On the extensional eigenvalues of graphs

•In this paper, we first propose the extensional eigenvalues of graphs, which generalizes almost all other types of eigenvalues of graphs.•We present some basic properties of extensional eigenvalues, which also hold for classical eigenvalues.•Our method provides a consistent method that is valid for...

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Bibliographic Details
Published in:Applied mathematics and computation 2021-11, Vol.408, p.126365, Article 126365
Main Authors: Cheng, Tao, Feng, Lihua, Liu, Weijun, Lu, Lu
Format: Article
Language:English
Subjects:
Extensional eigenvalue
Extensional eigenvector
Rayleigh quotient
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Summary:•In this paper, we first propose the extensional eigenvalues of graphs, which generalizes almost all other types of eigenvalues of graphs.•We present some basic properties of extensional eigenvalues, which also hold for classical eigenvalues.•Our method provides a consistent method that is valid for all types of graph matrices.•This paper may open a new door in spectral graph theory. 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∈Rn such that Mx=λKx, 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.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2021.126365