incidence matrix with respect to an orientation
IncidenceMatrixWithRespectToAnOrientation
2007-05-14 12:10:10
2007-05-14 12:13:44
Definition
orientation
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Let $G$ be a finite graph with $n$ vertices, $\{v_1, \ldots, v_n\}$
and $m$ edges, $\{e_1, \ldots, e_m\}$.
For each edge $e = (v_i,v_j)$ of $G$ choose one vertex
to be the positive end and the other to be the negative end. In this way,
we assign an \emph{orientation} to $G$. The \emph{\PMlinkescapetext{incidence matrix}}
of $G$ with respect an orientation is an $n \times m$ matrix
$D=(d_{ij})$
where
\begin{displaymath}
d_{ij} = \left\{ \begin{array}{ll}
+1 & \textrm{if $v_i$ is the positive end of $e_j$} \\
-1 & \textrm{if $v_i$ is the negative end of $e_j$} \\
0 & \textrm{otherwise}.
\end{array} \right.
\end{displaymath}