The minimal spectral radius with given independence number

In this paper, we determine the graphs which have the minimal spectral radius among all the connected graphs of order $n$ and the independence number $\lceil\frac{n}{2}\rceil-1.$Comment: 14 page

Log BPS numbers of log Calabi-Yau surfaces

Let $(S,E)$ be a log Calabi-Yau surface pair with $E$ a smooth divisor. We define new conjecturally integer-valued counts of $\mathbb{A}^1$-curves in $(S,E)$. These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along $E$ via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.Comment: 49 pages, 2 figure