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A signed graph is a pair, say (G , σ) , where G is the underlying graph and σ : E (G) → { − 1 , + 1 } is a sign function on the edges of G. In this paper we present the complete characterization of cospectral mates of negative even cycles, that is the signed even cycles with only one negative edge. In [Spectral characterizations of signed cycles, Linear Algebra Appl. 553 (2018) 307–327], the authors characterized the cospectral mates of negative even cycles but they missed a few of them. Here, we characterize all cospectral mates of negative even cycles. [ABSTRACT FROM AUTHOR]

A signed graph is a pair like ( G , σ ) , where G is the underlying graph and σ : E ( G ) → { − 1 , + 1 } is a sign function on the edges of G . In this paper we study the spectral determination problem for signed n -cycles ( C n , σ ) with respect to the adjacency spectrum and the Laplacian spectrum. In particular, for the Laplacian spectrum, we prove that balanced odd cycles and unbalanced cycles, denoted, respectively, by C 2 n + 1 + and C n − , are uniquely determined by their Laplacian spectra (i.e., they are DLS). On the other hand, we determine all Laplacian cospectral mates of the balanced even cycles C 2 n + , so that we show that C 2 n + is not DLS. The same problem is then considered for the adjacency spectrum, hence we prove that odd signed cycles, namely, C 2 n + 1 + and C 2 n + 1 − , are uniquely determined by their (adjacency) spectrum (i.e., they are DS). Moreover, we find cospectral mates for the even signed cycles C 2 n + and C 2 n − , and we show that, except the signed cycle C 4 − , even signed cycles are not DS and we provide almost all cospectral mates. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new graph operation called subdivision vertex-edge join (denoted by G 1 S ▹ ( G 2 V ∪ G 3 E ) for short), and then the adjacency spectrum, the Laplacian spectrum and the signless Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) are respectively determined in terms of the corresponding spectra for a regular graph G 1 and two arbitrary graphs G 2 and G 3 . All the above can be viewed as the generalizations of the main results in [X. Liu, Z. Zhang, Bull. Malays. Math. Sci. Soc., 2017:1⁻17]. Furthermore, we also determine the normalized Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) whenever G i are regular graphs for each index i = 1 , 2 , 3 . As applications, we construct infinitely many pairs of A-cospectral mates, L-cospectral mates, Q-cospectral mates and L -cospectral mates. Finally, we give the number of spanning trees, the (degree-)Kirchhoff index and the Kemeny’s constant of G 1 S ▹ ( G 2 V ∪ G 3 E ) , respectively.

Abstract: A graph is said to be DLS, if there is no other non-isomorphic graph with the same Laplacian spectrum. Let be a DLS graph. We show that is DLS if is disconnected. If is connected, it is proved that is DLS under certain conditions. Applying this result, we prove that is DLS if is a tree on vertices or a unicyclic graph on vertices. [Copyright &y& Elsevier]

In this paper, we introduce a new graph operation called subdivision vertex-edge join (denoted by G 1 S ▹ (G 2 V ∪ G 3 E) for short), and then the adjacency spectrum, the Laplacian spectrum and the signless Laplacian spectrum of G 1 S ▹ (G 2 V ∪ G 3 E) are respectively determined in terms of the corresponding spectra for a regular graph G 1 and two arbitrary graphs G 2 and G 3 . All the above can be viewed as the generalizations of the main results in [X. Liu, Z. Zhang, Bull. Malays. Math. Sci. Soc., 2017:1–17]. Furthermore, we also determine the normalized Laplacian spectrum of G 1 S ▹ (G 2 V ∪ G 3 E) whenever G i are regular graphs for each index i = 1 , 2 , 3 . As applications, we construct infinitely many pairs of A-cospectral mates, L-cospectral mates, Q-cospectral mates and L -cospectral mates. Finally, we give the number of spanning trees, the (degree-)Kirchhoff index and the Kemeny's constant of G 1 S ▹ (G 2 V ∪ G 3 E) , respectively. [ABSTRACT FROM AUTHOR]