The loop hafnian

Section author: Nicolás Quesada <nicolas@xanadu.ai>

Graphs with loops

In the previous section we introduced the hafnian as a way of counting the number of perfect matchings of a loopless graph. The loop hafnian does the same for graphs with loops. Before defining the loop hafnian let us introduce graphs with loops. A graph will still be an ordered pair \((V,E)\) of vertices and edges but now we will allow edges of the form \((i,i)\) where \(i \in V\). We can now define adjacency matrices in the same way as we did before i.e. if \((i,j) \in E\) then \(M_{i,j}=1\) and otherwise \(M_{i,j}=0\).

Consider the following graph

_images/loop_fig1.svg

for which we have \(V = \{1,2,3,4,5,6 \}\), the edges are \(E=\{(1,1),(1,4),(2,4),(2,5),(3,4),(3,5),(3,6),(5,5) \}\) and the adjacency matrix is

\[\begin{split}\bm{A}'' = \begin{bmatrix} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{bmatrix}.\end{split}\]

Note that there are now nonzero elements in the diagonal indicating that vertices 1 and 5 have loops.

Once we allow for loops we have more options for making perfect matchings. For example for the graph shown above there are now 2 perfect matchings, illustrated in blue in the following figure

_images/loop_fig2.svg

As was done before for the hafnian we introduce the set of single pair matchings \(\text{SPM}(n)\) as the set of perfect matchings of a graph of size \(n\). For \(n=4\) we have

\[\begin{split}\text{SPM}(4) = \big\{ (0,1)(2,3),\ (0,2)(1,3),\ (0,3),(1,2),\ (0,0)(1,1)(2,3), \ (0,1)(2,2)(3,3),\\ (0,0)(2,2)(1,3),\ (0,2)(1,1)(3,3),\ (0,0)(3,3)(1,2),\ (0,3)(1,1)(2,2),\ (0,0)(1,1)(2,2)(3,3)\big\}.\end{split}\]

For a graph with 4 vertices they are

_images/fig2.svg

Note that there is a one to one correspondence (a bijection) between the elements in \(\text{SPM}(n)\) and the number of matchings of a graph with \(n\) vertices \(H(n)\). To see why this is the case, note that any element of \(\text{SPM}(n)\) can be converted into a matching by removing all the vertices that are loops. For example, to the following element \((0,0)(2,2)(1,3)\) we associate the matching \((1,2)\). Note that this mapping is one-to-one since, given a matching, we can always add as loops all the other vertices that are not part of the matching. Using this bijection we conclude that the number of elements in \(\text{SPM}(n)\) is (see The On-Line Encyclopedia of Integer Sequences)

\[|\text{SPM}(n)| = T(n),\]

where \(T(n)\) is the \(n^{\text{th}}\) telephone number.

Note that in general for given graph size \(n\) there a lot more single pair matching that there are perfect matchings. Their ratio goes like [4]

\[\frac{\text{SPM}(n)}{\text{PMP}(n)} = \frac{T(n)}{(n-1)!!} \sim e^{\sqrt{n}}.\]

The loop hafnian

We will also be interested in a generalization of the hafnian function where we will now allow for adjacency matrices that have loops. This new function we call the loop hafnian (lhaf). As explained before, the weight associated with said loops will be allocated in the diagonal elements of the adjacency matrix \(\bm{A}\) (which were previously ignored in the definition of the hafnian). To account for the possibility of loops we generalized the set of perfect matching permutations PMP to the single-pair matchings (SPM). Thus we define [4]

\[\lhaf(\bm{A}) = \sum_{M \in \text{SPM}(n)} \prod_{\scriptscriptstyle (i,j) \in M} A_{i,j}.\]

The lhaf of a \(4 \times 4\) matrix \(\bm{B}\) is

\[\begin{split}\lhaf(\bm{B}) =& B_{0,1} B_{2,3}+B_{0,2}B_{1,3}+B_{0,3} B_{1,2}\\ &+ B_{0,0} B_{1,1} B_{2,3}+B_{0,1} B_{2,2} B_{3,3}+B_{0,2}B_{1,1}B_{3,3}\nonumber\\ &+ B_{0,0} B_{2,2} B_{1,3}+B_{0,0}B_{3,3}B_{1,2}+B_{0,3} B_{1,1} B_{2,2}\nonumber\\ &+ B_{0,0} B_{1,1} B_{2,2} B_{3,3}. \nonumber\end{split}\]

Finally, let us comment on the scaling properties of the \(\haf\) and \(\lhaf\). Unlike the hafnian, the loop hafnian is not homogeneous in its matrix entries, i.e.

\[\begin{split}\haf(\mu \bm{A}) &= \mu ^{n/2} \haf(\bm{A}) \text{ but},\\ \lhaf(\mu \bm{A}) &\neq \mu ^{n/2} \lhaf(\bm{A}).\end{split}\]

where \(n\) is the size of the matrix \(\bm{A}\) and \(\mu \geq 0\). However if we split the matrix \(\bm{A}\) in terms of its diagonal \(\bm{A}_{\text{diag}}\) part and its offdiagonal part \(\bm{A}_{\text{off-diag}}\)

\[\bm{A} = \bm{A}_{\text{diag}}+\bm{A}_{\text{off-diag}},\]

then it holds that [4]

\[\lhaf(\sqrt{\mu} \bm{A}_{\text{diag}}+ \mu \bm{A}_{\text{off-diag}}) = \mu^{n/2} \lhaf(\bm{A}_{\text{diag}}+ \bm{A}_{\text{off-diag}}) =\mu^{n/2} \lhaf(\bm{A}).\]

One can use the loop hafnian to count the number of matchings of a loopless graph by simply calculating the loop hafnian of its adjacency matrix adding ones in its diagonal.

Finally, if \(\bm{A}_{\text{direct sum}} = \bm{A}_1 \oplus \bm{A}_2\) is a block diagonal matrix then

\[\text{lhaf}\left(\bm{A}_{\text{direct sum}}\right) = \text{lhaf}\left( \bm{A}_1 \oplus \bm{A}_2 \right) = \text{lhaf}\left( \bm{A}_1 \right) \text{lhaf}\left( \bm{A}_2 \right).\]

As for the hafnian, this identity tell us that the number of perfect matchings of a graph that is made of two disjoint subgraphs is simply the product of the number of perfect matchings of the two disjoint subgraphs.