2026/07/19 by Vaibhav Suvagiya · 1 citation
#math.CO #cs.DM #math.SP
For the circulant graph Cn(1,2) with n≥10 even, the \F2 system requiring every quadrilateral to be unbalanced is consistent and its solutions form exactly four switching classes. We show that the class containing the signing which is +1 on step-1 edges and (-1)i on step-2 edges has spectrum \±2√(cos2θk+cos2 2θk)\ and spectral radius exactly 2√2, well below the Kesten bound 2√3; that the quadrilateral system is equivalent to alternating triangle fluxes, so that the four classes are coordinatized by (τ0,α) and the spectral radius depends only on the Hamilton-cycle holonomy α; and that the two twisted classes attain ρ-(n)=2√(cos2(π/n)+cos2(2π/n))<2√2. Exhaustive enumeration of all 2n+1 switching classes for n∈\8,10,12,14,16,18\ shows that ρ-(n) is the global minimum in every case, and we conjecture this for all even n; the lower bound is a flux-minimization statement in the sense of Lieb's flux-phase theorem. For odd n the quadrilateral system is inconsistent.