Graph (1A)
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Some class of graphs (1) Complete graph A complete graph is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges. Connected graph In an undirected graph, an unordered pair of vertices {x, y} is called connected if a path leads from x to y. Otherwise, the unordered pair is called disconnected. Bipartite graph A bipartite graph is a graph in which the vertex set can be partitioned into two sets, W and X, so that no two vertices in W share a common edge and no two vertices in X share a common edge. Alternatively, it is a graph with a chromatic number of 2. https://en.wikipedia.org/wiki/graph_(discrete_mathematics) Graph (5A) 3
Complete Graphs https://en.wikipedia.org/wiki/complete_graph Graph (5A) 4
Connected Graphs This graph becomes disconnected when the right-most node in the gray area on the left is removed This graph becomes disconnected when the dashed edge is removed. With vertex 0 this graph is disconnected, the rest of the graph is connected. https://en.wikipedia.org/wiki/connectivity_(graph_theory) Graph (5A) 5
Bipartite Graphs Example of a bipartite graph without cycles A complete bipartite graph with m = 5 and n = 3 A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph. https://en.wikipedia.org/wiki/bipartite_graph Graph (5A) 6
Complete Graphs https://en.wikipedia.org/wiki/gallery_of_named_graphs Graph (5A) 7
Complete Bipartite Graphs https://en.wikipedia.org/wiki/gallery_of_named_graphs Graph (5A) 8
Star Graphs https://en.wikipedia.org/wiki/gallery_of_named_graphs Graph (5A) 9
Wheel Graphs https://en.wikipedia.org/wiki/gallery_of_named_graphs Graph (5A) 10
Some class of graphs (2) Planar graph A planar graph is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect. Cycle graph A cycle graph or circular graph of order n 3 is a graph in which the vertices can be listed in an order v1, v2,, vn such that the edges are the {vi, vi+1} where i = 1, 2,, n 1, plus the edge {vn, v1}. Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph. Tree A tree is a connected graph with no cycles. https://en.wikipedia.org/wiki/graph_(discrete_mathematics) Graph (5A) 11
Planar Graphs A planar graph and its dual https://en.wikipedia.org/wiki/planar_graph Graph (5A) 12
Cycle Graphs https://en.wikipedia.org/wiki/cycle_graph https://en.wikipedia.org/wiki/gallery_of_named_graphs Graph (5A) 13
Tree Graphs https://en.wikipedia.org/wiki/cycle_graph Graph (5A) 14
Hypercube A hypercube can be defined by increasing the numbers of dimensions of a shape: 0 A point is a hypercube of dimension zero. 1 If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one. 2 If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square. 3 If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube. 4 If one moves the cube one unit length into the fourth dimension, it generates a 4- dimensional unit hypercube (a unit tesseract). Tesseract https://en.wikipedia.org/wiki/hypercube Graph (5A) 15
Gray Code Tesseract https://en.wikipedia.org/wiki/gray_code Graph (5A) 16
Adjacency Lists https://en.wikipedia.org/wiki/adjacency_list Graph (5A) 17
Incidence Matrix https://en.wikipedia.org/wiki/incidence_matrix Graph (5A) 18
Adjacency Matrix https://en.wikipedia.org/wiki/adjacency_matrix Graph (5A) 19
Hamiltonian Path https://en.wikipedia.org/wiki/path_(graph_theory) Graph (5A) 20
Minimum Spanning Tree https://en.wikipedia.org/wiki/minimum_spanning_tree Graph (5A) 21
Seven Bridges of Königsberg The problem was to devise a walk through the city that would cross each of those bridges once and only once. https://en.wikipedia.org/wiki/seven_bridges_of_k%c3%b6nigsberg Graph (5A) 22
Shortest path problem https://en.wikipedia.org/wiki/shortest_path_problem Graph (5A) 23
Traveling salesman problem https://en.wikipedia.org/wiki/travelling_salesman_problem Graph (5A) 24
Euler Cycle Any connected graph with even degree vertices An Euler cycle A proof by induction on the number of edges in G A connected graph G with even degree vertices only and k edges (k < n) An Euler cycle Assume this is true A connected graph G with even degree vertices only and n edges An Euler cycle Then this holds true Johnsonbough, Discrete Mathematics Graph (5A) 25
Euler Cycle Any connected graph with even degree vertices which has n edge An Euler cycle Any connected graph with even degree vertices which has n-1 edge Any connected graph with even degree vertices which has n-2 edge An Euler cycle An Euler cycle Any connected graph with even degree vertices which has 2 edge Any connected graph with even degree vertices which has 1 edge An Euler cycle An Euler cycle Johnsonbough, Discrete Mathematics Graph (5A) 26
Euler Cycle Base Cases n = 0 edge n = 1 edge all even degree vertices n = 2 edge an Euler cycle Johnsonbough, Discrete Mathematics Graph (5A) 27
Euler Cycle decrease the number of edges by one A connected graph G with even degree vertices only and n edges (k < n) e 1 e 2 v 1 v 2 v 3 all even degree vertices P: a path from v to v 1 A connected graph G' with even degree vertices only and n-1 edges (k < n) P': a portion of the path P that are in G' e v 1 v 2 v 3 all even degree vertices Johnsonbough, Discrete Mathematics Graph (5A) 28
Euler Cycle a path from v to v 1 Case 1: P ends at v 1 Case 2: P ends at v 2 Case 3: P ends at v 3 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 P P P v v v v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 P ' P ' P ' v v v Johnsonbough, Discrete Mathematics Graph (5A) 29
Euler Cycle F B H E D C I J G ABCDEFGHIJK A K en.wikipedia.org Graph (5A) 30
Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 2 components Graph (5A) 31
Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 1 component Graph (5A) 32
Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 2 components Graph (5A) 33
Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 1 component Graph (5A) 34
References [1] http://en.wikipedia.org/ [2]