Chapter 23 first version
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Antti H S Laaksonen
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luku23.tex
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luku23.tex
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@ -607,19 +607,17 @@ X^n \cdot
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\]
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\end{samepage}
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\section{Verkot ja matriisit}
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\section{Graphs and matrices}
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\subsubsection{Polkujen määrä}
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\subsubsection{Counting paths}
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Matriisipotenssilla
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on mielenkiintoinen vaikutus
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verkon vierusmatriisin sisältöön.
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Kun $V$ on painottoman verkon vierusmatriisi,
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niin $V^n$ kertoo,
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montako $n$ kaaren pituista polkua
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eri solmuista on toisiinsa.
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The powers of an adjacency matrix of a graph
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have an interesting property.
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When $V$ is an adjacency matrix of an unweighted graph,
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the matrix $V^n$ contains the numbers of paths of
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$n$ edges between the nodes in the graph.
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Esimerkiksi verkon
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For example, for the graph
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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@ -639,7 +637,7 @@ Esimerkiksi verkon
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\path[draw,thick,->,>=latex] (6) -- (5);
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\end{tikzpicture}
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\end{center}
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vierusmatriisi on
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the adjacency matrix is
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\[
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V= \begin{bmatrix}
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0 & 0 & 0 & 1 & 0 & 0 \\
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@ -650,7 +648,7 @@ V= \begin{bmatrix}
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0 & 0 & 1 & 0 & 1 & 0 \\
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\end{bmatrix}.
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\]
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Nyt esimerkiksi matriisi
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Now, for example, the matrix
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\[
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V^4= \begin{bmatrix}
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0 & 0 & 1 & 1 & 1 & 0 \\
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@ -661,26 +659,26 @@ V^4= \begin{bmatrix}
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0 & 0 & 1 & 1 & 1 & 0 \\
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\end{bmatrix}
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\]
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kertoo, montako 4 kaaren pituista polkua
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solmuista on toisiinsa.
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Esimerkiksi $V^4[2,5]=2$,
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koska solmusta 2 solmuun 5 on olemassa
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4 kaaren pituiset polut
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contains the numbers of paths of 4 edges
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between the nodes.
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For example, $V^4[2,5]=2$,
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because there are two paths of 4 edges
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from node 2 to node 5:
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$2 \rightarrow 1 \rightarrow 4 \rightarrow 2 \rightarrow 5$
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ja
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and
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$2 \rightarrow 6 \rightarrow 3 \rightarrow 2 \rightarrow 5$.
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\subsubsection{Lyhimmät polut}
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\subsubsection{Shortest paths}
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Samantapaisella idealla voi laskea painotetussa verkossa
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kullekin solmuparille,
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kuinka pitkä on lyhin $n$ kaaren pituinen polku solmujen välillä.
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Tämä vaatii matriisitulon määritelmän muuttamista
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niin, että siinä ei lasketa polkujen yhteismäärää
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vaan minimoidaan polun pituutta.
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Using a similar idea in a weighted graph,
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we can calculate for each pair of nodes the shortest
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path between them that contains exactly $n$ edges.
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To calculate this, we have to define matrix multiplication
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in another way, so that we don't calculate the number
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of paths but minimize the length of a path.
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\begin{samepage}
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Tarkastellaan esimerkkinä seuraavaa verkkoa:
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As an example, consider the following graph:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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@ -702,10 +700,10 @@ Tarkastellaan esimerkkinä seuraavaa verkkoa:
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\end{center}
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\end{samepage}
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Muodostetaan verkosta vierusmatriisi, jossa arvo
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$\infty$ tarkoittaa, että kaarta ei ole,
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ja muut arvot ovat kaarten pituuksia.
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Matriisista tulee
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Let's construct an adjacency matrix where
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$\infty$ means that an edge doesn't exist,
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and other values correspond to edge weights.
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The matrix is
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\[
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V= \begin{bmatrix}
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\infty & \infty & \infty & 4 & \infty & \infty \\
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@ -717,18 +715,20 @@ V= \begin{bmatrix}
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\end{bmatrix}.
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\]
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Nyt voimme laskea matriisitulon kaavan
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Instead of the formula
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\[
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AB[i,j] = \sum_{k=1}^n A[i,k] \cdot B[k,j]
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\]
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sijasta kaavalla
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we now use the formula
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\[
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AB[i,j] = \min_{k=1}^n A[i,k] + B[k,j],
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\]
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eli summa muuttuu minimiksi ja tulo summaksi.
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Tämän seurauksena matriisipotenssi
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selvittää lyhimmät polkujen pituudet solmujen
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välillä. Esimerkiksi
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for matrix multiplication, so we calculate
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a minimum instead of a sum,
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and a sum of elements instead of a product.
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After this modification,
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matrix powers can be used for calculating
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shortest paths in the graph:
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\[
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V^4= \begin{bmatrix}
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@ -740,19 +740,20 @@ V^4= \begin{bmatrix}
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\infty & \infty & 12 & 13 & 11 & \infty \\
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\end{bmatrix}
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\]
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eli esimerkiksi lyhin 4 kaaren pituinen polku
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solmusta 2 solmuun 5 on pituudeltaan 8.
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Tämä polku on $2 \rightarrow 1 \rightarrow 4 \rightarrow 2 \rightarrow 5$.
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For example, the shortest path of 4 edges
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from node 2 to node 5 has length 8.
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This path is
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$2 \rightarrow 1 \rightarrow 4 \rightarrow 2 \rightarrow 5$.
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\subsubsection{Kirchhoffin lause}
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\subsubsection{Kirchhoff's theorem}
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\index{Kirchhoffin lause@Kirchhoffin lause}
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\index{virittxvx puu@virittävä puu}
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\index{Kirchhoff's theorem}
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\index{spanning tree}
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\key{Kirchhoffin lause} laskee
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verkon virittävän puiden määrän
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verkosta muodostetun matriisin determinantin avulla.
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Esimerkiksi verkolla
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\key{Kirchhoff's theorem} provides us a way
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to calculate the number of spanning trees
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in a graph as a determinant of a special matrix.
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For example, the graph
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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@ -766,7 +767,7 @@ Esimerkiksi verkolla
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\path[draw,thick,-] (1) -- (4);
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\end{tikzpicture}
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\end{center}
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on kolme virittävää puuta:
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has three spanning trees:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1a) at (1,3) {$1$};
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@ -800,12 +801,12 @@ on kolme virittävää puuta:
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%\path[draw,thick,-] (1c) -- (4c);
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\end{tikzpicture}
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\end{center}
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\index{Laplacen matriisi@Laplacen matriisi}
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Muodostetaan verkosta \key{Laplacen matriisi} $L$,
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jossa $L[i,i]$ on solmun $i$ aste ja
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$L[i,j]=-1$, jos solmujen $i$ ja $j$ välillä on kaari,
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ja muuten $L[i,j]=0$.
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Tässä tapauksessa matriisista tulee
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\index{Laplacean matrix}
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We construct a \key{Laplacean matrix} $L$,
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where $L[i,i]$ is the degree of node $i$
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and $L[i,j]=-1$ if there is an edge between
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nodes $i$ and $j$, and otherwise $L[i,j]=0$.
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In this graph, the matrix is as follows:
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\[
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L= \begin{bmatrix}
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@ -813,14 +814,14 @@ L= \begin{bmatrix}
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-1 & 1 & 0 & 0 \\
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-1 & 0 & 2 & -1 \\
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-1 & 0 & -1 & 2 \\
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\end{bmatrix}.
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\end{bmatrix}
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\]
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Nyt virittävien puiden määrä on determinantti
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matriisista, joka saadaan poistamasta matriisista $L$
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jokin rivi ja jokin sarake.
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Esimerkiksi jos poistamme ylimmän rivin ja
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vasemman sarakkeen, tuloksena on
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The number of spanning trees is the same as
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the determinant of a matrix that is obtained
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when we remove any row and any column from $L$.
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For example, if we remove the first row
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and column, the result is
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\[ \det(
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\begin{bmatrix}
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@ -829,12 +830,12 @@ vasemman sarakkeen, tuloksena on
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0 & -1 & 2 \\
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\end{bmatrix}
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) =3.\]
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Determinantista tulee aina sama riippumatta siitä,
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mikä rivi ja sarake matriisista $L$ poistetaan.
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The determinant is always the same,
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regardless of which row and column we remove from $L$.
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Huomaa, että Kirchhoffin lauseen erikoistapauksena on
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luvun 22.5 Cayleyn kaava, koska
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täydellisessä $n$ solmun verkossa
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Note that a special case of Kirchhoff's theorem
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is Cayley's formula in Chapter 22.5,
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because in a complete graph of $n$ nodes
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\[ \det(
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\begin{bmatrix}
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