软件设计8

aph:
G=(V，E), G'=(V'，E'), V ‘ is a subset of V; E is a subset of E‘,meanwhile ,the associative points of edges in E’ also belong to V’,then G’ is a subset of G. such as V'={v1，v2，v3}，E'={(vl，v2)，(v2，v4)}; ， ， V={v1，v2，v3 ,v,4} ，E'={(vl，v2)，(v2，v4),(v3,v4),(v4,v1)}; ， obviously,G’ is not a subset of G.
Graph
• The relationship between vertex n and edge e: • (1)if G is an undigraph, then: • 0≤e≤n(n-1)/2 an undiagraph with exactly n(n1)/2 edges is an undirected complete graph. • (2)if G is an digraph,then: • 0≤e≤n(n-1) • an diagraph with exactly n(n-1) edges is an directed complete graph. • Complete graph has the most edges,any pair of vertexes has an edge. • • Degree of vertex: • (1)degree of vertex in undigraph: • D(v)-----the number of edges associative to the node. • (2) degree of vertex in digraph: • Indegree/ID---the number of edges with the end-point V in digraph; • Outdegree/OD—the number of edges with the start-point V; • (3)D(V) in a diagraph • D(V)=ID(V)+OD(V); • NOTE: either in digraph or undigraph exist such relation: • E=(1/2) ∑D(Vi) (1≦i≦n) ≦≦
Care and diligence bring luck．
细心勤奋，幸福之本。
Great works are performed not by strength but by perseverance．
成大事不是靠力量大小，而是靠坚忍不拔。 成大事不是靠力量大小，而是靠坚忍不拔。
There are two kinds of persons,one is frigaround; the other is living a meaningful life;in the view of the former,life is like a sleep on an tender&warm doss,which gives him satisfaction;whereas in the point of the latter,life means establishing great achievements from which the person enjoys his blessedness. • 世界上有两种人，一种人，虚度年华；另一种人，过 世界上有两种人，一种人，虚度年华；另一种人，
• Arc (vi，vj ) is same as(vj，vi).
Digraph description
G1=(V,E) • V(G1)={v1，v2，v3} E(G1)={<v1，v2>，<v2，v1>，<v2，v3>}
• Digraph description
G1=(V,E) • V(G1)={v1，v2，v3} E(G1)={(v1，v2 ) ，(v2，v3),(v3，v1 )}
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Graph
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Disconnected graph
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Built tree,obviously built tree is not unique,how to create a min built tree? A min built tree is one that has the minmum weight sum of all the edges.
• Strong connected graph:
• In digraph V(G),any two different points have path from Vi to Vj.
Graph
• Strong connected component: • Digraph’s max strong connected subgraph. • Note:①strong connected graph has only one strong connected component –itself; ② non-strong connected digraph has many strong connected components. • Network: graph with weight of all the edges.
• Strong connected vertex:
• In digraph,if there is a path from vertex V to W,and there also exists a path from W to V,then V and W are strong connected.
着有意义的生活。在第一种人眼里，生活就是一场睡 着有意义的生活。在第一种人眼里， 眠，如果这场睡眠在他看来，是睡在既柔和又温暖的 如果这场睡眠在他看来， 床铺上，那他便十分心满意足；在第二种人眼里，可 床铺上，那他便十分心满意足；在第二种人眼里， 以说，生活就是建立丰功伟绩……人就在这个功绩中 以说， 享到自己的幸福。 －－别林斯基 享到自己的幸福。 －－别林斯基
Graph
• Path length:the edge number of path. • Simple path:the path has different points except that Vp is
identical to Vq . Yes : vl，v2，v3，v4 // No:v1，v2，v4，v1，v3;
Graph
• Example:
v1 v2 Digraph ;non-strong connected graph; It has two strong connected components. v1 v3 v4 v3 IN(V)? OD(V) TD(V)? Analysis the relationship between degree sum and edge sum? v4
Graph
• Digraph: • A graph with directed edges from tail to head.
arc<vi，vj>is different from<vj，vi>.
• Undigraph:
• A graph with undirected edges from tail to head.
• Connectivity between vertexes:
• In undigraph, if there is path between Vi and Vj, both from vi to vj and from vj to Vi; • Connected graph:if any two different vertexes are connected, • this graph is an connected graph. • Built tree of connected graph: a minimum connected subgraph containing all the nodes of the connected graph. e=n-1,when e>n-1, the subgraph is not a min connected subgraph; when e>n1,the subgraph is not connected. •
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Graph
• • • • • Note: (1)The relationship between vertex pair : (x,y) is different from <x,y>; (2) no permission self-cycle in diagram; (3) no repetition of edges between two vertexes, such diagram is called multi-edge diagram.
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Graph
A C F G L D N K E H B A non-connected graph with three connected components. How to judge the number of connected components? ----From the number of DFS call in adjacency list.
Graph