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 s...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(−2,2),which are S(1,2,2),S(1,2,3),S(1,2,4)and S(1,1,n−3)for n≥4.In this paper,our focus is the problem of spectral determination of them.We prove that S(1,2,2),S(1,2,3),S(1,2,4)and S(1,1,n−3)for n≠8,10,11,13,16 are determined by their adjacency spectra,and characterize all signed graphs which are non-switching isomorphic and adjacency cospectral with S(1,1,n−3)for other cases.Further,we show that S(1,2,2),S(1,2,3),S(1,2,4)and S(1,1,n−3)for n≠4 are determined by their Laplacian spectra,and we characterize all signed graphs which are non-switching isomorphic and Laplacian cospectral to S(1,1,1).展开更多
基金Supported by the Natural Science Foundation of Xinjiang Uygur Autonomous Region(No.2024D01C41)NSFC(Nos.12361071 and 11901498)+1 种基金Tianshan Talent Training Program,China(No.2024TSYCCX0013)Basic Scientific Research in Universities of Xinjiang Uygur Autonomous Region,China(No.XJEDU2025P001).
文摘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(−2,2),which are S(1,2,2),S(1,2,3),S(1,2,4)and S(1,1,n−3)for n≥4.In this paper,our focus is the problem of spectral determination of them.We prove that S(1,2,2),S(1,2,3),S(1,2,4)and S(1,1,n−3)for n≠8,10,11,13,16 are determined by their adjacency spectra,and characterize all signed graphs which are non-switching isomorphic and adjacency cospectral with S(1,1,n−3)for other cases.Further,we show that S(1,2,2),S(1,2,3),S(1,2,4)and S(1,1,n−3)for n≠4 are determined by their Laplacian spectra,and we characterize all signed graphs which are non-switching isomorphic and Laplacian cospectral to S(1,1,1).
