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For an equal-part biregular graph with part size $n$ and degree $\delta n$, let $M$ be its biadjacency matrix and $H=M-\delta J$, where $J$ is the all-ones matrix. The constant row and column sums make $H$ annihilate constants on both sides. Since $M{M}^{\mathsf{T}}={\delta }^{2}nJ+H{H}^{\mathsf{T}}$ and the two summands have zero products, their squared matrix trace splits as displayed. The bipartite four-cycle density counts the squared matrix trace of $M{M}^{\mathsf{T}}$ divided by $n^4$.

Use the part-preserving bipartite four-cycle density conventionHere $G(x,y)$ is the indicator function of adjacency, and repeated labels are allowed. This is the homomorphism density convention appropriate to the four-cycle norm. We derive the needed counting and operator norm facts directly.

Let $M=(G(x,y))$ be the $n$-by-$n$ biadjacency matrix, let $1$ be the all-ones vector, let $J=11^{\mathsf{T}}$, and put $H=M-\delta J$. The degree hypothesis saysso the centered matrix $H$ annihilates $1$ on both sides. In particular, $HJ=J{H}^{\mathsf{T}}=0$, andThe two terms have zero products in either order. Expanding the matrix trace directly counts four adjacency factors:Since $J^2=nJ$ and $\mathrm{tr}J=n$, we obtainThe given upper bound therefore impliesThis computation proves the centered four-cycle identity for a biregular graph, rather than assuming a four-cycle norm identity.

The matrix $Q=H{H}^{\mathsf{T}}$ is a positive semidefinite matrix. By the spectral theorem for real symmetric matrices it has nonnegative eigenvalues ${\lambda }_j$, andMoreover, for the Euclidean norm,Indeed $\Vert H^{\mathsf{T}}v{\Vert }_2^2=v^{\mathsf{T}}Qv$, whose maximum on unit vectors is ${\lambda }_{\max }$; the Cauchy-Schwarz inequality characterization $\Vert H{\Vert }_{\mathrm{op}}={\sup }_{\Vert u{\Vert }_2=\Vert v{\Vert }_2=1}|u^{\mathsf{T}}Hv|$ also shows that $H$ and $H^{\mathsf{T}}$ have the same operator norm. Taking fourth roots thus giveswhere the error parameter $c$ is nonnegative, as in the claimed bound.

Finally use the indicator vectors $a=1_A$, $b=1_B$. Their Euclidean norms are $\sqrt{|A|}$ and $\sqrt{|B|}$, so the Cauchy-Schwarz inequality and the operator norm estimate yieldThis is the required discrepancy bound. It also holds when either subset is empty or $\delta =0$. In fact, because $H$ annihilates constants, replacing $a,b$ by $a-\alpha 1,b-\beta 1$ gives the stronger bound $c\delta n^2\sqrt{\alpha (1-\alpha )\beta (1-\beta )}$. This is the spectral discrepancy bound for a biregular graph.

A perfect matching is a matching in a graph that covers every vertex exactly once. Its edges have no shared endpoints, and every vertex is an endpoint of one selected edge. It is useful when a collection must be divided into disjoint pairs with no element left over.

In a bipartite graph with equally many vertices on its two sides, a perfect matching can assign every person one allowed job and every job one person. The example under permanent of a matrix has three people and three jobs, with exactly three valid assignments. The blue edges below show those three perfect matchings; a crossing in the drawing is not an extra vertex.

Every selected edge covers two vertices, so a graph with an odd integer number of vertices has no perfect matching. An even integer number is necessary but does not guarantee existence: an isolated vertex cannot be covered. A perfect matching also need not be unique. For example, the complete bipartite graph $K_{2,2}$ has two perfect matchings, corresponding to the two ways of assigning two people to two jobs.

Counting all perfect matchings is a different question from finding one. For a zero-one biadjacency matrix of a balanced bipartite graph, the count is its permanent of a matrix. The nonzero determinant criterion for a perfect matching supplies an existence test, but cancellation means that a zero determinant does not rule one out.

The permanent of a matrix adds the products obtained by choosing one entry in each row and one in each column. For an $n\times n$ matrix $A=(a_{ij})$ with entries in the real numbers or complex numbers,Here each permutation $\sigma$ chooses column $\sigma (i)$ in row $i$, and the symmetric group $S_n$ lists every such bijection. The formula resembles the Leibniz formula for determinants, but omits the sign of a permutation. For a two-by-two matrix,The absence of alternating signs makes the permanent of a matrix particularly useful for counting allowed assignments.

For a zero-one matrix, let rows represent people, columns represent jobs, and $a_{ij}=1$ mean that person $i$ can do job $j$. This is the biadjacency matrix of a bipartite graph. A product in the permanent is one exactly when all the selected assignments are allowed; otherwise it is zero. Each surviving permutation assigns every person a different job, which is a perfect matching. Thus the permanent counts perfect matchings. This interpretation and the definition are given in Yufei Zhao, Probabilistic Methods in Combinatorics, §10.2, pp. 178–179.

For a fully enumerated example, takeAmong the six permutations of three columns, exactly three give nonzero products: $(1,2,3)$, $(1,3,2)$ and $(2,1,3)$. These tuples list the chosen column in each successive row. Each product is one, so $\mathrm{perm}A=3$. Their permutation signs are $+1,-1,-1$, giving $\det A=-1$. The diagram displays all three allowed assignments.

A zero determinant need not mean that no assignment exists. The two-by-two all-ones matrix has permanent $2$ and determinant $0$: its two assignments cancel in the determinant. The nonzero determinant criterion for a perfect matching is sufficient for existence, but the converse fails for a fixed zero-one matrix.

For nonnegative entries, the permanent instead sums assignment weights, with each assignment weighted by the product of its selected entries. It is then a total weight rather than necessarily an integer count. With arbitrary entries in the real numbers or complex numbers, even the permanent can have cancellation; the counting interpretation specifically uses zero-one entries. The permanent as probability of distinct independent choices gives another useful nonnegative interpretation.

There are $n!$ terms in the defining sum, linking direct evaluation to the factorial. For the all-ones $n\times n$ matrix, every term is one and the permanent is exactly $n!$; the identity matrix has permanent one. Finding one allowed assignment and counting all allowed assignments are different questions. Their relationship to computational complexity theory was studied in L. G. Valiant, The complexity of computing the permanent, Theoretical Computer Science 8(2), 189–201, 1979. The elementary sum is useful for small examples, while the existing dense permanent approximation describes a more specialized way of estimating large counts.

Let $H=M-\delta J$ be the centered biadjacency matrix of an equal-part biregular graph, and put $s=\Vert H{\Vert }_{\mathrm{op}}$. For subsets $A,B$ of the two parts, their edge discrepancy is $1_A^{\mathsf{T}}H{1}_B$. The Cauchy-Schwarz inequality proves the displayed bound. If the part size is $n$, centering the two indicator vectors improves it to $sn\sqrt{\alpha (1-\alpha )\beta (1-\beta )}$. The centered four-cycle identity for a biregular graph shows that $t_4(G)\le {\delta }^4(1+c^4)$ implies $s\le c\delta n$ for $c\ge 0$.