Y15-11-graph - 1 Graphs SM Yiu(7 th edition chapter 10...

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1 Graphs SM Yiu (7 th edition: chapter 10 except 10.6)
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2 What is a Graph? A simple graph G = (V, E) consists of ¾ V , the set of vertices (nodes, points) , and ¾ E , the set of edges (lines ) Undirected graphs are graphs where edges have no direction ( unordered pairs of V). Example: V = {a, b, c, d, e} E = {{a,b}, {a, c}, {c, d}, {c, e}, {d, e}} Directed graphs are graphs where edges have directions, i.e., E consists of ordered pairs of V Example: V = {u, x, y, z} E = {(x, y), (x, z), (y, u), (z, x), (z, u)} a b c d e z x y u
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3 Simple Graphs and Multigraphs Simple graphs - at most one edge between any pair of vertices. Multi - graphs - having multiple edges between the same pair of vertices . Pseudographs – multiple edges + self-loops If x, y are two vertices, and e = {x, y} is in E, then ¾ x and y are adjacent ¾ e is incident with x and y ¾ x and y are the endpoints of e Degree (v) = number of edges incident with vertex v ¾ deg (y)=2, deg (u)=2, deg (x)=3, deg (z)=5 z x y u
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4 Complementary graph Let G = (V, E) be a graph, the complementary graph, denoted as G’ = (V’, E’), is defined as V’ = V; (u, v) E’ iff (u, v) E. (a) Draw the complementary graph of the graph on the right. a b c d e a b c d e Ans: the red edges (b) Let G = (V, E), where |V| = 5, |E| = 8, what is the number of edges in G’? |E’| = 10 – 8 = 2.
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5 Graph properties: Degrees By pigeon hole principle, there exist two vertices with the same degree. [ Why? ] For an undirected graph G = (V, E): e.g. sum of degrees = 3+2+5+2 = 12, |E| = 6 Proof: every edge is counted twice in counting degrees. Fact: Any undirected graph has an even number of odd- degree vertices (as the sum of degrees is even) For a directed graph G = (V, E): | E | (v) deg (v) deg V v V v = = + | E | 2 deg(v) V v = z x y u
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6 Some Special Graphs Cycle graphs C n : K 3 K 4 K 6 C 3 C 4 C 6 Complete graphs K n : edges joining every two vertices |V| = n |E| = C n,2 = n(n-1)/2 Degree = n-1 |V| = n, |E| = n Degree = 2 Diameter = maximum “distance” (length of shortest path) between two vertices Diameter = 1 Diameter = ¬ n/2 ¼
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7 Bipartite Graphs A graph is bipartite if the vertices V can be V 1 V 2 partitioned into two disjoint sets V 1 , V 2 such that every edge connects a vertex in V 1 to a vertex in V 2 i.e. no edge within V 1 , no edge within V 2 A complete bipartite graph K m,n is a bipartite graph with all possible edges between a set of m vertices and a set of n vertices. It has mn edges. ¾ e.g. K 3,4 has 12 edges Number of edges in a bipartite graph |V 1 | |V 2 |
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8 Bipartite Graphs Are these graphs bipartite? A graph is bipartite iff the graph contains no odd cycles. [Proof?] a b c f d e a b c e f d
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9 Graph Isomorphism A simple graph G = (V, E) consists of V = the set of vertices E = the set of edges Are these two graphs “the same”? Although they look different, they represent the same adjacency relationships, i.e., set of edges Example: f(a) = x, f(b) = w, f(c) = z, f(d) = y Two graphs G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) are isomorphic if there is a 1-1 correspondence f: V 1 V 2 such that ¾ (u, v) E 1 if and only if (f(u), f(v)) E 2 .
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