The line graphs of lollipop graphs are determined by their spectra

Haemers et al. [W.H. Haemers, X.G. Liu, Y.P. Zhang, Spectral characterizations of lollipop graphs, Linear Algebra Appl. 428 (2008) 2415–2423] first investigated the spectral characterizations of the so called lollipop graphs with order n, which is obtained by identifying a vertex of a cycle with order g and a pendent vertex of a path with order l. For the graphs with least eigenvalue at least -2, Cvetković and Lepović [D. Cvetković, M. Lepović, Cospectral graphs with least eigenvalue at least -2, Publ. Inst. Math., Nouv. Sér. 78(92) (2005) 51–63] introduced the discriminant and the star value of a graph, whose relations are investigated in this paper. Employing this relation and other techniques, we prove that all line graphs of lollipop graphs are determined by their adjacency spectra.