# Graph Theory Summary Notes

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Self-made revision notes for Module CS1231 Discrete Structure
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CS1231 Graph Theory Summary Notes YANG LU 1 CS1231 Discrete Structure Chapter 11 Graph Theory Summary Notes Contents Section 1 Terminology   Undirected Graph   Directed Graph   Complete Graphs   Complete Bipartite Graph   Subgraphs   Vertex Degree   The Handshake Theorem   Section 2 Connectivity   Section 3 Euler Tour    Section 4 Hamilton Cycles   Section 5 Graph Representation   Last Edit: 18 th  October 2014 Section 1 Terminology Undirected Graph 1.   A pseudograph    =,,    consists of    a non-empty vertex   set   of vertices    an edge   set   of edges    an incidence function       : →, | ,∈ . [NOTE: vertex set   must be non-empty] 2.   Edges are undirected   in an undirected graph. 3.   An edge e  is a loop   if    ⅇ=,=  for some ∈ . 4.   Two distinct edges ⅇ   and ⅇ   are multiple   (or parallel) edges if    ⅇ  =  ⅇ   . 5.   A simple graph   is a pseudograph with no loops and parallel edges 6.   A multigraph   is a pseudograph with no loops. 7.   An edge e  is incident   with vertices u  and v  (vice versa) if    ⅇ=, . The edge e  is said to connect   its endpoints u  and v . 8.   Two vertices are adjacent   (or neighbours) if they are incident with a common edge. 9.   Two edges are called adjacent   if they are incident with a common vertex.  CS1231 Graph Theory Summary Notes YANG LU 2 Directed Graph 10.   A directed multigraph    =,,    consists of    a non-empty vertex   set   of vertices    an edge   set   of edges    an incidence function       : →,| ,∈   [NOTE: difference with undirected graph here is the use of notation of round brackets instead of curly brackets   ordered pair is used   indicate the importance of sequence] 11.   The directed edge e  start at initial vertex u   and end at terminal   (or end) vertex    v    if    ⅇ=, . 12.   A directed edge e  is a loop   if    ⅇ=,  for some ∈ . 13.   The directed edges e 1   and e 2  are multiple directed edges if    ⅇ  =  ⅇ   . 14.   A simple directed graph   is a directed multigraph with no loops and multiple directed edges. Complete Graphs 15.   A  complete graph   on n vertices, denoted by  K  n , is a simple graph in which every two distinct vertices area adjacent.  Number of edges in complete graph  Kn  = |  |= 2= −   Complete Bipartite Graph 16.   A complete    bipartite graph   on (m,n)  vertices, denoted b  K  m,n , is a simple graph with    ( , )=  ,  ,...,   ∪  ,  …        ( , )  = { {  ′   }  ⅈ=1…;=1…    Number of edges between u  vertices on the top or bottom = ( , )=×   Subgraphs 17.   A graph  H   is a subgraph of a graph G  if    ⊆      ⊆      ∀ⅇ∈ (   ⅇ=  ⅇ)   Vertex Degree 18.   The degree      of a vertex v  in an undirected graph G  is the number if edges incident with v , each loop counting as 2 edges. A vertex of degree 0 is isolated  .  CS1231 Graph Theory Summary Notes YANG LU 3 The Handshake Theorem THEOREM 1  Let G be an undirected graph. Then   = 2|| ∈   COROLLARY 1 Sum of degrees of all vertices of G is even. PROOF : Combinational Proof 1)   List each edge and label its endpoints with the vertex names. 2)   The number of times each vertex name is used is the vertex degree. 3)   All vertex names appear 2||  times, by STEP 1. 4)   This is also the sum of all the vertex degrees, by STEP 2. Q.E.D. COROLLARY 2  In an undirected graph, the number of vertices of odd degree is even. PROOF : Direct Proof 1)   Let V  1  and V  2  be the sets of vertices of odd and even degree in G , respectively. 2)   Therefore,    ∈  +    ∈  =    ∈  3)   Therefore RHS is even, by Theorem 1. 4)   Therefore LHS is even by STEP 3. 5)   ∑   ∈   is even. 6)   Therefore ∑  = ∈    ∈  1+ ∑1 ∈   and |  |=∑1 ∈  , |  |  is even. Q.E.D.  CS1231 Graph Theory Summary Notes YANG LU 4 Section 2 Connectivity 1.   A walk   of length n in an undirected graph G  is a finite alternating sequence of vertices and edges of G  such that,    ⅇ      ⅇ  …  −  ⅇ          ⅇ   connects its endpoints  −  and     for  I    = 1, … n      vertices v o  and v n  are the srcin   and terminus   respectively    v 1   … v n-1  are internal vertices       if in a simple graph a walk can be specified uniquely by its vertex sequence 2.   A trail   is a walk with distinct edges. [no repeated edges] 3.   A path   is a trail with distinct vertice. [no repeated vertices except for srcin or terminus] 4.   An undirected graph G  is connected   if there is a walk between every pair of distinct vertices of G . 5.   An undirected graph  H   is a connected component   of the undirected graph G  if     H is a subgraph of G,       H is connected,     No connected subgraph of G  has  H   as a proper subgraph. [Contains maximum number of edges] THEOREM 2 There is a path between every pair of distinct vertices of a connected undirected  graph PROOF : Direct Proof 1)   Let u  and v  be 2 distinct vertices of G . There is at least one walk between u  and v , by definition. 2)   Choose the walk of least length. 3)   This walk of least length is a path. PROOF : Contrapositive Proof 1)   Let   ,  …   be the vertex sequence of a walk between distinct vertices v 0  and v n . 2)   By definition, this walk exists because the graph is connected. If the walk is not a path, then   =   for some i and  j  with  0≤ⅈ≤ . 3)   Therefore, there is a walk from    to    of shorter length with vertex sequence   … −    …   obtained by deleting the edges corresponding to the vertex sequence   … − .

Jul 23, 2017

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Jul 23, 2017
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