圖論學習筆記 Part 1

Graph Theory (Part 1)

學習的視頻來自 Sarada Herke 的Youtube 空間

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01 Seven Bridges of Konigsberg

• solved by Euler in 1735;

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02 Definition of a Graph

• A graph G is an ordered pair G=(V,E) where V is a set of elements and E is a set of 2-subsets of V;

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03 Examples of Graphs

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04 Families of Graphs

• A complete graph Kn on n vertices is a simple graph with an edge between every pair of vertices
• empty graph
• bipartite graph: A graph whose vertex set can be partitioned into 2 sets V1 and V2 such that every edge uv∈E has u∈V1,v∈V2 : Km,n
• path Pn : A path Pn is a graph whose vertices can be arranged in a sequence.
• cycle Cn A cycle Cn is a graph whose vertices can be arranged in a cyclic sequence; n>=3

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05 Connected and Regular Graphs

• Connected Graph: A graph is connected if for every pair of distance vertices u,v∈V(G) there is a path from u to v in G ;

• Regular Graph:

  • neighborhood: the (open) neighborhood of v in G is NG(v)={u|uv∈E(G)}
    
    • close neighborhood: NG[v]=NG(v)∪{v}
    • the degree of a vertex: degG(v)=|NG(v)|
  • regular: A graph G is r-regular if degG(v)=r for all v∈V(G)
• min degree: δ(G)
• max degree: Δ(G)

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06 Sum of Degrees is ALWAYS Twice the Number of Edges

• Theory: In any graph G, ∑u∈V(G)deg(v)=2|E(G)| (顯而易見)
• Corollary: In any graph, there are an even number of odd degree vertices;

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07 Adjacency Matrix and Incidence Matrix

• Adjacency Matrix : 和數據結構中描述一樣
  
  • row’s sum = degree
  • row 和 column 表示的都是 vertex
• Incidence matrix
  
  • row 代表 vertex
  • column 代表 edge
  • row 的和是相應vertex的degree

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08 Basic Problem Set (part 1/2)

• Let G be a simple graph and m=|E(G)| , show that m≤Ckn
• Prove that every path is bipartite.

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09 Basic Problem Set (part 2/2)

• For k=0,1,2 , characterize the k-regular graphs;
  
  • k = 0: empty graph
  • k = 1: disjoint union of K2
  • k = 2: disjoint union of cycles of any lengths
• Let G be a bipartite graph with partite sets X and Y. Prove that ∑v∈Xdeg(v)=∑v∈Ydeg(v)
  
  • they both equals to |E(G)|
• True or False? If u,v,w∈V(G) and there is an even length path from u to v and there is an even path from v to w, then there is an even length path form u to w.
  
  • False