HW 3 (Due Wednesday Feb 6) Create slide(s) for your 1 minute presentation on a graph theory application. Make sure your slide(s) include (1) Define the problem (2) What do the vertices represent (3) What do the edges represent (4) What can graph theory say about your real-life problem? Can you formally state the graph theory problem(s)? Use large font (best minimum = 24 point, 18 OK) Figures are helpful. INCLUDE YOUR NAME and affiliation.

Application: Assign classes to professors Isabel Darcy University of Iowa, Mathematics/AMCS/Informatics http://homepage.divms.uiowa.edu/~idarcy MATH: 1020: 0EXW MATH: 2700: 0081 MATH: 4060: 0001 MATH: 5700: 0AAA MATH: 2550: 0230 MATH: 2550: 0235 Example: UI’s mathbio group (Spr 2018)

Application: Assign classes to professors Example: UI’s mathbio group (Spr 2018) MATH: 1020: 0EXW MATH: 2700: 0081 MATH: 4060: 0001 MATH: 5700: 0AAA MATH: 2550: 0230 MATH: 2550: 0235

Application: Assign classes to professors Example: UI’s mathbio group (Spr 2018) MATH: 1020: 0EXW MATH: 2700: 0081 MATH: 4060: 0001 MATH: 5700: 0AAA MATH: 2550: 0230 MATH: 2550: 0235 A vertex represents either a math professor or a section of a math course

Application: Assign classes to professors Example: UI’s mathbio group (Spr 2018) MATH: 1020: 0EXW MATH: 2700: 0081 MATH: 4060: 0001 MATH: 5700: 0AAA MATH: 2550: 0230 MATH: 2550: 0235 An edge connects a math professor to a section of a math course that professor would like to teach

Application: Assign classes to professors Example: UI’s mathbio group (Spr 2018) MATH: 1020: 0EXW MATH: 2700: 0081 MATH: 4060: 0001 MATH: 5700: 0AAA MATH: 2550: 0230 MATH: 2550: 0235 Graph theory problem: Select a subset of the edges so that each vertex representing a course section has degree 1 and each vertex representing a professor has degree 0, 1, or 2.

Application: Assign classes to professors Problem description: Math professors at UI are asked to provide an ordered list of classes that they would like to teach in a particular semester. The goal is to assign classes to these professors which fit their preferences as much as possible. Vertices: The set of professors union the set of classes. I.e., each math professor is represented by a vertex and each section of a math class is represented by a vertex. That is a vertex will represent either a math professor or a section of a math class. Edges: An edge is drawn between a vertex representing a math professor and all sections of a math class if that professor has listed that math class as one of the courses they would like to teach.

Application: Assign classes to professors Example: UI’s mathbio group (Spr 2018) 2 1 1 2 MATH: 1020: 0EXW MATH: 2700: 0081 MATH: 4060: 0001 MATH: 5700: 0AAA MATH: 2550: 0230 MATH: 2550: 0235 Math professors at UI are asked to provide an ordered list of classes that they would like to teach in a particular semester. Weighted graph

Application: Assign classes to professors Example: UI’s mathbio group (Spr 2018) 2 1 1 2 MATH: 1020: 0EXW MATH: 2700: 0081 MATH: 4060: 0001 MATH: 5700: 0AAA MATH: 2550: 0230 MATH: 2550: 0235 Math professors at UI are asked to provide an ordered list of classes that they would like to teach in a particular semester. Weighted graph What is the real problem?

Bipartite graphs In a simple graph G, if V can be partitioned into two disjoint sets V1 and V2 such that every edge in the graph connects a vertex in V1 and a vertex V2 (so that no edge in G connects either two vertices in V1 or two vertices in V2) Application example: Representing Relations Representation example: V1 = {v1, v2, v3} and V2 = {v4, v5, v6}, v4 v1 v5 v2 v6 v3 V2 V1

Graph with 7 nodes and 16 edges Graphs Graph with 7 nodes and 16 edges Nodes / Vertices Undirected Edges Directed https://www.csc2.ncsu.edu/faculty/nfsamato/practical-graph-mining-with-R/

https://www.distributed-systems.net/index.php/books/gtcn/

Definitions – Graph Type Edges Multiple Edges Allowed ? Loops Allowed ? Simple Graph undirected No Multigraph Yes Depends on book (yes for us) Pseudograph Directed Graph directed Directed Multigraph Modified from https://utdallas.edu/~praba/graph.ppt

Simple graphs – special cases Complete graph: Kn, is the simple graph that contains exactly one edge between each pair of distinct vertices. K2 K1 K4 K3 Modified from https://utdallas.edu/~praba/graph.ppt

Bipartite graphs Modified from https://utdallas.edu/~praba/graph.ppt In a simple graph G, if V can be partitioned into two disjoint sets V1 and V2 such that every edge in the graph connects a vertex in V1 and a vertex V2 (so that no edge in G connects either two vertices in V1 or two vertices in V2) Application example: Representing Relations Representation example: V1 = {v1, v2, v3} and V2 = {v4, v5, v6}, v1 v2 v3 v4 v5 v6 V1 V2 https://www.distributed-systems.net/index.php/books/gtcn/

Complete Bipartite graphs Km,n is the graph that has its vertex set portioned into two subsets of m and n vertices, respectively There is an edge between two vertices if and only if one vertex is in the first subset and the other vertex is in the second subset. K2,3 K3,3 Modified from https://utdallas.edu/~praba/graph.ppt

N(1) = {2, 5} N(2) = {1, 3, 5} Degree of vertex 1 is 4 https://en.wikipedia.org/wiki/Adjacency_matrix Degree of vertex 1 is 4 https://www.distributed-systems.net/index.php/books/gtcn/

If every vertex has the same degree, the graph is called regular. A sequence is graphic iff it is the degree sequence for a simple graph. If every vertex has the same degree, the graph is called regular. In a k-regular graph each vertex has degree k. Thus its degree sequence is [k, k, …, k] https://www.distributed-systems.net/index.php/books/gtcn/

https://www.distributed-systems.net/index.php/books/gtcn/

The complement of a graph G, denoted as G is the graph obtained from G by removing all its edges and joining exactly those vertices that were not adjacent in G. It should be clear that if we take a graph G and its complement G “together,” we obtain a complete graph. https://www.distributed-systems.net/index.php/books/gtcn/

Adjacency matrix: A[i, j] = the number of edges joining vertex vi and vj. https://en.wikipedia.org/wiki/Adjacency_matrix https://www.distributed-systems.net/index.php/books/gtcn/

Adjacency matrix: A[i, j] = the number of edges joining vertex vi and vj. 1 1

M[i, j] = the number of times that edge ej is incident with vertex vi. Incidence matrix: M[i, j] = the number of times that edge ej is incident with vertex vi. 2 1 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 OR https://en.wikipedia.org/wiki/Adjacency_matrix 2 1 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 0 OR … 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0

(A) (B) (C) B https://www.csc2.ncsu.edu/faculty/nfsamato/practical-graph-mining-with-R/

Graph Isomorphism C (A) (B) (C) Which graphs are isomorphic? https://www.csc2.ncsu.edu/faculty/nfsamato/practical-graph-mining-with-R/

Reachability: exploring a maze      https://www.cs.upc.edu/~jordicf/Teaching/AP2/pptx/10_Graphs_Connectivity.pptx L K B F H G E C J I D A D G H A C B F E I J K L Which vertices of the graph are reachable from a given vertex? Graphs © Dept. CS, UPC

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Walk: Vertices may repeat. Edges may repeat (Open or Closed) https://math.stackexchange.com/questions/655589/what-is-difference-between-cycle-path-and-circuit-in-graph-theory Walk: Vertices may repeat. Edges may repeat (Open or Closed) Trail : Vertices may repeat. Circuit: Vertices may repeat. Edges cannot repeat Edges cannot repeat (Open) (Closed) Path: Vertices cannot repeat. Cycle: Vertices cannot repeat. Edges cannot repeat Edges cannot repeat 645234523 45125 https://wiki.sagemath.org/interact/graph_theory 6451 451234 https://en.wikipedia.org/wiki/File:6n-graf.svg

645234523 45125 6451 451234 Walk: Vertices may repeat. https://math.stackexchange.com/questions/655589/what-is-difference-between-cycle-path-and-circuit-in-graph-theory Walk: Vertices may repeat. Edges may repeat (Open or Closed) Trail : Vertices may repeat. Circuit: Vertices may repeat. Edges cannot repeat Edges cannot repeat (Open) (Closed) Path: Vertices cannot repeat. Cycle: Vertices cannot repeat. Edges cannot repeat Edges cannot repeat Path  Trail  Walk Cycle  Circuit  Closed Walk 645234523 45125 https://wiki.sagemath.org/interact/graph_theory 6451 451234 https://en.wikipedia.org/wiki/File:6n-graf.svg

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D G H A C B F E I J K L D G H A C B F E I J K L D G H A C B F E I J K L