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.
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