Abstract

We are interested in a signed graph G˙ which admits a decomposition into a totally disconnected (i.e., without edges) star complement and a signed graph S˙ induced by the star set. In this study we derive certain properties of G˙ ; for example, we prove that the number of (distinct) eigenvalues of S˙ does not exceed the number of those of G˙ . Some particular cases are also considered.

Highlights

  • Given a graph G = (V (G), E(G)), let σ : E(G) → {1, −1}

  • We recall from [3, 6] the existence of a spectral tool for constructing comparatively large graphs with specified spectral properties from their smaller parts, known as star complements; these can be defined in the case of signed graphs as well

  • A connected signed line graph has two eigenvalues if and only if the corresponding signed root graph is switching equivalent to (i) the star K1,n−1, (ii) the negative quadrangle or the signed multigraph obtained by inserting the negative edge between the vertices of degree 2 of the path with four vertices, (iii) the complete graph Kn, or (iv) a connected signed doubled regular graph with n vertices, in all cases, for n ≥ 3

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Summary

Introduction

By virtue of Theorem 2.1, if μ is not an eigenvalue of H (with k vertices), there exists a signed graph Gwith Has a star complement for μ if and only if bi, bi = μ and bi, bj ∈ {1, 0, −1}, (2.2) A triangle with positive edges is a star complement for −2 in the signed graph of Figure 1 (with spectrum 23, (−2)3 ).

Results
Conclusion
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