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.