Let sim_1 and sim_2 be equivalence relations on [n]=1,…,n, let P_1 and P_2 be the groups of permutations preserving their respective equivalence classes, and let A be an n× n Hermitian positive semidefinite matrix. Define f_sim_1,sim_2(A)=∑_sigma_1∈ P_1∑_sigma_2∈ P_2∏_i=1^n A_sigma_1(i),sigma_2(i). Matrix-analysis conjecture. Under these hypotheses, operatornameRe(f_sim_1,sim_2(A))≥| P_1 cap P_2|∏_i=1^n A_ii. The paper states that this conjecture implies the tree real-part conjecture; no proof or resolution is supplied.

Matrix-analysis inequality implying the tree real-part conjecture

From papers

Let $\sim_1$ and $\sim_2$ be equivalence relations on $[n]=\{1,\ldots,n\}$ , let $P_1$ and $P_2$ be the groups of permutations preserving their respective equivalence classes, and let $A$ be an $n\times n$ Hermitian positive semidefinite matrix. Define

f_{\sim_1,\sim_2}(A)=\sum_{\sigma_1\in P_1}\sum_{\sigma_2\in P_2}\prod_{i=1}^n A_{\sigma_1(i),\sigma_2(i)}.

Matrix-analysis conjecture. Under these hypotheses,

\operatorname{Re}(f_{\sim_1,\sim_2}(A))\geq\lvert P_1\cap P_2\rvert\prod_{i=1}^n A_{ii}.

The paper states that this conjecture implies the tree real-part conjecture; no proof or resolution is supplied.

Progress summary

Open

No public discussion or published progress on this conjecture was found.

No public discussion or published progress addressing this conjecture was found in the retrieved sources.

Current status (as of August 2026): The conjecture appears open, with no recorded public activity or resolution.

Sources & referencesView supporting material

Primary source

Joseph Malkoun, “Finite graphs and configurations of points”, arXiv:2508.13472 (2026).

matrix-analysis positive-semidefinite-matrices

1 week ago

For equivalence relations ~₁,~₂ on [n], let P₁,P₂ be their class-preserving permutation groups and define f(A)=Σ_{σ∈P₁,τ∈P₂} ∏{j=1}^n A{σ(j),τ(j)}. Consider the Hermitian matrix B = [[1, 1/2, 7i/10], [1/2, 1, 1/2], [-7i/10, 1/2, 1]]. Its leading principal minors are 1, 3/4, and 1/100, so B is strictly positive definite. On one three-element block let ~₁ have classes {1,2},{3} and ~₂ have classes {1},{2,3}. Then P₁={e,(12)}, P₂={e,(23)}, P₁∩P₂={e}, and direct evaluation of the four terms gives f(B)=1+1/4+1/4+7i/40=(60+7i)/40.

Now let n=42 and take A=B⊕⋯⊕B with fourteen blocks. On each block use the same two equivalence relations, independently. Their class-preserving groups are products of the block groups, their intersection is the identity, every diagonal entry of A is 1, and the defining double permutation sum factors: f(A)=((60+7i)/40)^14. Exact integer arithmetic gives Re (60+7i)^14 = −475148137808619635065249, 40^14 = 26843545600000000000000. Consequently Re f(A)=−475148137808619635065249/26843545600000000000000 < 0 < 1 = |P₁∩P₂| ∏{j=1}^{42} A{jj}. Thus the conjecture fails even for a strictly positive definite Hermitian correlation matrix. The source separately discusses a related disconnected-power mechanism for graph amplitudes, but its matrix conjecture has no connectedness restriction; this counterexample concerns that matrix conjecture and does not disprove the separate conjecture about trees.

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Shivam Patel · 22 hours ago