In mathematics, the hafnian of an adjacency matrix of a graph is the number of perfect matchings in the graph. It was so named by Eduardo R. Caianiello "to mark the fruitful period of stay in Copenhagen (Hafnia in Latin)."^[1]

The hafnian of a ${\displaystyle 2n\times 2n}$ symmetric matrix is computed as

${\displaystyle \operatorname {haf} (A)={\frac {1}{n!2^{n}}}\sum _{\sigma \in S_{2n}}\prod _{j=1}^{n}A_{\sigma (2j-1),\sigma (2j)},}$

where ${\displaystyle S_{2n}}$ is the symmetric group on ${\displaystyle [2n]=\{1,2,...,2n\}}$.^[2]

Equivalently,

${\displaystyle \operatorname {haf} (A)=\sum _{M\in {\mathcal {M}}}\prod _{\scriptscriptstyle (u,v)\in M}A_{u,v}}$

where ${\displaystyle {\mathcal {M}}}$ is the set of all 1-factors (perfect matchings) on the complete graph ${\displaystyle K_{2n}}$, namely the set of all ${\displaystyle (2n)!/(n!2^{n})}$ ways to partition the set ${\displaystyle \{1,2,\dots ,2n\}}$ into ${\displaystyle n}$ subsets of size ${\displaystyle 2}$.^[3][4]

The definition of the hafnian is similar to that of the Pfaffian, but differs in that the signatures of the permutations are not taken into account. Thus the relationship of the hafnian to the Pfaffian is the same as relationship of the permanent to the determinant.^[5]

Furthermore, the permanent and the hafnian are related as ${\displaystyle \operatorname {per} (A)=\operatorname {haf} {\begin{pmatrix}0&A\\A^{T}&0\end{pmatrix}}}$.^[3]

If ${\displaystyle A}$ is a Hermitian positive semi-definite ${\displaystyle n\times n}$ matrix and ${\displaystyle B}$ is a complex symmetric ${\displaystyle n\times n}$ matrix, then we have the inequality

${\displaystyle \operatorname {haf} {\begin{pmatrix}B&A\\{\overline {A}}&{\overline {B}}\end{pmatrix}}\geq 0,}$

where ${\displaystyle {\overline {A}}}$ denotes the complex conjugate of ${\displaystyle A}$.^[6] A simple way to see this when ${\displaystyle {\begin{pmatrix}A&B\\{\overline {B}}&{\overline {A}}\\\end{pmatrix}}}$ is positive semi-definite is to observe that, by Wick's probability theorem, ${\displaystyle \operatorname {haf} {\begin{pmatrix}B&A\\{\overline {A}}&{\overline {B}}\\\end{pmatrix}}=\mathbb {E} \left[\left|X_{1}\dots X_{n}\right|^{2}\right]}$ when ${\displaystyle \left(X_{1},\dots ,X_{n}\right)}$ is a complex normal random vector with mean ${\displaystyle 0}$, covariance matrix ${\displaystyle A}$ and relation matrix ${\displaystyle B}$.

Generating function

Let ${\displaystyle S={\begin{pmatrix}A&C\\C^{T}&B\end{pmatrix}}=S^{T}}$ be an arbitrary complex symmetric ${\displaystyle 2m\times 2m}$ matrix composed of four ${\displaystyle m\times m}$ blocks ${\displaystyle A=A^{T}}$, ${\displaystyle B=B^{T}}$, ${\displaystyle C}$ and ${\displaystyle C^{T}}$. Let ${\displaystyle z_{1},\ldots ,z_{m}}$ be a set of ${\displaystyle m}$ independent variables, and let ${\displaystyle Z={\begin{pmatrix}0&{\textrm {diag}}(z_{1},z_{2},\ldots ,z_{m})\\{\textrm {diag}}(z_{1},z_{2},\ldots ,z_{m})&0\end{pmatrix}}}$ be an antidiagonal block matrix composed of entries ${\displaystyle z_{j}}$ (each one is presented twice, one time per nonzero block). Let ${\displaystyle I}$ denote the identity matrix. Then the following identity holds:^[5]

${\displaystyle {\frac {1}{\sqrt {\det {\big (}I-ZS{\big )}}}}=\sum _{\{n_{k}\}}\operatorname {haf} {\tilde {S}}(\{n_{k}\})\prod _{k=1}^{m}{\frac {z_{k}^{n_{k}}}{n_{k}!}}}$

where the right-hand side involves hafnians of ${\displaystyle {\Big (}2\sum _{k}n_{k}{\Big )}\times {\Big (}2\sum _{k}n_{k}{\Big )}}$ matrices ${\displaystyle {\tilde {S}}(\{n_{k}\})={\begin{pmatrix}{\tilde {A}}(\{n_{k}\})&{\tilde {C}}(\{n_{k}\})\\{\tilde {C}}^{T}(\{n_{k}\})&{\tilde {B}}(\{n_{k}\})\\\end{pmatrix}}}$, whose blocks ${\displaystyle {\tilde {A}}(\{n_{k}\})}$, ${\displaystyle {\tilde {B}}(\{n_{k}\})}$, ${\displaystyle {\tilde {C}}(\{n_{k}\})}$ and ${\displaystyle {\tilde {C}}^{T}(\{n_{k}\})}$ are built from the blocks ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$ and ${\displaystyle C^{T}}$ respectively in the way introduced in MacMahon's Master theorem. In particular, ${\displaystyle {\tilde {A}}(\{n_{k}\})}$ is a matrix built by replacing each entry ${\displaystyle A_{k,t}}$ in the matrix ${\displaystyle A}$ with a ${\displaystyle n_{k}\times n_{t}}$ block filled with ${\displaystyle A_{k,t}}$; the same scheme is applied to ${\displaystyle B}$, ${\displaystyle C}$ and ${\displaystyle C^{T}}$. The sum ${\displaystyle \sum _{\{n_{k}\}}}$ runs over all ${\displaystyle k}$-tuples of non-negative integers, and it is assumed that ${\displaystyle \operatorname {haf} {\tilde {S}}(\{n_{k}=0|k=1\ldots m\})=1}$.

The identity can be proven^[5] by means of multivariate Gaussian integrals and the Wick (or Isserlis) theorem.

The expression in the left-hand side, ${\displaystyle 1{\Big /}{\sqrt {\det {\big (}I-ZS{\big )}}}{\Big .}}$, is in fact a multivariate generating function for a series of hafnians, and the right-hand side constitutes its multivariable Taylor expansion in the vicinity of the point ${\displaystyle z_{1}=\ldots =z_{m}=0.}$ As a consequence of the given relation, the hafnian of a symmetric ${\displaystyle 2m\times 2m}$ matrix ${\displaystyle S}$ can be represented as the following mixed derivative of the order ${\displaystyle m}$:

${\displaystyle \operatorname {haf} S={\bigg (}\prod _{k=1}^{m}{\frac {\partial }{\partial z_{k}}}{\bigg )}{\Bigg .}{\frac {1}{\sqrt {\det {\big (}I-ZS{\big )}}}}{\Bigg \vert }_{z_{j}=0}.}$

The hafnian generating function identity written above can be considered as a hafnian generalization of MacMahon's Master theorem, which introduces the generating function for matrix permanents and has the following form in terms of the introduced notation:

${\displaystyle {\frac {1}{\det {\big (}I-{\textrm {diag}}(z_{1},z_{2},\ldots ,z_{m})C{\big )}}}=\sum _{\{n_{k}\}}\operatorname {per} {\tilde {C}}(\{n_{k}\})\prod _{k=1}^{m}{\frac {z_{k}^{n_{k}}}{n_{k}!}}}$

Note that MacMahon's Master theorem comes as a simple corollary from the hafnian generating function identity due to the relation ${\displaystyle \operatorname {per} C=\operatorname {haf} {\begin{pmatrix}0&C\\C^{T}&0\end{pmatrix}}}$.

Loop hafnian

The loop hafnian of an ${\displaystyle m\times m}$ symmetric matrix is defined as

${\displaystyle \operatorname {lhaf} (A)=\sum _{M\in {\mathcal {M}}^{\prime }}\prod _{\scriptscriptstyle (u,v)\in M}A_{u,v}}$

where ${\displaystyle {\mathcal {M}}^{\prime }}$ is the set of all perfect matchings of the complete graph on ${\displaystyle m}$ vertices with loops, i.e., the set of all ways to partition the set ${\displaystyle \{1,2,\dots ,m\}}$ into subsets of size ${\displaystyle 2}$ or size ${\displaystyle 1}$ (treated as a pair ${\displaystyle (u,u)}$).^[7] Thus the loop hafnian depends on the diagonal entries of the matrix, whereas the hafnian does not.^[7] Furthermore, the loop hafnian can be non-zero even when ${\displaystyle m}$ is odd.

The loop hafnian of the matrix obtained by taking the adjacency matrix of a graph and setting the diagonal elements to 1 is equal to the total number of matchings in the graph (perfect or non-perfect),^[7] also known as its Hosoya index.

By Wick's probability theorem, the hafnian and loop hafnian of a real ${\displaystyle m\times m}$ symmetric matrix can equivalently be defined as

${\displaystyle \operatorname {haf} (A)=\mathbb {E} _{\left(X_{1},\dots ,X_{m}\right)\sim {\mathcal {N}}\left(0,A+\lambda I\right)}\left[X_{1}\dots X_{m}\right]}$

and

${\displaystyle \operatorname {lhaf} (A)=\mathbb {E} _{\left(X_{1},\dots ,X_{m}\right)\sim {\mathcal {N}}\left(\operatorname {diag} \left(A\right),A+\lambda I\right)}\left[X_{1}\dots X_{m}\right],}$

where ${\displaystyle \lambda }$ is any number large enough to make ${\displaystyle A+\lambda I}$ positive semi-definite (since this expectation does not depend on ${\displaystyle \lambda }$).

Computation

Computing the hafnian of a (0,1)-matrix is #P-complete, because computing the permanent of a (0,1)-matrix is #P-complete.^[5][7]

The hafnian of a ${\displaystyle 2n\times 2n}$ matrix can be computed in ${\displaystyle O(n^{3}2^{n})}$ time.^[7]

If the entries of a matrix are non-negative, then its hafnian can be approximated to within an exponential factor in polynomial time.^[8]

See also

• Permanent
• Pfaffian
• Boson sampling

References