Erdös Matrix: Extremal Bistochastic Theory

• Erdös matrices are extremal bistochastic matrices defined by the equality between their Frobenius norm squared and maximal permutation trace, bridging combinatorial and spectral theory.
• They exhibit unique structural properties such as rational entries, finite classification per matrix size, and distinct zero-pattern invariants that aid in precise enumeration.
• Algorithmic construction of Erdös matrices utilizes permutation expansions and Gram matrix systems, efficiently generating solutions up to moderate dimensions.

An Erdös matrix refers to two distinct but well-established mathematical notions—one rooted in extremal combinatorial matrix theory (arising from questions of Erdős and the Marcus–Ree inequality for bistochastic matrices), and the other denoting the weighted adjacency matrix ensemble of sparse Erdös–Rényi random graphs in spectral and quantum physics contexts. Both classes are characterized by sharp structural and spectral properties, distinct enumeration techniques, and specialized invariants. This article systematically presents the combinatorial (“Erdös matrix” of Marcus–Ree type) theory, with precise definitions, classification results, and enumeration algorithms; connections to related matrix classes (e.g., RCDS matrices, Birkhoff polytope faces); and illustrative examples in low dimension.

1. Definition and Characterization

Let $A\in\mathbb{R}^{n\times n}$. A bistochastic (or doubly stochastic) matrix satisfies:

• $A_{ij}\ge 0$ for all $i,j$,
• $\sum_{j=1}^n A_{ij} = 1$ for all rows $i$,
• $\sum_{i=1}^n A_{ij} = 1$ for all columns $j$.

The Frobenius norm squared is given by

$\|A\|_F^2 = \sum_{i,j=1}^n A_{ij}^2.$

The maximal trace (maxtrace) over all permutations $\sigma\in S_n$ (the symmetric group) is

$$\mathrm{maxTrace}(A) = \max_{\sigma\in S_n}\sum_{i=1}^n A_{i,\sigma(i)}.$$

The Marcus–Ree inequality states

$$\|A\|_F^2 \le \mathrm{maxTrace}(A) \quad \text{for all bistochastic } A.$$

A bistochastic $A$ is called an Erdös matrix if equality holds: $\|A\|_F^2 = \mathrm{maxTrace}(A),$ equivalently $\Delta(A) = \mathrm{maxTrace}(A) - \|A\|_F^2 = 0$ (Tripathi, 2024, Karmakar et al., 4 Dec 2025, Kushwaha et al., 12 Mar 2025).

2. Structural Properties and Permutation Expansion

Any bistochastic matrix is a convex combination of permutation matrices by the Birkhoff–von Neumann theorem: $$A = \sum_{i=1}^m x_i P_i,\quad P_i\text{ permutation matrices},\ x_i>0,\ \sum_i x_i=1.$$ For an Erdös matrix, these coefficients must solve a specific linear system involving the Gram matrix $M_{ij} = \langle P_i, P_j\rangle_F$: $$M\mathbf{x} = \langle M\mathbf{x}, \mathbf{x}\rangle\, \mathbf{1}_m.$$ If $\{P_1,\dots,P_m\}$ is linearly independent, set $\mathbf{y} = M^{-1}\mathbf{1}_m$, then $$\mathbf{x} = \mathbf{y}/\langle \mathbf{1}_m, \mathbf{y}\rangle$$ (Tripathi, 2024, Kushwaha et al., 12 Mar 2025). All Erdös matrices arise as unique barycenters of such linearly independent sets.

Notably, every Erdös matrix has rational entries, as $M$ is integer-valued, making $M^{-1}$ and thus $\mathbf{x}$ and $A$ rational (Tripathi, 2024).

3. Enumeration, Finiteness, and Zero-Pattern Uniqueness

There are only finitely many $n\times n$ Erdös matrices for each $n$, with an explicit upper bound: $$\left| \{A\in B_n: \|A\|_F^2 = \mathrm{maxTrace}(A)\} \right| \leq \sum_{j=1}^{(n-1)^2+1} \binom{n!}{j}.$$ This follows from Carathéodory’s theorem and uniqueness of the barycenter for each affinely independent subset of permutation matrices (Tripathi, 2024, Kushwaha et al., 12 Mar 2025).

Zero-patterns (“skeletons”): Given $M$, its skeleton is the binary matrix $S$ with $S_{ij} = 1$ if $M_{ij}\neq 0$, $0$ otherwise. Each skeleton supports at most one Erdös matrix, and if $\skel(E_1) < \skel(E_2)$ entrywise, then $\mathrm{maxTrace}(E_1) > \mathrm{maxTrace}(E_2)$. Thus each admissible skeleton determines at most one Erdös matrix (Karmakar et al., 4 Dec 2025).

4. Algorithmic Construction

Enumeration of all Erdös matrices up to equivalence can be carried out as follows (Tripathi, 2024, Karmakar et al., 4 Dec 2025):

1. List all linearly independent subsets $\{P_1,\ldots,P_m\}\subset P_n$ with $1\leq m\leq (n-1)^2+1$.
2. For each:
   • Form the Gram matrix $M$.
   • Solve $M\mathbf{y}=\mathbf{1}$; normalize as $$\mathbf{x} = \mathbf{y}/\langle \mathbf{1},\mathbf{y} \rangle$$.
   • Form $A=\sum_i x_iP_i$; verify $A\ge0$, sums = $1$, and $\|A\|_F^2 = \mathrm{maxTrace}(A)$.
   • Factor out permutation equivalences.

A complementary approach is enumeration by skeletons: For each skeleton, list compatible permutations, solve the Gram system, and verify the bistochastic and equality constraints (Karmakar et al., 4 Dec 2025). In practice, this is effective for $n\le6$; the number of classes grows rapidly.

5. Connections to RCDS Matrices and Related Matrix Classes

An RCDS matrix (“restricted common diagonal sum”) is a bistochastic matrix for which all “inner” permutation traces (on the skeleton) are equal. Every Erdös matrix is RCDS (with the common inner trace equal to both $\|E\|_F^2$ and maxtrace), and an RCDS bistochastic matrix $E$ is Erdös if and only if no “outer” trace exceeds the inner trace (Karmakar et al., 4 Dec 2025). This links the extremal structure of Erdös matrices to properties of the Birkhoff polytope’s faces and to Brualdi–Dahl’s RCDS classification.

6. Explicit Examples and Enumerative Data

Small-Dimension Erdös Matrices

$n$	Number of Classes	Representative Matrices
2	2	$I_2$, $J_2 = \frac12 \begin{pmatrix}1&1\1&1\end{pmatrix}$
3	6	$I_3$, $J_3=\frac13\text{all 1s}$, $I_1\oplus J_2$, $T$, $S$, $R$
4	41	See (Kushwaha et al., 12 Mar 2025), examples with various denominators

Explicit form for $n=3$ (see (Tripathi, 2024, Karmakar et al., 4 Dec 2025, Kushwaha et al., 12 Mar 2025)): $T = \frac12\left(3J_3 - I_3\right) = \begin{pmatrix}0&\tfrac12&\tfrac12\\tfrac12&0&\tfrac12\\tfrac12&\tfrac12&0\end{pmatrix}, \quad S = \begin{pmatrix}0&\tfrac12&\tfrac12\\tfrac12&\tfrac14&\tfrac14\\tfrac12&\tfrac14&\tfrac14\end{pmatrix}, \quad R = \begin{pmatrix}\tfrac35&0&\tfrac25\0&\tfrac35&\tfrac25\\tfrac25&\tfrac25&\tfrac15\end{pmatrix}$

For $n=4$, a typical matrix (from the complete list of 41 classes) is: $E_4 = \frac1{43}\begin{pmatrix} 2&7&15&19\7&12&0&24\15&0&28&0\19&24&0&0 \end{pmatrix}$ (Karmakar et al., 4 Dec 2025, Kushwaha et al., 12 Mar 2025).

7. Open Problems and Extensions

• Precise enumeration: The growth rate of the number of non-equivalent Erdös matrices as $n$ increases is open; lower bound $p(n)$ (number of partitions of $n$), but actual asymptotics are unknown (Kushwaha et al., 12 Mar 2025).
• Efficient generation: For $n>6$ the combinatorial explosion of skeletons and permutation subsets presents significant computational obstacles.
• Alpha-Erdös matrices: For fixed $n$ and $\alpha\in(0,(n-1)/4)$, there are uncountably many symmetric matrices $A$ with $\mathrm{maxTrace}(A)-\|A\|_F^2=\alpha$; only for $\alpha=0$ or $(n-1)/4$ is the solution set finite or unique (Kushwaha et al., 12 Mar 2025).
• Infinite-dimensional and kernel analogues: The Marcus–Ree inequality generalizes to infinite bistochastic arrays and continuous bistochastic kernels on $[0,1]^2$ (Kushwaha et al., 12 Mar 2025).

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The theory of Erdös matrices occupies a central position at the intersection of combinatorial optimization, matrix extremal theory, and polytope geometry. It provides a complete description of extremal bistochastic matrices for the Marcus–Ree bound, exact structural and enumeration theory, and links to broader classes such as RCDS and Birkhoff polytope faces. Recent results have resolved small-dimensional cases completely and established rationality and finiteness for each $n$, with algorithmic and enumerative methods extending steadily to higher-dimensions and related matrix families (Tripathi, 2024, Karmakar et al., 4 Dec 2025, Kushwaha et al., 12 Mar 2025).