c02_graph_conc_handout

c02_graph_conc_handout - Graph concepts Graphs are made up...

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1 Graph concepts Graphs are made up by vertices (nodes) and edges (links) . An edge connects two vertices, or a vertex with itself – loop. AC, AC - multiple edges BB – loop The shape of the graph does not matter, only the way the nodes are connected to each other. Simple graph - does not have loops (self-edges) and does not have multiple identical edges. Further reading: http://www.utm.edu/departments/math/graph/glossary.html Symmetrical and directed graphs Two distinct types of edges: symmetrical and directed (also called arcs). Two different graph frameworks: graph, digraph = directed graph. In the digraph framework a symmetrical edge means the superposition of two opposite directed edges. Node degrees Node degree: the number of edges connected to the node. In directed networks we can define an in- degree and ou degree The (total) degree is the sum of in 4 k i = and out-degree . The (total) degree is the sum of in- and out-degree. Source : a node with in-degree = 0. Sink : a node with out-degree = 0. E.g. A, F are sources, B is a sink. 2 k in C = 1 k out C = 3 = C k Average degree N – the number of nodes in the graph = N i i k N k 1 1 N N Q: What is the relation between the number of edges in a (non-directed) graph and the sum of node degrees? How about in a directed graph? out in 1 i out i out 1 i in i in k k , k N 1 k , k N 1 k = = =
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2 Statistics of node degrees Average degree The degree distribution gives the N E 2 k N 1 k N 1 i i = = ) k ( P N E k k out in = = fraction of nodes that have edges. Similarly / gives the fraction of nodes that have in-degree / out-degree . Ex. Calculate the degree distributions of the graphs in the left. k ) k ( P in in k ) k ( P out out k Paths and circuits Adjacent nodes (vertices) – there is an edge joining them. In the digraph framework the adjacency is only defined in the direction of the arrow. Path: a sequence of nodes in which each node is adjacent to the next one each node is adjacent to the next one. Edges can be part of a path only once. In the digraph framework a symmetrical edge can be used once in one direction and once in the opposite direction. Circuit : a path that starts and ends at the same vertex . Cycle : a circuit that does not revisit any nodes. Ex. Give examples of circuits and cycles in the above graph Connectivity of undirected graphs Connected (undirected) graph: any two vertices can be joined by a path. A disconnected graph is made up by two or more connected components. Bridge: if we erase it, the graph becomes disconnected. Connectivity of directed graphs Strongly connected directed graph: has a path from each node to every other node and vice versa (e.g. AB path and BA path). Weakly connected directed graph: it is connected if we disregard the edge directions. Strongly connected components can be identified, but not every node is part of a nontrivial strongly connected component.
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c02_graph_conc_handout - Graph concepts Graphs are made up...

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