The document discusses Kirchhoff's matrix tree theorem which provides a formula for calculating the number of spanning trees in a finite graph based on the graph's Laplacian matrix. It provides an example of applying the theorem to calculate that a particular graph has 29 spanning trees by taking the determinant of the Laplacian matrix with a row and column removed. Finally, it discusses several applications that rely on building and analyzing spanning trees, such as routing algorithms, power networks, and telecommunications networks.

1. S.SWATHI SUNDARI, M.Sc.,M.Phil.,
Assistant Professor of Mathematics
V.V.Vanniaperumal College for Women
Virudhunagar

2. George Gustav Robert
Kirchhoff was a German
physician and mathematician.
He is occupied especially
spectroscopy, electrical
engineering and
thermodynamics.
 He formulated the two famous
laws for the calculation of the
distribution of electric currents
in a network of wires.

3. Kirchhoff’s matrix tree theorem gives a
formula for the number of spanning trees of a
finite graph in terms of a matrix derived from
that graph.

4. Suppose that T = (V,E) is a finite graph consisting of
n + 1 vertices labelled y1, y2, · · · , yn, yn+1.
• undirected
• connected
• no multiple edges
Note that yi ∼ yj are nearest neighbours if (yi, yj ) ∈ E.

5. The graph Laplacian L is the matrix L = D − A, where
D is the degree matrix and A is the adjacency matrix.

6. Suppose that L{k} denotes the submatrix of L obtained by
deleting row k and column k corresponding to vertex yk
Statement:
If Ω = {spanning trees of T}, then
det[L{1}] = det[L{2}] = · · · = det[L{n}= det[L{n+1}]
and that these are equal to |Ω|, the number of
spanning trees of T.

7.  This graph has 29 spanning trees.
 To see this, consider deg(y4) in the spanning tree.

8.  This graph has 29 spanning trees. For example, using
MTT, det[L{5}] = 29.

9.  Several path finding algorithms, including Dijkstra’s
algorithm and the A* search algorithm, internally
build a spanning trees as an intermediate step in
solving problem.
In order to minimize the cost of power networks,
wiring connections, pipings, automatic speech
recognition, etc., -- and we often use algorithms that
gradually build a spanning tree s intermediate steps
in the process of finding the minimum spanning tree.
The Internet and telecommunications-Transmission
links, Open shortest path, Augmented tree-based
routing– required to remember a spanning tree.