Median eigenvalues of bipartite graphs

For a graph $G$ of order $n$ and with eigenvalues $\lambda_1\geqslant\cdots\geqslant\lambda_n$, the HL-index $R(G)$ is defined as $R(G) ={\max}\left\{|\lambda_{\lfloor(n+1)/2\rfloor}|, |\lambda_{\lceil(n+1)/2\rceil}|\right\}.$ We show that for every connected bipartite graph $G$ with maximum degree $\Delta\geqslant3$, $R(G)\leqslant\sqrt{\Delta-2}$ unless $G$ is the the incidence graph of a projective plane of order $\Delta-1$. We also present an approach through graph covering to construct infinite families of bipartite graphs with large HL-index

Tropical secant graphs of monomial curves

The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is represented by a balanced graph. Our main result is an explicit construction of that graph. As a consequence, we obtain algorithms to effectively compute the multidegree and Chow polytope of an arbitrary projective monomial curve. This generalizes an earlier degree formula due to Ranestad. The combinatorics underlying our construction is rather delicate, and it is based on a refinement of the theory of geometric tropicalization due to Hacking, Keel and Tevelev.Comment: 30 pages, 8 figures. Major revision of the exposition. In particular, old Sections 4 and 5 are merged into a single section. Also, added Figure 3 and discussed Chow polytopes of rational normal curves in Section

Logarithmic Moduli Spaces for Surfaces of Class VII

In this paper we describe logarithmic moduli spaces of pairs (S,D) consisting of minimal surfaces S of class VII with positive second Betti number b_2 together with reduced divisors D of b_2 rational curves. The special case of Enoki surfaces has already been considered by Dloussky and Kohler. We use normal forms for the action of the fundamental group of the complement of D and for the associated holomorphic contraction germ from (C^2,0) to (C^2,0).Comment: Minor correction of the dimension of the moduli spac