Regular and Maximal Graphs with Prescribed Tripartite Graph as a Star Complement

Let G be a graph of order n andμbe an adjacency eigenvalue of G with multiplicity k≥1.A star complement H forμin G is an induced subgraph of G of order n-k with no eigenvalueμ,and the subset X=V(G-H)is called a star set forμin G.The star complement provides a strong link between graph structure and linear algebra.In this paper,the authors characterize the regular graphs with K2,2,s(s≥2)as a star complement for all possible eigenvalues,the maximal graphs with K2,2,s as a star complement for the eigenvalueμ=1,and propose some questions for further research.

Sharp bounds for spectral radius of nonnegative weakly irreducible tensors

We obtain the sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor.By using the technique of the representation associate matrix of a tensor and the associate directed graph of the matrix,the equality cases of the bounds are completely characterized by graph theory methods.Applying these bounds to a nonnegative irreducible matrix or a connected graph(digraph),we can improve the results of L.H.You,Y.J.Shu,and P.Z.Yuan[Linear Multilinear Algebra,2017,65(1):113-128],and obtain some new or known results.Applying these bounds to a uniform hypergraph,we obtain some new results and improve some known results of X.Y.Yuan,M.Zhang,and M.Lu[Linear Algebra Appl.,2015,484:540-549].Finally,we give a characterization of a strongly connected/c-uniform directed hypergraph,and obtain some new results by applying these bounds to a uniform directed hypergraph.

Character Modelling with Sketches and ODE-Based Shape Creation

Character models have enormous applications in industry.Efficient creation of detailed character models is an important topic.This paper proposes a new and easy-to-use technique to quickly create detailed character models from sketches.The proposed technique consists of two main components:primitive deformer and shape generators.With this technique,2D silhouette contours of a character model are drawn or extracted from an image or sketch.Then,proper geometric primitives are selected and aligned with the corresponding 2D silhouette contours.After that,a primitive deformer is used to create a base mesh and three shape generators are used to add 3D details to the base mesh.The primitive deformer and three shape generators are developed from ODE-driven deformations.The primitive deformer deforms the aligned geometric primitives to exactly match the 2D silhouette contours in one view plane and obtains a base mesh of a character model consisting of deformed primitives.The shape generators are used to add 3D details to the base mesh by creating local 3D models.The experimental results demonstrate that the new technique can quickly create detailed 3D character models from sketches with few manual operations.The new technique is physics-based and easy to learn and use.