An Extremal Spectral Problem for Triangle-Free Graphs Arising from Quantum Transport

Abstract

For a graph $G$ of order $n$ with adjacency matrix $A$, let $F_G(t)$ be the average of $|(\exp(-\ii tA))_{vu}|^2$ over distinct ordered vertex pairs. Under the dense scaling $t=τ/n$, the quantities $n^2F_G(τ/n)$ lead to a graphon functional $Φ_τ$ whose leading term is $τ^2$ times the edge density and whose remaining terms form a weighted alternating series of even cycle densities. For $0\leτ\leτ_{\mathrm c}$, we determine the exact maximum of $Φ_τ$ over all triangle-free graphons. The balanced complete bipartite graphon $B_1$ is the unique maximizer, up to weak isomorphism, when $0<τ\leτ_{\mathrm c}$, where $τ_{\mathrm c}$ is the unique positive solution of \[ τ_{\mathrm c}=4\sin(τ_{\mathrm c}/2), \qquad τ_{\mathrm c}\approx3.79099, \] and the maximum equals $4(1-\cos(τ/2))$. This threshold is sharp: $B_1$ is not globally optimal for $τ>τ_{\mathrm c}$. For $τ>τ_{\mathrm c}$, the unique maximizer within the bipartite class, up to weak isomorphism, is the balanced bipartite graphon $B_{q_τ}$, where $q_τ\in(0,1)$; the unrestricted maximization problem beyond $τ_{\mathrm c}$ remains open. We also prove an explicit edge density deficit bound and quantitative cut distance stability, uniform for $τ$ in compact subintervals of $(0,τ_{\mathrm c})$, together with qualitative cut distance stability on compact subintervals of $(0,τ_{\mathrm c}]$. The corresponding finite triangle-free extremal values converge locally uniformly to the graphon maximum, with an $O(n^{-1})$ error uniformly on $[0,τ_{\mathrm c}]$. The proof uses a coefficient criterion for spectral graphon functionals and combines a sixth-degree spectral minorant with a four-vertex inequality and a six-vertex moment inequality; the latter is established by an exact rational flag algebra certificate.