In this paper, we consider the Turán problems on {1, 3}-hypergraphs. We prove that a {1, 3}-hypergraph is degenerate if and only if it's His a hypergraph with vertex set V = [5] and edge set E = {{2}, {3}, {1, 2, 4}, {1, 3, 5}, {1, 4, 5}}. Using this result, we further prove that for any finite set R of distinct positive integers, except the case R = {1, 2}, there always exist non-trivial degenerate R-graphs. We also compute the Turán densities of some small {1, 3}-hypergraphs.

For r ≥ 3, let fr : [0, ∞) → [1, ∞) be the unique analytic function such that fr( k r ) = k−1 r−1 for any k ≥ r − 1. We prove that the spectral radius of an r-uniform hypergraph H with e edges is at most fr(e). The equality holds if and only if e = k r for some positive integer k and H is the union of a complete r-uniform hypergraph K r k and some possible isolated vertices. This result generalizes the classical Stanley's theorem on graphs. √ 1+8e−1 2. The equality holds if and only if e = k 2 and G is the union of the complete graph K k and some isolated vertices. Friedland [7] proved a bound which is tight on the complete graph with one, two, or three edges removed or the complete graph with one edge added. Rowlinson [17] finally confirmed Brualdi and Hoffman's conjecture, and proved that G e attains the maximum spectral radius among all graphs with e edges.

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Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straightline drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tanglegram is the minimum crossing number over all such drawings and is related to biologically relevant quantities, such as the number of times a parasite switched hosts.Our main results for tanglegrams which parallel known theorems for crossing numbers are as follows. The removal of a single matching edge in a tanglegram with n leaves decreases the tangle crossing number by at most n − 3, and this is sharp. Additionally, if γ(n) is the maximum tangle crossing number of a tanglegram with n leaves, we prove 1 2 n 2 (1 − o(1)) ≤ γ(n) < 1 2 n 2 . Further, we provide an algorithm for computing non-trivial lower bounds on the tangle crossing number in O(n 4 ) time. This lower bound may be tight, even for tanglegrams with tangle crossing number Θ(n 2 ).

In this paper, we generalize Lin-Lu-Yau's Ricci curvature to weighted graphs and give a simple limit-free definition. We prove two extremal results on the sum of Ricci curvatures for weighted graph.A weighted graph G = (V, E, d) is an undirected graph G = (V, E) associated with a distance function d : E → [0, ∞). By redefining the weights if possible, without loss of generality, we assume that the shortest weighted distance between u and v is exactly d(u, v) for any edge uv. Now consider a random walk whose transitive probability from an vertex u to its neighbor v (a jump move along the edge uv) is proportional to wuv := F (d(u, v))/d(u, v) for some given function F (•). We first generalize Lin-Lu-Yau's Ricci curvature definition to this weighted graph and give a simple limit-free representation of κ(x, y) using a so called * -coupling functions. The total curvature K(G) is defined to be the sum of Ricci curvatures over all edges of G. We proved the following theorems: ifBoth equalities hold if and only if d is a constant function plus the girth is at least 6.In particular, these imply a Gauss-Bonnet theorem for (unweighted) graphs with girth at least 6, where the graph Ricci curvature is defined geometrically in terms of optimal transport.