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

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## Abstract

In this paper, we prove that a simple graph *G* of order sufficiently large *n* with the minimal degree \(\delta (G)\ge k\ge 2\) is Hamilton-connected except for two classes of graphs if the number of edges in *G* is at least \(\frac{1}{2}(n^2-(2k-1)n + 2k-2)\). 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.

### Keywords

Hamilton-connected Minimum degree The number of edges Spectral radius Signless Laplacian spectral radius### Mathematics Subject Classification

05C50 05C35## Notes

### Acknowledgements

The authors would like to thank the referee for very constructive suggestions and comments on this paper and providing the reference Yu et al. (2017) which independently obtains part similar results.

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