Journal
MATHEMATICS
Volume 9, Issue 16, Pages -Publisher
MDPI
DOI: 10.3390/math9161990
Keywords
signed graph; adjacency matrix; balanced signed graph; switching equivalent graphs; eigenvalue bounds
Categories
Funding
- CONICYT-PFCHA/Doctorado Nacional, Chile [2017-21170391]
- 2019 National Science Foundation grant DMS [1757603]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1757603] Funding Source: National Science Foundation
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This paper examines the basic properties of the matrix A(alpha)(G(σ)), including its positive semidefiniteness and some bounds on its eigenvalues, particularly deriving lower and upper bounds on its largest eigenvalue.
For alpha is an element of [0,1], let A(alpha)(G(sigma)) = alpha D(G) + (1 - alpha)A(G(sigma)), where G is a simple undirected graph, D(G) is the diagonal matrix of its vertex degrees and A(G(sigma)) is the adjacency matrix of the signed graph G(sigma) whose underlying graph is G. In this paper, basic properties of A(alpha)(G(sigma)) are obtained, its positive semidefiniteness is studied and some bounds on its eigenvalues are derived-in particular, lower and upper bounds on its largest eigenvalue are obtained.
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