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 $t=\tau/n$, the quantities $n^2F_G(\tau/n)$ lead to a graphon functional $\Phi_\tau$ whose leading term is $\tau^2$ times the edge density and whose remaining terms form a weighted alternating series of even cycle densities. For $0\le\tau\le\tau_{\mathrm c}$, we determine the exact maximum of $\Phi_\tau$ over all triangle-free graphons. The balanced complete bipartite graphon $B_1$ is the unique maximizer, up to weak isomorphism, when $0<\tau\le\tau_{\mathrm c}$, where $\tau_{\mathrm c}$ is the unique positive solution of \[ \tau_{\mathrm c}=4\sin(\tau_{\mathrm c}/2), \qquad \tau_{\mathrm c}\approx3.79099, \] and the maximum equals $4(1-\cos(\tau/2))$. This threshold is sharp: $B_1$ is not globally optimal for $\tau>\tau_{\mathrm c}$. For $\tau>\tau_{\mathrm c}$, the unique maximizer within the bipartite class, up to weak isomorphism, is the balanced bipartite graphon $B_{q_\tau}$, where $q_\tau\in(0,1)$; the unrestricted maximization problem beyond $\tau_{\mathrm c}$ remains open. We also prove an explicit edge density deficit bound and quantitative cut distance stability, uniform for $\tau$ in compact subintervals of $(0,\tau_{\mathrm c})$, together with qualitative cut distance stability on compact subintervals of $(0,\tau_{\mathrm c}]$. The corresponding finite triangle-free extremal values converge locally uniformly to the graphon maximum, with an $O(n^{-1})$ error uniformly on $[0,\tau_{\mathrm c}]$. 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.