Erratum to “The lollipop graph is determined by its Q-spectrum”

AbstractThe proof of Theorem 3.3 in [Y. Zhang, X. Liu, B. Zhang, X. Yong, The lollipop graph is determined by its Q-spectrum, Discrete Math. 309 (2009) 3364–3369] is not correct. Actually, in the development of PA(G1)(λ), the authors missed several products into addition, which makes the rest of the proof invalid. Note that the statement of the theorem is true. Here, we give a correct proof

A lower bound for the first Zagreb index and its application

For a graph G, the first Zagreb index is defined as the sum of the squares of the vertices degrees. By investigating the connection between the first Zagreb index and the first three coefficients of the Laplacian characteristic polynomial, we give a lower bound for the first Zagreb index, and we determine all corresponding extremal graphs. By doing so, we generalize some known results, and, as an application, we use these results to study the Laplacian spectral determination of graphs with small first Zagreb index

Spectral characterizations of signed lollipop graphs

Let Γ=(G,σ) be a signed graph, where G is the underlying simple graph and σ:E(G)→{+,-} is the sign function on the edges of G. In this paper we consider the spectral characterization problem extended to the adjacency matrix and Laplacian matrix of signed graphs. After giving some basic results, we study the spectral determination of signed lollipop graphs, and we show that any signed lollipop graph is determined by the spectrum of its Laplacian matrix