The (signless) Laplacian spectral radii of c-cyclic graphs with n vertices, girth g and k pendant vertices

Let Γ_g(n,k;c) denote the class of c-cyclic graphs with n vertices, girth g ≥ 3 and k ≥ 1 pendant vertices. In this paper, we determine the unique extremal graph with largest signless Laplacian spectral radius and Laplacian spectral radius in the class of connected c-cyclic graphs with n ≥ c(g - 1) + 1 vertices, girth g and at most n-c(g - 1) - 1 pendant vertices, respectively, and the unique extremal graph with largest signless Laplacian spectral radius of Γ_g(n,k;c) when n ≥ c(g - 1) + k + 1 and c ≥ 1, and we also identify the unique extremal graph with largest Laplacian spectral radius in Γ_g(n,k;c) in the case c ≥ 1 and either n ≥ c(g - 1) + k + 1 and g is even or n ≥ 1/2(g - 1)k + cg and g is odd. Our results extends the corresponding results of [Sci. Sin. Math. 2010;40:1017–1024, Electron. J. Combin. 2011; 18:p.183, Comput. Math. Appl. 2010;59:376–381, Electron. J. Linear Algebra. 2011;22:378–388 and J. Math. Res. Appl. 2014;34:379–391].