# Quelle:Nm/Brandes Erlebach 2005

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 Autor Ulrik Brandes, Thomas Erlebach Titel Chapter 2 Fundamentals Sammlung Network Analysis: Methodological Foundations Herausgeber Ulrik Brandes, Thomas Erlebach Ort Berlin Heidelberg Verlag Springer Jahr 2005 ISBN 978-3-540-24979-5 ISSN 0302-9743 URL http://www.inf.uni-konstanz.de/algo/publications/be-f-05.pdf Literaturverz. no Fußnoten no Fragmente 4

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 Zuletzt bearbeitet: 2012-04-26 07:59:44 Fiesh

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Graphs can be undirected or directed. The adjacency matrix of an undirected graph (as shown in Figure 3.2) is symmetric. An undirected edge joining vertices $u, v \in V$ is denoted by $\{u, v\}$.

In directed graphs, each directed edge (arc) has an origin (tail) and a destination (head). An edge with origin $u \in V$ is represented by an order pair $(u, v)$. As a shorthand notation, an edge $\{u, v\}$ can also be denoted by $uv$. It is to note that, in a directed graph, $uv$ is short for $(u, v)$, while in an undirected graph, $uv$ and $vu$ are the same and both stands for $\{u, v\}$. Graphs that can have directed as well undirected edges are called mixed graphs, but such graphs are encountered rarely.

Graphs can be undirected or directed. In undirected graphs, the order of the endvertices of an edge is immaterial. An undirected edge joining vertices $u, v \in V$ is denoted by $\{u, v\}$. In directed graphs, each directed edge (arc) has an origin (tail) and a destination (head). An edge with origin $u \in V$ and destination $v \in V$ is represented by an ordered pair $(u, v)$. As a shorthand notation, an edge $\{u, v\}$ or $(u, v)$ can also be denoted by $uv$. In a directed graph, $uv$ is short for $(u, v)$, while in an undirected graph, $uv$ and $vu$ are the same and both stand for $\{u, v\}$. [...]. Graphs that can have directed edges as well as undirected edges are called mixed graphs, but such graphs are encountered rarely [...]
 Anmerkungen The source is not mentioned anywhere in the thesis. The definitions given here are certainly standard and don't need to be quoted. However, Nm uses for several passages the same wording as the source. Note also that "An edge with origin $u \in V$ is represented by an order pair $(u, v)$" is a curious abbreviation of the statement "An edge with origin $u \in V$ and destination $v \in V$ is represented by an ordered pair $(u, v)$" in the source. Sichter (Hindemith). WiseWoman

 Zuletzt bearbeitet: 2012-04-24 23:07:40 WiseWoman

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In both undirected and directed graphs', we may allow the edge set E to contain the same edge several times, that is, E can be a multiset. If an edge occurs several times in E, the copies of that edge are called parallel edges. Graphs with parallel edges are also called multigraphs. A graph is called simple, if each of its edges is contained in E only once, i.e., if the graph does not have parallel edges. An edge joining a vertex to itself, i.e., and edge whose end [vertices are identical, is called a loop.] In both undirected and directed graphs, we may allow the edge set E to contain the same edge several times, i.e., E can be a multiset. If an edge occurs several times in E, the copies of that edge are called parallel edges. Graphs with parallel edges are also called multigraphs. A graph is called simple, if each of its edges is contained in E only once, i.e., if the graph does not have parallel edges. An edge joining a vertex to itself, i.e., an edge whose endvertices are identical, is called a loop.
 Anmerkungen The source is not given. The definitions given here are certainly standard and don't need to be referenced. Nm, however, copied the formulation of those definitions word for word. Sichter (Hindemith), WiseWoman

 Zuletzt bearbeitet: 2014-01-11 10:20:53 Hindemith

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A graph is called loop-free if it has no loops.

A graph $G' = (V' ,E')$ is a subgraph of the graph $G = (V, E)$ if $V' \subseteq V$ and $E' \subseteq E$. It is a (vertex-)induced graph if $E'$ contains all edges $e \in E$ that join vertices in $V'$.

Often it is useful to associate numerical values (weights) with the edges or vertices of a graph $G = (V, E)$. The graph having weights is known as weighted graph. Here we only discuss edge weights. Edge weights can be represented as a function $\omega : E \to \mathbb{R}$ that assigns each edge $e \in E$, a weight $\omega(e)$. Depending on the context, edge weights can describe various properties such as strength of interaction, or similarity. One usually tries to indicate the characteristics of the edge weights by the choice of the name of the function. For most purposes, an unweighted graph $G = (V, E)$ is equivalent to a weighted graph with unit edge weights $\omega(e) = 1$ for all $e \in E$.

[Page 8, line 14]

A graph is called loop-free if it has no loops.

[Page 9, line 4-6]

A graph $G' = (V' ,E')$ is a subgraph of the graph $G = (V, E)$ if $V' \subseteq V$ and $E' \subseteq E$. It is a (vertex-)induced subgraph if $E'$ contains all edges $e \in E$ that join vertices in $V'$.

[Page 8, line 16ff]

Weighted graphs. Often it is useful to associate numerical values (weights) with the edges or vertices of a graph $G = (V, E)$. Here we discuss only edge weights. Edge weights can be represented as a function $\omega : E \to \mathbb{R}$ that assigns each edge $e \in E$ a weight $\omega(e)$.

Depending on the context, edge weights can describe various properties such as [...] strength of interaction, or similarity. One usually tries to indicate the characteristics of the edge weights by the choice of the name for the function. [...] For most purposes, an unweighted graph $G = (V, E)$ is equivalent to a weighted graph with unit edge weights $\omega(e) = 1$ for all $e \in E$.

 Anmerkungen The source is not given. The definitions given here are certainly standard and don't need to be referenced. The author however copies not only the definitions but also their formulation word for word. The symbol for the real numbers used as the domain of the weight function is reproduced only as a box in the thesis, indicating that this character was not available in the font used. Sichter (Hindemith), WiseWoman