The number of edges, spectral radius and Hamilton-connectedness of graphs

In this paper, we prove that a simple graph G of order sufficiently large n with the minimal degree $$\delta (G)\ge k\ge 2$$δ(G)≥k≥2 is Hamilton-connected except for two classes of graphs if the number of edges in G is at least $$\frac{1}{2}(n^2-(2k-1)n + 2k-2)$$12(n2-(2k-1)n+2k-2). In addition, this result is used to present sufficient spectral conditions for a graph with large minimum degree to be Hamilton-connected in terms of spectral radius or signless Laplacian spectral radius, which extends the results of (Zhou and Wang in Linear Multilinear Algebra 65(2):224–234, 2017) for sufficiently large n. Moreover, we also give a sufficient spectral condition for a graph with large minimum degree to be Hamilton-connected in terms of spectral radius of its complement graph.