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

## 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.

## References

- Benediktovich VI (2016) Spectral condition for Hamiltonicity of a graph. Linear Algebra Appl 494:70–79MathSciNetCrossRefMATHGoogle Scholar
- Berge C (1976) Graphs and hypergraphs (Minieka E, trans.) 2nd revised edn, vol 6. Elsevier, AmsterdamGoogle Scholar
- Bondy JA, Chvátal V (1976) A method in graph theory. Discrete Math 15(2):111–135MathSciNetCrossRefMATHGoogle Scholar
- Brouwer AE, Haemers WH (2012) Spectra of graphs. Universitext. Springer, New YorkCrossRefMATHGoogle Scholar
- Csikvári P (2009) On a conjecture of V. Nikiforov. Discrete Math 309:4522–4526MathSciNetCrossRefMATHGoogle Scholar
- Erdős P, Gallai T (1959) On maximal paths and circuits of graphs. Acta Math Acad Sci Hungar 10:337–356MathSciNetCrossRefMATHGoogle Scholar
- Feng LH, Yu GH (2009) On three conjectures involving the signless Laplacian spectral radius of graphs. Publ Inst Math (Beograd) 85:35–38MathSciNetCrossRefMATHGoogle Scholar
- Fiedler M, Nikiforov V (2010) Spectral radius and Hamiltonicity of graphs. Linear Algebra Appl 432:2170–2173MathSciNetCrossRefMATHGoogle Scholar
- Ge J, Ning B (2016) Spectral radius and Hamiltonicity of graphs and balanced bipartite graphs with large minimum degree. arXiv:1606.08530v3
- Godsil C, Royle G (2001) Algebraic graph theory. Graduate texts in mathematics. Springer, New YorkCrossRefMATHGoogle Scholar
- Ho TY, Lin CK, Tan JJM et al (2010) On the extremal number of edges in hamiltonian connected graphs. Appl Math Lett 23:26–29MathSciNetCrossRefMATHGoogle Scholar
- Li B, Ning B (2016) Spectral analogues of Erdős and Moon–Moser’s theorems on Hamilton cycles. Linear Multilinear Algebra 64(11):2252–2269MathSciNetCrossRefMATHGoogle Scholar
- Liu R, Shiu WC, Xue J (2015) Sufficient spectral conditions on Hamiltonian and traceable graphs. Linear Algebra Appl 467:254–266MathSciNetCrossRefMATHGoogle Scholar
- Lu M, Liu H, Tian F (2012) Spectral radius and Hamiltonian graphs. Linear Algebra Appl 437:1670–1674MathSciNetCrossRefMATHGoogle Scholar
- Nikiforov V (2002) Some inequalities for the largest eigenvalue of a graph. Comb Probab Comput 11:179–189MathSciNetCrossRefMATHGoogle Scholar
- Nikiforov V (2016) Spectral radius and Hamiltonicity of graphs with large minimum degree. Czechoslovak Math J 66(141):925–940MathSciNetCrossRefMATHGoogle Scholar
- Ning B, Ge J (2015) Spectral radius and Hamiltonian properties of graphs. Linear Multilinear Algebra 63(8):1520–1530MathSciNetCrossRefMATHGoogle Scholar
- Pósa L (1964) On the circuits of finite graphs. Magyar Tud Akad Mat Kutató Int Közl 8:355–361MathSciNetMATHGoogle Scholar
- Prasolov VV (2001) Polynomials. MTsNMO, MoscowGoogle Scholar
- Read RC, Wilson RJ (1998) An Atlas of graphs. Oxford University Press, OxfordMATHGoogle Scholar
- Tomescu I (1983) On Hamiltonian-connected regular Graphs. J Graph Theory 7:429–436MathSciNetCrossRefMATHGoogle Scholar
- Yu GD, Fan YZ (2013) Spectral conditions for a graph to be Hamilton-connected. Appl Mech Mater 336–338:2329–2334CrossRefGoogle Scholar
- Yu GD, Fang Y, Fan YZ, et al (2017) Spectral radius and Hamiltonicity of graphs. arXiv:1705.01683v1
- Yu GD, Ye ML, Cai GX et al (2014) Signless Laplacian spectral conditions for Hamiltonicity of graphs. J Appl Math 2014:6 (Article ID 282053)MathSciNetGoogle Scholar
- Zhou B (2010) Signless Laplacian spectral radius and Hamiltonicity. Linear Algebra Appl 432:566–570MathSciNetCrossRefMATHGoogle Scholar
- Zhou QN, Wang LG (2017) Some sufficient spectral conditions on Hamilton-connected and traceable graphs. Linear Multilinear Algebra 65(2):224–234MathSciNetCrossRefMATHGoogle Scholar