a simple graph consists of a nonempty set of vertices and

A simple graph consists of , a nonempty set of vertices, and , a - PowerPoint PPT Presentation

P P


  1. �✁ ❪ ❭ ◆ ❙ ▼ ❘ ❪ ❭ ◆ ❙ ❫ ❭ ❭ ❘ ❳ ❖ ❩ ❲ P ❯ ❨ ❘ ❲ P ❯ ❫ ✰✱✲ ❚ ❃ ✼ ✱ ❃ ❇ ❆ ✳ ❅ ❀ ❁❄ ✷ ✵ ❂ ✳ ❀❁ ✿ ✾ ✽ ✶ ✻✼ ✺ ✹ ✸ ✶✷ ✵ ✴ ◆ ◗ ❃ ✱ ✂✄ ☎ �✁ ✼ ✹ ❈ ✺ ✱ ❃ ✹ ✼ ❃ ✝ ❇ ❆ ✳ ❅ ❀ ❁❄ ✷ ✵ ❃ ❂ ❀❁ ✄ ✆ ❙ ◗ ◗ ❖ ❘ ◗ P ❖ ◆ ▼ ▲ ❑ ❖ ❖ ❋ ❘ ◗ P ❖ ◆ ▼ ▲ ❑ ✂ ❍ ☎ ● ✹ ✱ ✾ ▼ ✵ ✴ ✳ ✰✱✲ ❑ ❲ ◆ ✐ ❤ ❣ ❢ ❭ ✸ ❡ ❭ ❘ ◗ P ❖ ◆ ▼ ▲ ❑ ❲ ✶✷ ✹ ❑ ✳ ✹ ❈ ✺ ✱ ❃ ✹ ✼ ✱ ❃ ❇ ❆ ❅ ✺ ❀ ❁❄ ✷ ✵ ❃ ❂ ❀❁ ✿ ✾ ✽ ✶ ✻✼ ❲ ❳ ✺ ❋ ❜ ❛ ❊ ✁ ❵ ☎ ✆ ❴ ❍ ☎ ● ✆ ❝❞ ✝ ✄ ✆ ☎ ✁ ❋ ❊ ❉ ✻ ✹ ❈ ❛ ❑ ❲ ❳ P ❯ ❚ ❲ ❯ ❲ ❯ ❭ ❲ ❯ ❭ ❲ ❯ P ▼ ❭ ▲ ❳ ❲ P ❯ ❚ ▲ ▲ ❲ ✿ ✆ ✽ ❃ ✆ ☎ ✁ ❋ ❊ ❉ ❈ ✹ ❈ ✺ ✱ ✹ ✝ ✼ ✱ ❃ ❇ ❆ ✳ ❅ ❀ ❁❄ ✷ ✵ ❃ ✄ ✆ ❀❁ ❖ ❖ ❘ ◗ P ❖ ◆ ▼ ▲ ❑ ❖ ◗ ✶ ❋ ❘ ◗ P ❖ ◆ ▼ ▲ ❑ ✂ ❍ ☎ ● ❂ ✿ ❙ ✠ ✘ ✗ ✖ ✔✕ ✓ ✒ ✑ ✏ ✎ ☛ ✡ ✂ ✕ ✆ ☎ ✟ ✝ ✄ ✟ ☎ ✝ ✞ ✆ ✆✝ ☎ ✙ ✚ ✾ ✭ ✽ ✶ ✻✼ ✺ ✹ ✸ ✶✷ ✵ ✴ ✳ ✰✱✲ ✮✯ ✪✬ ✔ ✩✪✫ ★ ✧ ✤ ✗ ✦ ✤✥ ✗ ✣ ✢ ✜ ✛ ◗ ✺ ❙ ❘ ▼ ▼ ▲ ❚ ❯ ❑ P ❯ ❫ ❘ ❭ ◆ ❳ ❙ ▼ ▼ ❳ ❘ ◆ ❭ ◗ ❳❨ ❖ ◗ ❚ ❚ ❯ P ❘ ❲ ◗ ❖ ❯ P ❲ ❩ P ❖ ❳ ❙ ◆ ❯ ❩ ❖ P ✹ ❪ P ❖ ❩ ❲ ❯ ▼ ❳❨ ❲ P ✺ ❚ ✻✼ ◗ ❬ ❯ ❲ ❳ ❭ ✰✱✲ ◆ ❙ ✳ ✴ ❫ ❭ ❪ ✸ ✵ ❭ ✶✷ ■✍❏ ☞✍✌ A simple graph consists of , a nonempty set of vertices, and , a set of unordered pairs of distinct elements of called edges. A multigraph consists of a set of vertices, a set of edges, and a function from to . The edges ❚❱❯ and are called multiple or parallel edges if . A pseudograph consists of a set of vertices, a set of edges, and a function from to . An edge is a loop if for some . ❚❱❯ ■✍❏ ■✍❏ Two vertices and in an undirected graph are called adjacent (or neighbors ) in if is an edge of . If , the edge is A directed graph consists of a set of vertices and a set of ❚❱❯ called incident with the vertices and . The edge is also said to edges that are ordered pairs of elements of . connect and . The vertices and are called endpoints of the edges . A directed multigraph consists of a set of vertices, a set The degree of a vertex in an undirected graph is the number of edges incident with it, except that a loop at a vertex contributes twice to the of edges, and a function from to . The edges and are multiple edges if . degree of that vertex. The degree of the vertex is denoted by deg( ). The Handshaking Theorem : Let be an undirected graph with edges. Then deg ❘❦❥ Theorem : An undirected graph has an even number of vertices of odd degree.

  2. � ⑧ ▼ ❚ ⑧ ⑦ ❧ ❪ P ⑦ ⑦ ❳ ⑧ ⑨ ▼ ❯ ⑧ ⑦ ❘ ❭ ❲ P ❙ ◆ ❭ ❫ ❘ ▼ ❪ ⑧ ◆ ❫ ❳ P ❥ ❥ ❥ ❙ ▼ ⑧ ❳ ❲ ⑧ ⑦ ❧ ❪ ❑ ⑥ ❯ ⑥ ❥ ❭ ❪ P ❭ ❫ P ❥ P ❥ ⑨ P P ⑧ ❪ P ❥ ❥ ❥ ⑧ ❥ ⑦ ⑧ ❪ P ⑧ ❫ P P ❪ ❥ ❇ ✵ ✷ ❁❄ ❀ ❅ ✳ ❆ ❃ ✁ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ❃ ❀❁ ② ✵ ① ❖ ❪ ❙ ✰✱✲ ✳ ✴ ✶✷ ✿ ✸ ✹ ✺ ✻✼ ✶ ✽ ✾ ✹ ③ ⑧ ⑦ ❫ P ❥ ❥ ❥ P ❭ ❙ P ◆ ❭ ❪ ❘ ▼ ⑨ P ❭ ❪ ❛ ✁ ❊ ❊ ✆ ✄ ✝ ✁ ④ ✝ ❭ ❞ ⑤ ❑ ⑥ ❯ ❲ ⑥ ❥ P ❫ ✁ ③ ❛ ❊ ❊ ✆ ✄ ✝ ④ ✹ ✁ ✝ ❞ ❶ ❑ ❑ ❑ ⑩ ❈ ① ✳ ❂ ❃ ✵ ✷ ❁❄ ❀ ❅ ❆ ✺ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✇ ① ✿ ❃ ✷ ❁❄ ❀ ❅ ✳ ❆ ❇ ✱ ❃ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ✵ ❂ ✇ ✶✷ ✇ ① ❑ ✰✱✲ ✳ ✴ ✵ ✸ ❀❁ ✹ ✺ ✻✼ ✶ ✽ ✾ ✿ ❀❁ ✾ ❭ ❘ ▼ ◆ ⑧ ❪ P ⑧ ❫ P ❫ ❥ ❥ ❥ ❙ ◆ ❭ ⑦ ❘ ❭ ▼ ◆ ⑦ ❙ ◆ ❭ ❪ ❘ ▼ ⑧ ◆ ⑨ P ⑧ ❪ ❘ P ❙ ❘ ◆ ✽ ✳ ❥ ❥ ❥ P ⑧ ⑦ ✰✱✲ ✴ ❪ ✵ ✶✷ ✸ ✹ ✺ ✻✼ ✶ P ⑧ ⑧ ⑧ ⑦ ❧ ❪ P ⑧ ⑦ ❘ ⑨ P ▼ ❯ ⑧ ⑦ ▼ ❲ ⑧ ⑨ ✇ ❂ ▲ ✵ ✽ ✾ ✿ ❀❁ ❂ ❃ ✷ ✻✼ ❁❄ ❀ ❅ ✳ ❆ ❇ ✶ ✺ ✱ ◗ ♠ ◆ ❲ ❘ ▼ ❨ ❨ ✹ ❥ ✰✱✲ ✳ ✴ ✵ ✶✷ ✸ ❃ ✼ ❤ ❖ ✂ q q ❑ ▲ ❖ ❪ ❛ ❖ ❫ ❖ ❪ ❖ ❫ ▲ ❊ ✁ ✹ ♦ ❃ ✱ ✺ ❈ ✹ ♥ ❛ ✝ ☎ ✆ � ✆ ♣ ❊ ✁ ✐ ❣ ❪ ▲ ❑ ◆ ❯ P ❲ ❘ ❯ ❛ ❲ ❲ ❯ ❯ ◆ ❯ P ❝❞ ❜ ❘ ● ☎ ✆ ✄ ✝ ✆ ❋ ☎ ❛ ❍ ❴ ✆ ☎ ❵ ✁ ❊ ❲ ❲ ❢ ❢ ▼ ◆ ❖ P ◗ ❘ ❣ ❑ ❤ ✐ ❧ ◆ ❲ ❘ ▼ ▲ ❲ ◆ ❧ ❯ P ❲ ❘ ❑ ❲ ◆ ❘ ❲ ❘ ❲ ❲ ♠ ◆ ❲ ❖ q ❃ P P ◗ ❫ ❖ ✈ ❪ ❖ ❫ ❘ ❘ ❫ ◗ ❫ ◗ ❖ ◆ ▼ ❫ ▲ ❘ ❪ ◗ P ❪ ❖ ❫ ❫ ❪ ❪ ❪ ▲ ✈ ▲ ❪ ❪ ▲ ▼ ◆ ❖ ❪ P ◗ ❫ ❖ ❖ ❘ ▲ ▲ ❪ ▲ ❫ ▼ ◆ ❫ ◗ ✈ ▼ ◆ ▲ ✇ ◗ P ❫ ✇ ① ▲ ❪ ❙ ◆ ❘ ❖ ❖ ◆ ▼ ▲ ❑ ❙ ❫ ◆ ① ❘ ❘ ❑ ❪ ✉ s ❘ t t P ✉ ◗ ❖ s ❙ ◆ ❑ ▼ r ❖ ■✍❏ When is an edge of the graph with directed edges, is said to be A simple graph is is called bipartite if its vertex can be partitioned adjacent to and is said to be adjacent from . The vertex is into two disjoint nonempty sets and such that every edge in the called the initial vertex of , and is called the terminal or graph connects a vertex in and a vertex in (so that no edge in end vertex of . The initial vertex and terminal vertex of a loop connects either two vertices in or two vertices in . are the same. A subgraph of a graph is a graph where In a graph with directed edges the in-degree of a vertex , denoted by and . deg , is the number of edges with as their terminal vertex. The out-degree of , denoted by deg , is the number of edges with The union of two simple graphs and is the as their initial vertex. simple graph with vertex set and edge set . The union of and is denoted by . Theorem: Let be a graph with directed edges. Then The simple graphs and are isomorphic if there deg deg is a one-to-one and onto function from to with the property that and are adjacent in if and only if and are adjacent in , for all and in . Such a function is called an isomorphism . A path of lengh from to , where is a positive integer, in an An undirected graph is called connected if there is a path between every undirected graph is a sequence of edges of the graph pair of distinct vertices of the graph. such that , ❚❱⑧ ❚❱⑧ where and . When the graph is simple, we denote this path by its vertex sequence . The path is a circuit if it Theorem: There is a simple path between every pair of distinct vertices of begins and ends at the same vertex. The path or circuit is said to a connected undirected graph. pass through or traverse the vertices . A path or circuit is simple if it does not contain the same edge more than once. A directed graph is strongly connected if there is a path from to and from to whenever and are vertices in the graph. A path of lengh from to in a directed multigraph, where is a positive integer, is a sequence of edges of the graph such that A directed graph is weakly connected if there is a path between any two , where vertices in the underlying undirected graph. and . When there are no multiple edges in the graph, we denote this path by its vertex sequence . The path is a circuit or cycle if it begins and ends at the same vertex. A path or circuit is simple if it does not contain the same edge more than once.

  3. ❷ ❅ ✶ ✽ ✾ ✿ ❀❁ ❂ ❃ ✵ ✷ ❁❄ ❀ ✳ ✺ ❆ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ✻✼ ✹ ✟ ❑ ❜ ✆ ☎ ③ ✁ ☎ ✄ ✟ ✁ ✝ ✂ ❸ ✸ ▲ ▲ ❑ ✰✱✲ ✳ ✴ ✵ ✶✷ An Euler circuit in a graph is a simple circuit containing every edge of . Theorem: A connected multigraph has an Euler circuit if and only if each of its vertices has even degree.

Recommend


More recommend