On the Spectral Characterization of Starlike Trees Based on Signed Graphs

A signed graph is determined by its adjacency spectrum (resp., Laplacian spectrum) if there is no other non-switching isomorphic signed graph having the same adjacency spectrum (resp., Laplacian spectrum). In particular, a starlike tree can also be interpreted as a signed graph. Oboudi [On the eigenvalues and spectral radius of starlike trees, Aequationes Math. 92 (2018) 683–694] characterized all starlike trees whose adjacency eigenvalues are all in the interval [Formula: see text], which are [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text]. In this paper, our focus is the problem of spectral determination of them. We prove that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text] are determined by their adjacency spectra, and characterize all signed graphs which are non-switching isomorphic and adjacency cospectral with [Formula: see text] for other cases. Further, we show that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text] are determined by their Laplacian spectra, and we characterize all signed graphs which are non-switching isomorphic and Laplacian cospectral to [Formula: see text].