The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$. Results obtained so far are rather fragmentary and often treated case by case. In this work, we shall unify all the known results on the subject by showing that the irrationality exponents of large classes of automatic numbers and Mahler numbers (which are transcendental) are exactly equal to $2$. Our classes contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the Fermat numbers, the regular paperfolding numbers, which have been previously considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new classes such as the Stern numbers and so on. Among other ingredients, our proofs use results on Hankel determinants obtained recently by Han.

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The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let T(n,D
l) be the tree which minimizes the spectral radius of all trees of order n with exactly l vertices of maximum degree D
. In this paper, T(n,D
l) is determined for D=3, and for l<= 3 and n large enough. It is proven that for sufficiently large n, T(n, 3, l) is a caterpillar with (almost) uniformly distributed legs, T(n,D
2) is a dumbbell, and T(n,
D,3) is a tree consisting of three distinct stars of order D
connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lov´asz and Pelik´an, Simi´c and To˘si´c, Wu, Yuan and Xiao, and Xu, Lin and Shu.

To characterize the complete structure of the Fucik spectrum of the p-Laplacian on higher dimensional domains is a long-standing problem. In this paper, we study the p-Laplacian with integrable potentials on an interval under the Dirichlet or the Neumann boundary conditions. Based on the strong continuity and continuous differentiability of solutions in potentials, we will give a comprehensive characterization of the corresponding Fucik spectra: each of them is composed of two trivial lines and a double-sequence of differentiable, strictly decreasing, hyperbolic-like curves; all asymptotic lines of these spectral curves are precisely described by using eigenvalues of the p-Laplacian with potentials; and moreover, all these spectral curves have strong continuity in potentials, i.e. as potentials vary in the weak topology, these spectral curves are continuously dependent on potentials in a certain sense.