Given an SO(3)-bundle with connection, the associated twosphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov [34]. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c1 = 0 which are never K¨ahler; e.g., we produce such manifolds with b1 = 0 = b3 and also with c2 · [ω] < 0, answering questions posed by Smith–Thomas–Yau [37]. Examples come from Riemannian geometry, via the Levi–Civita connection on +. The underlying six-manifold is then the twistor space and often the symplectic structure tames the Eells–Salamon twistor almost complex structure. Our second aim is to exploit this to deduce new results about minimal surfaces: if a certain curvature inequality holds, it follows that the space of minimal surfaces (with fixed topological invariants) is compactifiable; the minimal surfaces must also satisfy an adjunction inequality, unifying and generalising results of Chen–Tian [6]. One metric satisfying the curvature inequality is hyperbolic four-space H4. Our final aim is to show that the corresponding symplectic manifold is symplectomorphic to the small resolution of the conifold xw −yz = 0 in C4. We explain how this fits into a hyperbolic description of the conifold transition, with isometries of H4 acting symplectomorphically on the resolution and isometries of H3 acting biholomorphically on the smoothing.

The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let T(n,D
l) be the tree which minimizes the spectral radius of all trees of order n with exactly l vertices of maximum degree D
. In this paper, T(n,D
l) is determined for D=3, and for l<= 3 and n large enough. It is proven that for sufficiently large n, T(n, 3, l) is a caterpillar with (almost) uniformly distributed legs, T(n,D
2) is a dumbbell, and T(n,
D,3) is a tree consisting of three distinct stars of order D
connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lov´asz and Pelik´an, Simi´c and To˘si´c, Wu, Yuan and Xiao, and Xu, Lin and Shu.

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