Determine the extremal normalized least signless Laplacian eigenvalue for odd girth seven

Determine the exact value of \(\beta_7\), the supremum of \(q_{}(G)/|V(G)|\) over all finite graphs \(G\) with odd girth at least seven, thereby resolving whether the folded 7-cube lower bound is optimal and identifying the sharp upper coefficient.

Background

For every odd $g\ge 7$, the paper defines $\beta_g$ as the supremum of the least signless Laplacian eigenvalue divided by the number of vertices, taken over graphs of odd girth at least $g$. The main theorem establishes the upper bound $\beta_7<569/10000$, while the folded 7-cube provides the lower bound $\beta_7\ge 1/32$.

The folded 7-cube is a 7-regular graph on 64 vertices whose least signless Laplacian eigenvalue is 2, yielding the normalized value $1/32$; balanced blow-ups produce arbitrarily large examples with the same ratio. The exact extremal value between these bounds, and consequently the optimality of the paper’s upper coefficient, is left unresolved.