The Study Of Some Problems On Star Complement And Strong Positive Definite Tensor

The research on star complements originated independently in papers by Peter Rowlinson and M.N. Ellingham. If ? is an eigenvalue of G of multiplicity k,then a star set for ? in G is a subset X of V(G) such that |X| = k and the induced subgraph G - X does not have ? as an eigenvalue. The induced subgraph G - X is called a star complement for ? in G. The theory of star complement has been used to study graphs with least eigenvalue -2, structure properties of strongly regular graphs and the multiplicities of graph eigenvalues.In 2005, Liqun Qi gave the definition of the eigenvalue of tensor. In 2012, Jiayu Shao gave a definition of tensor product. In recent years, some papers about the adjacency tensor and Laplace tensor of hypergraph have appeared. The study of the spectral of tensor is at beginning,we consider the eigenvalue of tensor based on the strong positive tensor. The main results of this paper are as follows:(1) Give a new proof of the Reconstruction Theorem for star complement;(2) Generalize the definition of star complement with adjacency matrix to real symmetric matrix;(3) The upper bounds on eigenvalue multiplicity of a specific graph is given, and the limit case is characterized;(4) The upper bounds on eigenvalue multiplicity of trees are given, which generalized the related results, and the limit case is characterized;(5) The upper bounds on eigenvalue multiplicity of unicyclic graphs are given;(6) Generalize the definition of product of tensors;(7) The definition and Hadamard inequation of strong positive definite tensor is obtained.