves.
In fact, we can do even better. We can fully characterize discreteMorse functions with gradient flows.
![Page 19: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/19.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsCharacterizing Critical Cells
We call this oriented graph corresponding to a pair (Γ, f ) of agraph and it’s discrete Morse function the gradient flow Γf .
Theorem (A.T.)
1. edge is critical ⇐⇒ undirected in Γf
2. vertex is a critical ⇐⇒ sink in Γf
Another note to have in mind is critical vertices occur at the endof gradient curves.
In fact, we can do even better. We can fully characterize discreteMorse functions with gradient flows.
![Page 20: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/20.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsCharacterizing Critical Cells
We call this oriented graph corresponding to a pair (Γ, f ) of agraph and it’s discrete Morse function the gradient flow Γf .
Theorem (A.T.)
1. edge is critical ⇐⇒ undirected in Γf
2. vertex is a critical ⇐⇒ sink in Γf
Another note to have in mind is critical vertices occur at the endof gradient curves.
In fact, we can do even better. We can fully characterize discreteMorse functions with gradient flows.
![Page 21: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/21.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsCharacterizing Gradient Flow
Theorem (A.T)Let Γo be a directed graph and Γ be it’s underlying undirected graph. ThenΓo = Γf for some discrete Morse function f on Γ if and only if
1. no two edges share a tail, and
2. there are no directed loops.
If we denote by Morse(Γ) be the set of all discrete Morse functions that can bedefined on Γ, then
f ∼Γ g ⇐⇒ Γf∼= Γg
is an equivalence relation on Morse(Γ).
This turns out to be equivalent to Forman’s equivalence of discrete Morsefunctions on a graph, i.e. f is equivalent to g if and only if for every vertex andedge of Γ,
f (v) < f (e) ⇐⇒ g(v) < g(e).
![Page 22: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/22.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsCharacterizing Gradient Flow
Theorem (A.T)Let Γo be a directed graph and Γ be it’s underlying undirected graph. ThenΓo = Γf for some discrete Morse function f on Γ if and only if
1. no two edges share a tail, and
2. there are no directed loops.
If we denote by Morse(Γ) be the set of all discrete Morse functions that can bedefined on Γ, then
f ∼Γ g ⇐⇒ Γf∼= Γg
is an equivalence relation on Morse(Γ).
This turns out to be equivalent to Forman’s equivalence of discrete Morsefunctions on a graph, i.e. f is equivalent to g if and only if for every vertex andedge of Γ,
f (v) < f (e) ⇐⇒ g(v) < g(e).
![Page 23: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/23.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsCharacterizing Gradient Flow
Theorem (A.T)Let Γo be a directed graph and Γ be it’s underlying undirected graph. ThenΓo = Γf for some discrete Morse function f on Γ if and only if
1. no two edges share a tail, and
2. there are no directed loops.
If we denote by Morse(Γ) be the set of all discrete Morse functions that can bedefined on Γ, then
f ∼Γ g ⇐⇒ Γf∼= Γg
is an equivalence relation on Morse(Γ).
This turns out to be equivalent to Forman’s equivalence of discrete Morsefunctions on a graph, i.e. f is equivalent to g if and only if for every vertex andedge of Γ,
f (v) < f (e) ⇐⇒ g(v) < g(e).
![Page 24: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/24.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsWeak Morse Inequalities
Recall that the 0th Betti number of a graph, b0(Γ), is number ofconnected components, and the 1st Betti number, b1(Γ), is thenumber of independent cycles.
Additionally, the Eulercharacteristic χ(Γ) can be expressed as b0 − b1 = |V | − |E |.
Theorem (Weak Morse Inequalities, A.T.)
Let Γ be a simple and finite graph and f : Γc → R be a Morsefunction on the graph.Then
• b0(Γ) ≤ c0(f )
• b1(Γ) ≤ c1(f )
• χ(Γ) = c0(f )− c1(f )
Additionally, there have been numerous technical results concerningthe equivalence classes that are still being studied in more depth.
![Page 25: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/25.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsWeak Morse Inequalities
Recall that the 0th Betti number of a graph, b0(Γ), is number ofconnected components, and the 1st Betti number, b1(Γ), is thenumber of independent cycles. Additionally, the Eulercharacteristic χ(Γ) can be expressed as b0 − b1 = |V | − |E |.
Theorem (Weak Morse Inequalities, A.T.)
Let Γ be a simple and finite graph and f : Γc → R be a Morsefunction on the graph.Then
• b0(Γ) ≤ c0(f )
• b1(Γ) ≤ c1(f )
• χ(Γ) = c0(f )− c1(f )
Additionally, there have been numerous technical results concerningthe equivalence classes that are still being studied in more depth.
![Page 26: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/26.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsWeak Morse Inequalities
Recall that the 0th Betti number of a graph, b0(Γ), is number ofconnected components, and the 1st Betti number, b1(Γ), is thenumber of independent cycles. Additionally, the Eulercharacteristic χ(Γ) can be expressed as b0 − b1 = |V | − |E |.
Theorem (Weak Morse Inequalities, A.T.)
Let Γ be a simple and finite graph and f : Γc → R be a Morsefunction on the graph.
Then
• b0(Γ) ≤ c0(f )
• b1(Γ) ≤ c1(f )
• χ(Γ) = c0(f )− c1(f )
Additionally, there have been numerous technical results concerningthe equivalence classes that are still being studied in more depth.
![Page 27: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/27.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsWeak Morse Inequalities
Recall that the 0th Betti number of a graph, b0(Γ), is number ofconnected components, and the 1st Betti number, b1(Γ), is thenumber of independent cycles. Additionally, the Eulercharacteristic χ(Γ) can be expressed as b0 − b1 = |V | − |E |.
Theorem (Weak Morse Inequalities, A.T.)
Let Γ be a simple and finite graph and f : Γc → R be a Morsefunction on the graph.Then
• b0(Γ) ≤ c0(f )
• b1(Γ) ≤ c1(f )
• χ(Γ) = c0(f )− c1(f )
Additionally, there have been numerous technical results concerningthe equivalence classes that are still being studied in more depth.
![Page 28: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/28.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsWeak Morse Inequalities
Recall that the 0th Betti number of a graph, b0(Γ), is number ofconnected components, and the 1st Betti number, b1(Γ), is thenumber of independent cycles. Additionally, the Eulercharacteristic χ(Γ) can be expressed as b0 − b1 = |V | − |E |.
Theorem (Weak Morse Inequalities, A.T.)
Let Γ be a simple and finite graph and f : Γc → R be a Morsefunction on the graph.Then
• b0(Γ) ≤ c0(f )
• b1(Γ) ≤ c1(f )
• χ(Γ) = c0(f )− c1(f )
Additionally, there have been numerous technical results concerningthe equivalence classes that are still being studied in more depth.
![Page 29: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/29.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsWeak Morse Inequalities
Recall that the 0th Betti number of a graph, b0(Γ), is number ofconnected components, and the 1st Betti number, b1(Γ), is thenumber of independent cycles. Additionally, the Eulercharacteristic χ(Γ) can be expressed as b0 − b1 = |V | − |E |.
Theorem (Weak Morse Inequalities, A.T.)
Let Γ be a simple and finite graph and f : Γc → R be a Morsefunction on the graph.Then
• b0(Γ) ≤ c0(f )
• b1(Γ) ≤ c1(f )
• χ(Γ) = c0(f )− c1(f )
Additionally, there have been numerous technical results concerningthe equivalence classes that are still being studied in more depth.
![Page 30: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/30.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
ResultsWeak Morse Inequalities
Recall that the 0th Betti number of a graph, b0(Γ), is number ofconnected components, and the 1st Betti number, b1(Γ), is thenumber of independent cycles. Additionally, the Eulercharacteristic χ(Γ) can be expressed as b0 − b1 = |V | − |E |.
Theorem (Weak Morse Inequalities, A.T.)
Let Γ be a simple and finite graph and f : Γc → R be a Morsefunction on the graph.Then
• b0(Γ) ≤ c0(f )
• b1(Γ) ≤ c1(f )
• χ(Γ) = c0(f )− c1(f )
Additionally, there have been numerous technical results concerningthe equivalence classes that are still being studied in more depth.
![Page 31: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/31.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Future Directions
• Study the number of equivalence classes for a given graph Γ.
• The number depends on the automorphism group Aut(Γ), andhow it acts on a special set of colored-graphs relating to Γ.
• We know the number when the automorphism group is trivial,and are very close to obtaining it for an arbitrary graph.
• Expand from vertex-gluing to interface-gluing discrete Morsefunctions and their corresponding graphs.
• Study properties of Morse complexes MΓ(f ) and theirrelation to the equivalence classes.
• By observing the barycentric subdivision of simiplicialcomplexes and regular CW complexes, attempt to framegeneral discrete Morse theory in terms of these equivalenceclasses.
• Develop an analogue of discrete Morse theory for hypergraphs.
![Page 32: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/32.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Future Directions
• Study the number of equivalence classes for a given graph Γ.• The number depends on the automorphism group Aut(Γ), and
how it acts on a special set of colored-graphs relating to Γ.
• We know the number when the automorphism group is trivial,and are very close to obtaining it for an arbitrary graph.
• Expand from vertex-gluing to interface-gluing discrete Morsefunctions and their corresponding graphs.
• Study properties of Morse complexes MΓ(f ) and theirrelation to the equivalence classes.
• By observing the barycentric subdivision of simiplicialcomplexes and regular CW complexes, attempt to framegeneral discrete Morse theory in terms of these equivalenceclasses.
• Develop an analogue of discrete Morse theory for hypergraphs.
![Page 33: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/33.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Future Directions
• Study the number of equivalence classes for a given graph Γ.• The number depends on the automorphism group Aut(Γ), and
how it acts on a special set of colored-graphs relating to Γ.• We know the number when the automorphism group is trivial,
and are very close to obtaining it for an arbitrary graph.
• Expand from vertex-gluing to interface-gluing discrete Morsefunctions and their corresponding graphs.
• Study properties of Morse complexes MΓ(f ) and theirrelation to the equivalence classes.
• By observing the barycentric subdivision of simiplicialcomplexes and regular CW complexes, attempt to framegeneral discrete Morse theory in terms of these equivalenceclasses.
• Develop an analogue of discrete Morse theory for hypergraphs.
![Page 34: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/34.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Future Directions
• Study the number of equivalence classes for a given graph Γ.• The number depends on the automorphism group Aut(Γ), and
how it acts on a special set of colored-graphs relating to Γ.• We know the number when the automorphism group is trivial,
and are very close to obtaining it for an arbitrary graph.
• Expand from vertex-gluing to interface-gluing discrete Morsefunctions and their corresponding graphs.
• Study properties of Morse complexes MΓ(f ) and theirrelation to the equivalence classes.
• By observing the barycentric subdivision of simiplicialcomplexes and regular CW complexes, attempt to framegeneral discrete Morse theory in terms of these equivalenceclasses.
• Develop an analogue of discrete Morse theory for hypergraphs.
![Page 35: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/35.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Future Directions
• Study the number of equivalence classes for a given graph Γ.• The number depends on the automorphism group Aut(Γ), and
how it acts on a special set of colored-graphs relating to Γ.• We know the number when the automorphism group is trivial,
and are very close to obtaining it for an arbitrary graph.
• Expand from vertex-gluing to interface-gluing discrete Morsefunctions and their corresponding graphs.
• Study properties of Morse complexes MΓ(f ) and theirrelation to the equivalence classes.
• By observing the barycentric subdivision of simiplicialcomplexes and regular CW complexes, attempt to framegeneral discrete Morse theory in terms of these equivalenceclasses.
• Develop an analogue of discrete Morse theory for hypergraphs.
![Page 36: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/36.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Future Directions
• Study the number of equivalence classes for a given graph Γ.• The number depends on the automorphism group Aut(Γ), and
how it acts on a special set of colored-graphs relating to Γ.• We know the number when the automorphism group is trivial,
and are very close to obtaining it for an arbitrary graph.
• Expand from vertex-gluing to interface-gluing discrete Morsefunctions and their corresponding graphs.
• Study properties of Morse complexes MΓ(f ) and theirrelation to the equivalence classes.
• By observing the barycentric subdivision of simiplicialcomplexes and regular CW complexes, attempt to framegeneral discrete Morse theory in terms of these equivalenceclasses.
• Develop an analogue of discrete Morse theory for hypergraphs.
![Page 37: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/37.jpg)
Origins Focusing on Graphs Results Future Directions Appendix
Bibliography
I. Contreras, B. Xu.The Graph Laplacian and Morse InequalitiesarXiv:1704.08354
K. Knudson.Morse Theory: Smooth and Discrete.World Scientific Publishing Co. Pte. Ltd., 2015.
L. Schneps.Hodge Theory and Complex Algebraic Geometry ICambridge University Press, 2002
R. Forman.A User’s Guide to Discrete Morse Theory.Seminaire Lotharingien de Combinatoire, 2002.
S. Lang.Fundamentals of Differential GeometrySpringer Science Business Media, 2001
![Page 38: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/38.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Outline
Topology of GraphsChain ComplexesThe Graph LaplacianGraph Homology
Graph de Rham CalculusDifferentiationIntegrationMain Results
Graph Stokes’ TheoremGraph Hodge Decomposition
Future DirectionsSolving Eigenvector IntegrationThe Morse ComplexAnalyzing Graph DiffEqs
![Page 39: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/39.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Graph Chain Complexes
Definition (Chain Complex of a Graph)
The chain complex of a directed graph Γ = (V ,E ) is asequence of vector spaces paired with linear maps
0→ C|E | ∂1−→ C|V | → 0,
where ∂1 is the boundary operator given by the |V | × |E |incidence matrix I whose entries are
Iij =
1 edge ej enters vertex vi
−1 edge ej leaves vertex vi
0 otherwise
.
![Page 40: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/40.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
The Graph Laplacian
Definition (Graph Laplacian)
The even and odd graph Laplacians ∆+ and ∆− of an orientedgraph Γ are given by
∆+ := II ∗ : C|V | → C|V |
∆− := I ∗I : C|E | → C|E |.
Both matrices are positive semidefinite and symmetric (andtherefore diagonalizable).
Lemma∆+ is invariant under orientation. However, ∆− is not.
![Page 41: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/41.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
The Graph Laplacian
Definition (Graph Laplacian)
The even and odd graph Laplacians ∆+ and ∆− of an orientedgraph Γ are given by
∆+ := II ∗ : C|V | → C|V |
∆− := I ∗I : C|E | → C|E |.
Both matrices are positive semidefinite and symmetric (andtherefore diagonalizable).
Lemma∆+ is invariant under orientation. However, ∆− is not.
![Page 42: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/42.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Homology and Betti Numbers
Definition (Homology Groups)
The homology groups of a graph Γ are given by
H1(Γ) = ker(I ) = ker(∆−)
andH0(Γ) = ker(I ∗) = ker(∆+).
Theorem (Contreras-Xu)
Let b1 and b0 be the Betti numbers of Γ. Then
dim(H1(Γ)) = b1
dim(H0(Γ)) = b0.
![Page 43: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/43.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Homology and Betti Numbers
Definition (Homology Groups)
The homology groups of a graph Γ are given by
H1(Γ) = ker(I ) = ker(∆−)
andH0(Γ) = ker(I ∗) = ker(∆+).
Theorem (Contreras-Xu)
Let b1 and b0 be the Betti numbers of Γ. Then
dim(H1(Γ)) = b1
dim(H0(Γ)) = b0.
![Page 44: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/44.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Cochain Complexes and the Graph Differential
The graph cochain complex is simply the graph chain complexbut with the arrows reversed, and I replaced with I ∗:
0→ C|V | I∗−→ C|E | → 0.
By analogy with the differential operator on the de Rham complex,we may view I ∗ as a kind of graph differential operator.In vector calculus terminology, I ∗ serves as the graph gradient.
![Page 45: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/45.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Cochain Complexes and the Graph Differential
The graph cochain complex is simply the graph chain complexbut with the arrows reversed, and I replaced with I ∗:
0→ C|V | I∗−→ C|E | → 0.
By analogy with the differential operator on the de Rham complex,we may view I ∗ as a kind of graph differential operator.In vector calculus terminology, I ∗ serves as the graph gradient.
![Page 46: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/46.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Integration on Graphs
Definition (Vertex Integral)
Let f be a discrete function on the vertices of Γ. The vertexintegral of f over Γ is given by∫ •
∂Γf =
∑vi∈V
f (vi )dn(vi ),
where dn(vi ) is the number of incoming minus outgoing edges.
Definition (Edge Integral)
Let F be a discrete function on the edges of Γ. The edge integralof F over Γ is given by∫ −
ΓF =
∑ei∈E
F (ei ).
![Page 47: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/47.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Integration on Graphs
Definition (Vertex Integral)
Let f be a discrete function on the vertices of Γ. The vertexintegral of f over Γ is given by∫ •
∂Γf =
∑vi∈V
f (vi )dn(vi ),
where dn(vi ) is the number of incoming minus outgoing edges.
Definition (Edge Integral)
Let F be a discrete function on the edges of Γ. The edge integralof F over Γ is given by∫ −
ΓF =
∑ei∈E
F (ei ).
![Page 48: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/48.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Main Results
Theorem (A.R)
(Stokes’ Theorem for Graphs) Let f be a vertex function on anoriented graph Γ. Then ∫ •
∂Γf =
∫ −Γ
I ∗f .
Theorem (A.R)
(Graph Hodge Decomposition) Let Γ be an oriented graph andH0(Γ) = ker(∆+) = ker(I ∗) and H1(Γ) = ker(∆−) = ker(I ) its nthhomology group. Then
C|V | = H0(Γ)⊕ Im(I )
C|E | = H1(Γ)⊕ Im(I ∗).
![Page 49: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/49.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Main Results
Theorem (A.R)
(Stokes’ Theorem for Graphs) Let f be a vertex function on anoriented graph Γ. Then ∫ •
∂Γf =
∫ −Γ
I ∗f .
Theorem (A.R)
(Graph Hodge Decomposition) Let Γ be an oriented graph andH0(Γ) = ker(∆+) = ker(I ∗) and H1(Γ) = ker(∆−) = ker(I ) its nthhomology group. Then
C|V | = H0(Γ)⊕ Im(I )
C|E | = H1(Γ)⊕ Im(I ∗).
![Page 50: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/50.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Future Directions
1. What happens when we integrate an eigenvector with nonzeroeigenvalue of ∆+ or ∆−?
2. One may consider the Morse complex, the chain complex ofthe subgraph induced by the critical cells of a Morse graph.
2.1 For an alternate proof of the Morse Inequalities, seeContreras-Xu, ”The Graph Laplacian and Morse Inequalities.”
3. A more coherent theory of graph calculus can help answerquestions about graph differential equations, such as thegraph Schrodinger equation
∂ϕ
∂t= i∆ϕ.
![Page 51: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/51.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Future Directions
1. What happens when we integrate an eigenvector with nonzeroeigenvalue of ∆+ or ∆−?
2. One may consider the Morse complex, the chain complex ofthe subgraph induced by the critical cells of a Morse graph.
2.1 For an alternate proof of the Morse Inequalities, seeContreras-Xu, ”The Graph Laplacian and Morse Inequalities.”
3. A more coherent theory of graph calculus can help answerquestions about graph differential equations, such as thegraph Schrodinger equation
∂ϕ
∂t= i∆ϕ.
![Page 52: Graph Topology and Discrete Morse Theory - GitHub PagesDiscrete Morse theory was developed by Robin Forman around 2002, in his published work A Users Guide to Discrete Morse Theory](https://reader030.vdocuments.mx/reader030/viewer/2022040215/5edc26b9ad6a402d6666b1ff/html5/thumbnails/52.jpg)
Topology of Graphs Graph de Rham Calculus Future Directions
Future Directions
1. What happens when we integrate an eigenvector with nonzeroeigenvalue of ∆+ or ∆−?
2. One may consider the Morse complex, the chain complex ofthe subgraph induced by the critical cells of a Morse graph.
2.1 For an alternate proof of the Morse Inequalities, seeContreras-Xu, ”The Graph Laplacian and Morse Inequalities.”
3. A more coherent theory of graph calculus can help answerquestions about graph differential equations, such as thegraph Schrodinger equation
∂ϕ
∂t= i∆ϕ.