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Sharp spectral bounds for the vertex-connectivity of regular graphs

Author

Listed:
  • Wenqian Zhang

    (Shandong University of Technology)

  • Jianfeng Wang

    (Shandong University of Technology)

Abstract

Let G be a connected d-regular graph and $$\lambda _2(G)$$ λ 2 ( G ) be the second largest eigenvalue of its adjacency matrix. Mohar and O (private communication) asked a challenging problem: what is the best upper bound for $$\lambda _2(G)$$ λ 2 ( G ) which guarantees that $$\kappa (G) \ge t+1$$ κ ( G ) ≥ t + 1 , where $$1 \le t \le d-1$$ 1 ≤ t ≤ d - 1 and $$\kappa (G)$$ κ ( G ) is the vertex-connectivity of G, which was also mentioned by Cioabă. As a starting point, we determine a sharp bound for $$\lambda _2(G)$$ λ 2 ( G ) to guarantee $$\kappa (G) \ge 2$$ κ ( G ) ≥ 2 (i.e., the case that $$t =1$$ t = 1 in this problem), and characterize all families of extremal graphs.

Suggested Citation

  • Wenqian Zhang & Jianfeng Wang, 2023. "Sharp spectral bounds for the vertex-connectivity of regular graphs," Journal of Combinatorial Optimization, Springer, vol. 45(2), pages 1-12, March.
    • Handle: RePEc:spr:jcomop:v:45:y:2023:i:2:d:10.1007_s10878-023-00992-0
    DOI: 10.1007/s10878-023-00992-0
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    References listed on IDEAS

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    1. Liu, Ruifang & Lai, Hong-Jian & Tian, Yingzhi & Wu, Yang, 2019. "Vertex-connectivity and eigenvalues of graphs with fixed girth," Applied Mathematics and Computation, Elsevier, vol. 344, pages 141-149.
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