Let α be a non-negative real number, and let Θ(G,α) be the largest eigenvalue of A(G)+αD(G). Specially, Θ(G,0) and Θ(G,1) are called the spectral radius and signless Laplacian spectral radius of G, respectively. A graph G is said to be Hamiltonian (traceable) if it contains a Hamiltonian cycle (path), and a graph G is called Hamilton-connected if any two vertices are connected by a Hamiltonian path in G. The number of edges of G is denoted by e(G). Recently, the (signless Laplacian) spectral property of Hamiltonian (traceable, Hamilton-connected) graphs received much attention. In this paper, we shall give a general result for all these existed results. To do this, we first generalize the concept of Hamiltonian, traceable, and Hamilton-connected to s-suitable, and we secondly present a lower bound for e(G) to confirm the existence of s-suitable graphs. Thirdly, when 0≤α≤1, we obtain a lower bound for Θ(G,α) to confirm the existence of s-suitable graphs. Consequently, our results generalize and improve all these existed results in this field, including the main results of Chen et al. (2018), Feng et al. (2017), Füredi et al. (2017), Ge et al. (2016), Li et al. (2016), Nikiforov et al. (2016), Wei et al. (2019) and Yu et al. (2013, 2014).