Eigenpairs of adjacency matrices of balanced signed graphs

Author:

Chen Mei-Qin1

Affiliation:

1. Department of Mathematical Sciences, The Citadel , 171 Moultrie St , Charleston , SC, 29409 , United States

Abstract

Abstract In this article, we study eigenvalues λ \lambda and their associated eigenvectors x x of the adjacency matrices A A of balanced signed graphs. Balanced signed graphs were first introduced and studied by Harary to handle a problem in social psychology. Harary showed in 1953 that a signed graph is balanced if and only if its vertex set V V can be divided into two sets (either of which may be empty), X X and Y Y , so that each edge between the sets is negative and each within a set is positive. Based on this fundamental theorem for the balanced signed graphs, vertices of a balanced signed graph can be labeled in a way so that its adjacency matrix is well structured. Using this special structure, we find algebraically all eigenvalues and their associated eigenvectors of the adjacency matrix A A of a given balanced signed graph. We present in this study eigenpairs ( λ , x ) \left(\lambda ,x) of adjacency matrices of balanced signed graphs with some special structures.

Publisher

Walter de Gruyter GmbH

Reference22 articles.

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3. M. Andelić, T. Koledin, and Z. Stanić, A note on the eigenvalue free intervals of some classes of signed threshold graphs, Spec. Matrices 7 (2019), 218–225.

4. D. Cartwright and F. Harary, Structural balance: A generalization of Heider’s theory, Psychol Rev. 63 (1956), 277–293.

5. D. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs, Johann Ambrosius Barth, Heilderberg-Leipzig, 1995.

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