Kirchhoff's theorem

<h2 id="an-example-using-the-matrix-tree-theorem">An example using the matrix-tree theorem</h2>

First, construct the Laplacian matrix Q for the example <a href="/facts/Diamond_graph/8awrY0xS">diamond graph</a> G (see image on the right):

Q
        =
        
          [
          
            
              
                
                  2
                
                
                  −
                  1
                
                
                  −
                  1
                
                
                  0
                
              
              
                
                  −
                  1
                
                
                  3
                
                
                  −
                  1
                
                
                  −
                  1
                
              
              
                
                  −
                  1
                
                
                  −
                  1
                
                
                  3
                
                
                  −
                  1
                
              
              
                
                  0
                
                
                  −
                  1
                
                
                  −
                  1
                
                
                  2
                
              
            
          
          ]
        
        .
      
    
    {\displaystyle Q=\left[{\begin{array}{rrrr}2&-1&-1&0\\-1&3&-1&-1\\-1&-1&3&-1\\0&-1&-1&2\end{array}}\right].}

Next, construct a matrix Q* by deleting any row and any column from Q. For example, deleting row 1 and column 1 yields

Q
          
            ∗
          
        
        =
        
          [
          
            
              
                
                  3
                
                
                  −
                  1
                
                
                  −
                  1
                
              
              
                
                  −
                  1
                
                
                  3
                
                
                  −
                  1
                
              
              
                
                  −
                  1
                
                
                  −
                  1
                
                
                  2
                
              
            
          
          ]
        
        .
      
    
    {\displaystyle Q^{\ast }=\left[{\begin{array}{rrr}3&-1&-1\\-1&3&-1\\-1&-1&2\end{array}}\right].}

Finally, take the determinant of Q* to obtain t(G), which is 8 for the diamond graph. (Notice t(G) is the (1,1)-cofactor of Q in this example.)

<h2 id="proof-outline">Proof outline</h2>
(The proof below is based on the <a href="/facts/Cauchy%25E2%2580%2593Binet_formula/jbRCH0tj">Cauchy–Binet formula</a>. An elementary <a href="/facts/Mathematical_induction/KxAPqNrH">induction</a> argument for Kirchhoff's theorem can be found on page 654 of Moore (2011).<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>)
First notice that the Laplacian matrix has the property that the sum of its entries across any row and any column is 0. Thus we can transform any minor into any other minor by adding rows and columns, switching them, and multiplying a row or a column by −1. Thus the cofactors are the same up to sign, and it can be verified that, in fact, they have the same sign.
We proceed to show that the determinant of the minor M11 is the number of spanning trees. Let n be the number of vertices of the graph, and m the number of its edges. The incidence matrix E is an n-by-m matrix, which may be defined as follows: suppose that (i, j) is the kth edge of the graph, and that i < j. Then Eik = 1, Ejk = −1, and all other entries in column k are 0 (see <a href="/facts/Oriented_incidence_matrix/eMXGWQBw">oriented incidence matrix</a> for understanding this modified incidence matrix E). For the preceding example (with n = 4 and m = 5):

E
        =
        
          
            [
            
              
                
                  1
                
                
                  1
                
                
                  0
                
                
                  0
                
                
                  0
                
              
              
                
                  −
                  1
                
                
                  0
                
                
                  1
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  −
                  1
                
                
                  −
                  1
                
                
                  0
                
                
                  1
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  −
                  1
                
                
                  −
                  1
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle E={\begin{bmatrix}1&1&0&0&0\\-1&0&1&1&0\\0&-1&-1&0&1\\0&0&0&-1&-1\\\end{bmatrix}}.}

Recall that the Laplacian L can be factored into the product of the <a href="/facts/Incidence_matrix/eMXGWQBw">incidence matrix</a> and its <a href="/facts/Transpose/8wmsagGS">transpose</a>, i.e., L = EET. Furthermore, let F be the matrix E with its first row deleted, so that FFT = M11.
Now the <a href="/facts/Cauchy%25E2%2580%2593Binet_formula/jbRCH0tj">Cauchy–Binet formula</a> allows us to write

det
        
          (
          
            M
            
              11
            
          
          )
        
        =
        
          ∑
          
            S
          
        
        det
        
          (
          
            F
            
              S
            
          
          )
        
        det
        
          (
          
            F
            
              S
            
            
              
                T
              
            
          
          )
        
        =
        
          ∑
          
            S
          
        
        det
        
          
            (
            
              F
              
                S
              
            
            )
          
          
            2
          
        
      
    
    {\displaystyle \det \left(M_{11}\right)=\sum _{S}\det \left(F_{S}\right)\det \left(F_{S}^{\mathrm {T} }\right)=\sum _{S}\det \left(F_{S}\right)^{2}}

where S ranges across subsets of [m] of size n − 1, and FS denotes the (n − 1)-by-(n − 1) matrix whose columns are those of F with index in S. Then every S specifies n − 1 edges of the original graph, and it can be shown that those edges induce a spanning tree if and only if the determinant of FS is +1 or −1, and that they do not induce a spanning tree if and only if the determinant is 0. This completes the proof.

<h2 id="particular-cases-and-generalizations">Particular cases and generalizations</h2>
<h3>Cayley's formula</h3>
Main article: <a href="/facts/Cayley%2527s_formula/fwjWPJTf">Cayley's formula</a>
<a href="/facts/Cayley%2527s_formula/fwjWPJTf">Cayley's formula</a> follows from Kirchhoff's theorem as a special case, since every vector with 1 in one place, −1 in another place, and 0 elsewhere is an eigenvector of the Laplacian matrix of the complete graph, with the corresponding eigenvalue being n. These vectors together span a space of <a href="/facts/Dimension_(vector_space)/z8m02A3W">dimension</a> n − 1, so there are no other non-zero eigenvalues.
Alternatively, note that as Cayley's formula gives the number of distinct labeled trees of a complete graph Kn we need to compute any cofactor of the Laplacian matrix of Kn. The Laplacian matrix in this case is

[
            
              
                
                  n
                  −
                  1
                
                
                  −
                  1
                
                
                  ⋯
                
                
                  −
                  1
                
              
              
                
                  −
                  1
                
                
                  n
                  −
                  1
                
                
                  ⋯
                
                
                  −
                  1
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  −
                  1
                
                
                  −
                  1
                
                
                  ⋯
                
                
                  n
                  −
                  1
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle {\begin{bmatrix}n-1&-1&\cdots &-1\\-1&n-1&\cdots &-1\\\vdots &\vdots &\ddots &\vdots \\-1&-1&\cdots &n-1\\\end{bmatrix}}.}

Any cofactor of the above matrix is nn−2, which is Cayley's formula.

<h3>Kirchhoff's theorem for multigraphs</h3>
Kirchhoff's theorem holds for <a href="/facts/Multigraph/udmS0Cp0">multigraphs</a> as well; the matrix Q is modified as follows:

<ul><li>The entry qi,j equals −m, where m is the number of edges between i and j;</li>
<li>when counting the degree of a vertex, all <a href="/facts/Loop_(graph_theory)/jCuevIad">loops</a> are excluded.</li></ul>
Cayley's formula for a complete multigraph is mn−1(nn−1−(n−1)nn−2) by same methods produced above, since a simple graph is a multigraph with m = 1.

<h3>Explicit enumeration of spanning trees</h3>
Kirchhoff's theorem can be strengthened by altering the definition of the Laplacian matrix. Rather than merely counting edges emanating from each vertex or connecting a pair of vertices, label each edge with an <a href="/facts/Indeterminate_(variable)/R7QAHS4j">indeterminate</a> and let the (i, j)-th entry of the modified Laplacian matrix be the sum over the indeterminates corresponding to edges between the i-th and j-th vertices when i does not equal j, and the negative sum over all indeterminates corresponding to edges emanating from the i-th vertex when i equals j.
The determinant of the modified Laplacian matrix by deleting any row and column (similar to finding the number of spanning trees from the original Laplacian matrix), above is then a <a href="/facts/Homogeneous_polynomial/WUHL7Kct">homogeneous polynomial</a> (the Kirchhoff polynomial) in the indeterminates corresponding to the edges of the graph. After collecting terms and performing all possible cancellations, each <a href="/facts/Monomial/6VAPCYXJ">monomial</a> in the resulting expression represents a spanning tree consisting of the edges corresponding to the indeterminates appearing in that monomial. In this way, one can obtain explicit enumeration of all the spanning trees of the graph simply by computing the determinant.
For a proof of this version of the theorem see Bollobás (1998).<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h3>Matroids</h3>
The spanning trees of a graph form the bases of a <a href="/facts/Graphic_matroid/UkEwWWhl">graphic matroid</a>, so Kirchhoff's theorem provides a formula for the number of bases in a graphic matroid. The same method may also be used to determine the number of bases in <a href="/facts/Regular_matroid/veBC6AXk">regular matroids</a>, a generalization of the graphic matroids (Maurer 1976).

<h3>Kirchhoff's theorem for directed multigraphs</h3>
Kirchhoff's theorem can be modified to give the number of oriented spanning trees in directed multigraphs.
The matrix Q is constructed as follows:

<ul><li>The entry qi,j for distinct i and j equals −m, where m is the number of edges from i to j;</li>
<li>The entry qi,i equals the indegree of i minus the number of loops at i.</li></ul>
The number of oriented spanning trees rooted at a vertex i is the determinant of the matrix gotten by removing the ith row and column of Q

<h3>Counting spanning k-component forests</h3>
Kirchhoff's theorem can be generalized to count k-component spanning <a href="/facts/Forest_(graph_theory)/TCWk4Joy">forests</a> in an unweighted graph.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> A k-component spanning forest is a subgraph with k <a href="/facts/Component_(graph_theory)/5owVSXtY">connected components</a> that contains all vertices and is cycle-free, i.e., there is at most one path between each pair of vertices. Given such a forest F with connected components 
 
 
 
 
 F
 
 1
 
 
 ,
 …
 ,
 
 F
 
 k
 
 
 
 
 {\textstyle F_{1},\dots ,F_{k}}
 
, define its weight 
 
 
 
 w
 (
 F
 )
 =
 
 |
 
 V
 (
 
 F
 
 1
 
 
 )
 
 |
 
 ⋅
 ⋯
 ⋅
 
 |
 
 V
 (
 
 F
 
 k
 
 
 )
 
 |
 
 
 
 {\textstyle w(F)=|V(F_{1})|\cdot \dots \cdot |V(F_{k})|}
 
 to be the product of the number of vertices in each component. Then

∑
          
            F
          
        
        w
        (
        F
        )
        =
        
          q
          
            k
          
        
        ,
      
    
    {\displaystyle \sum _{F}w(F)=q_{k},}

where the sum is over all k-component spanning forests and 
 
 
 
 
 q
 
 k
 
 
 
 
 {\textstyle q_{k}}
 
 is the coefficient of 
 
 
 
 
 x
 
 k
 
 
 
 
 {\textstyle x^{k}}
 
 of the polynomial

(
        x
        +
        
          λ
          
            1
          
        
        )
        …
        (
        x
        +
        
          λ
          
            n
            −
            1
          
        
        )
        x
        .
      
    
    {\displaystyle (x+\lambda _{1})\dots (x+\lambda _{n-1})x.}

The last factor in the polynomial is due to the zero eigenvalue 
 
 
 
 
 λ
 
 n
 
 
 =
 0
 
 
 {\textstyle \lambda _{n}=0}
 
. More explicitly, the number 
 
 
 
 
 q
 
 k
 
 
 
 
 {\textstyle q_{k}}
 
 can be computed as

q
          
            k
          
        
        =
        
          ∑
          
            {
            
              i
              
                1
              
            
            ,
            …
            ,
            
              i
              
                n
                −
                k
              
            
            }
            ⊂
            {
            1
            …
            n
            −
            1
            }
          
        
        
          λ
          
            
              i
              
                1
              
            
          
        
        …
        
          λ
          
            
              i
              
                n
                −
                k
              
            
          
        
        .
      
    
    {\displaystyle q_{k}=\sum _{\{i_{1},\dots ,i_{n-k}\}\subset \{1\dots n-1\}}\lambda _{i_{1}}\dots \lambda _{i_{n-k}}.}

where the sum is over all n−k-element subsets of 
 
 
 
 {
 1
 ,
 …
 ,
 n
 }
 
 
 {\textstyle \{1,\dots ,n\}}
 
. For example

 
 
 
 
 
 
 
 
 q
 
 n
 −
 1
 
 
 
 
 
 =
 
 λ
 
 1
 
 
 +
 ⋯
 +
 
 λ
 
 n
 −
 1
 
 
 =
 
 t
 r
 
 Q
 =
 2
 
 |
 
 E
 
 |
 
 
 
 
 
 
 q
 
 n
 −
 2
 
 
 
 
 
 =
 
 λ
 
 1
 
 
 
 λ
 
 2
 
 
 +
 
 λ
 
 1
 
 
 
 λ
 
 3
 
 
 +
 ⋯
 +
 
 λ
 
 n
 −
 2
 
 
 
 λ
 
 n
 −
 1
 
 
 
 
 
 
 
 q
 
 2
 
 
 
 
 
 =
 
 λ
 
 1
 
 
 …
 
 λ
 
 n
 −
 2
 
 
 +
 
 λ
 
 1
 
 
 …
 
 λ
 
 n
 −
 3
 
 
 
 λ
 
 n
 −
 1
 
 
 +
 ⋯
 +
 
 λ
 
 2
 
 
 …
 
 λ
 
 n
 −
 1
 
 
 
 
 
 
 
 q
 
 1
 
 
 
 
 
 =
 
 λ
 
 1
 
 
 …
 
 λ
 
 n
 −
 1
 
 
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}q_{n-1}&=\lambda _{1}+\dots +\lambda _{n-1}=\mathrm {tr} Q=2|E|\\q_{n-2}&=\lambda _{1}\lambda _{2}+\lambda _{1}\lambda _{3}+\dots +\lambda _{n-2}\lambda _{n-1}\\q_{2}&=\lambda _{1}\dots \lambda _{n-2}+\lambda _{1}\dots \lambda _{n-3}\lambda _{n-1}+\dots +\lambda _{2}\dots \lambda _{n-1}\\q_{1}&=\lambda _{1}\dots \lambda _{n-1}\\\end{aligned}}}

Since a spanning forest with n−1 components corresponds to a single edge, the k = n−1 case states that the sum of the eigenvalues of Q is twice the number of edges. The k = 1 case corresponds to the original Kirchhoff theorem since the weight of every spanning tree is n.
The proof can be done analogously to the proof of Kirchhoff's theorem. An invertible 
 
 
 
 (
 n
 −
 k
 )
 ×
 (
 n
 −
 k
 )
 
 
 {\displaystyle (n-k)\times (n-k)}
 
 submatrix of the incidence matrix corresponds <a href="/facts/Bijection/j4gTuTmW">bijectively</a> to a k-component spanning forest with a choice of vertex for each component.
The coefficients 
 
 
 
 
 q
 
 k
 
 
 
 
 {\textstyle q_{k}}
 
 are up to sign the coefficients of the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> of Q.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/List_of_graph_theory_topics/AsnkodSB">List of topics related to trees</a></li>
<li><a href="/facts/BEST_theorem/MFMVJ0Xl">BEST theorem</a></li>
<li><a href="/facts/Markov_chain_tree_theorem/ktWfqIMD">Markov chain tree theorem</a></li>
<li><a href="/facts/Minimum_spanning_tree/kgkGmY9s">Minimum spanning tree</a></li>
<li><a href="/facts/Pr%25C3%25BCfer_sequence/0tmI33Mm">Prüfer sequence</a></li></ul>

<ul><li>Harris, John M.; Hirst, Jeffry L.; Mossinghoff, Michael J. (2008), Combinatorics and Graph Theory, <a href="/facts/Undergraduate_Texts_in_Mathematics/MTjjHqWD">Undergraduate Texts in Mathematics</a> (2nd ed.), Springer.</li>
<li>Maurer, Stephen B. (1976), "Matrix generalizations of some theorems on trees, cycles and cocycles in graphs", SIAM Journal on Applied Mathematics, 30 (1): 143–148, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1137%2F0130017">10.1137/0130017</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0392635">0392635</a>.</li>
<li>Tutte, W. T. (2001), Graph Theory, Cambridge University Press, p. 138, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-79489-3.</li>
<li>Chaiken, S.; Kleitman, D. (1978), "Matrix Tree Theorems", Journal of Combinatorial Theory, Series A, 24 (3): 377–381, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0097-3165%2878%2990067-5">10.1016/0097-3165(78)90067-5</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0097-3165">0097-3165</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">O'Toole, J.B. (1958). "On the Solution of the Equations Obtained from the Investigation of the Linear Distribution of Galvanic Currents". IRE Transactions on Circuit Theory. 5 (1): 4–7. doi:10.1109/TCT.1958.1086426. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Moore, Cristopher (2011). The nature of computation. Oxford England New York: Oxford University Press. ISBN 978-0-19-923321-2. OCLC 180753706. <a href="978-0-19-923321-2" target="_blank">978-0-19-923321-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Bollobás, Béla (1998). Modern graph theory. Graduate Texts in Mathematics. Vol. 184. New York: Springer. doi:10.1007/978-1-4612-0619-4. ISBN 978-0-387-98488-9. <a href="978-0-387-98488-9" target="_blank">978-0-387-98488-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Biggs, N. (1993). Algebraic Graph Theory. Cambridge University Press. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Kirchhoff's theorem open-in-new

Kirchhoff's theorem