Abstract
For α∈[0,1], let Aα(Gσ)=αD(G)+(1−α)A(Gσ), where G is a simple undirected graph, D(G) is the diagonal matrix of its vertex degrees and A(Gσ) is the adjacency matrix of the signed graph Gσ whose underlying graph is G. In this paper, basic properties of Aα(Gσ) 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.
Highlights
The theory of signed graphs has attracted the attention of several researchers in recent decades
Among the most recent contributions, we highlight the works of Stanicin [4,5,6,7], in which the largest eigenvalue of signed graphs is studied
The result follows from Theorem 7 using the fact that the smallest eigenvalue of the signless Laplacian matrix of a connected non-bipartite graph is positive
Summary
The theory of signed graphs has attracted the attention of several researchers in recent decades (see Zaslavsky’s dynamic survey [1] for a mathematical bibliography of signed graphs). Among the most recent contributions, we highlight the works of Stanicin [4,5,6,7], in which the largest eigenvalue of signed graphs is studied. A(Gσ) and L(Gσ) are both real symmetric matrices Their eigenvalues are real, and they, counting multiplicities, define the spectrum of Gσ and the Laplacian spectrum of Gσ, respectively. The first work on Aα(Gσ) is the contribution of Belardo et al [22], in which the results obtained in [11] on the multiplicity of α as an eigenvalue of Aα(G), when the unsigned graph G under study has pendant vertices, are extended to Aα(Gσ).
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