On the extensional eigenvalues of graphs

Abstract Assume that G is a graph on n vertices with associated symmetric matrix M and K a positive definite symmetric matrix of order n . If there exists 0 ≠ x ∈ R n such that M x = λ K x , then λ is called an extensional eigenvalue of G with respect to K . This concept generalizes some classic graph eigenvalue problems of certain matrices such as the adjacency matrix, the Laplacian matrix, the diffusion matrix, and so on. In this paper, we study the extensional eigenvalues of graphs. We develop some basic theories about extensional eigenvalues and present some connections between extensional eigenvalues and the structure of graphs.