5 - Structure and one eigenvalue

Summary

In Chapters 3 and 4 we have concentrated on the relation between the structure and spectrum of a graph. Here we discuss the connection between structure and a single eigenvalue, and for this the central notion is that of a star complement. In Section 5.1 we define star complements both geometrically and algebraically, and note their basic properties. In Section 5.2 we illustrate a technique for constructing and characterizing graphs by star complements. In Section 5.3 we use star complements to obtain sharp upper bounds on the multiplicity of an eigenvalue different from −1 or 0 in an arbitrary graph, and in a regular graph. In Section 5.4 we describe how star complements can be used to determine the graphs with least eigenvalue −2, and in Section 5.5 we investigate the role of certain star complements in generalized line graphs.

Star complements

Let G be a graph with vertex set V(G) ={1, …, n} and adjacency matrix A. Let {e₁, …, e_n} be the standard orthonormal basis of IRⁿ and let P be the matrix which represents the orthogonal projection of IRⁿ onto the eigenspace ε(μ) of A with respect to {e₁, …, e_n}. Since ε(μ) is spanned by the vectors P e_j (j =1, …, n) there exists X ⊆V(G) such that the vectors P e_j (j ∈ X) form a basis for ε(μ). Such a subset X of V(G) is called a star set for μ in G.