Eigenpairs of adjacency matrices of balanced signed graphs

Abstract

In this article, we study eigenvalues λ\lambda and their associated eigenvectors xx of the adjacency matrices AA 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 VV can be divided into two sets (either of which may be empty), XX and YY, 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 AA 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.

Keywords

• balanced signed graphs
• structured adjacency matrix of a signed graph
• eigenvalues and eigenvectors of an adjacency matrix
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