Off-Diagonal Commonality of a Shorter Even Cycle and a Longer Odd Cycle
Source: hexagonmath.org/abs/2611.00066v1
Let $n\ge4$ be even and let $\ell>n$ be odd. We determine the complete off-diagonal commonality region of the ordered pair $(C_n,C_\ell)$: it is the half-open interval $(0,a_{n,\ell}^{\star}]$ cut out by the unique root $a_{n,\ell}^{\star}\in(1/2,1)$ of $\frac{a^{n-1}\bigl(n+(\ell-n)a\bigr)}{\ell}=2^{1-n}.$ The upper endpoint is witnessed by the balanced disjoint union of two cliques. For the lower bound, cycle densities are expressed as traces of graphon operators. Rank-one eigenvalue majorisation separates the Perron eigenvalue from the rest of the spectrum, while the Hilbert--Schmidt norm…
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