Graph Theory - Connectivity of r-regular graphs

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SUMMARY

The minimum positive integer r for which there exists an r-regular graph G such that the spectral radius λ(G) is at least κ(G) + 2 is a key focus in graph theory. The discussion highlights the relationship between the connectivity κ(G) and the spectral radius λ(G) in r-regular graphs. Participants suggest exploring specific examples and theoretical frameworks to determine the exact value of r. This inquiry is essential for advancing understanding in the field of graph connectivity.

PREREQUISITES
  • Understanding of r-regular graphs
  • Familiarity with graph connectivity concepts, specifically λ(G) and κ(G)
  • Knowledge of spectral graph theory
  • Basic proficiency in mathematical proofs and graph theory terminology
NEXT STEPS
  • Research the properties of r-regular graphs in detail
  • Explore the implications of spectral radius in graph theory
  • Investigate existing literature on connectivity in graphs
  • Examine case studies that illustrate the relationship between λ(G) and κ(G)
USEFUL FOR

Mathematicians, graph theorists, and students studying advanced graph theory concepts, particularly those interested in the properties of regular graphs and their connectivity.

Gh0stZA
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Hello everyone.

Find the minimum positive integer r for which there exists an r-regular graph G such that λ(G) ≥ κ(G) + 2

All help appreciated.
 
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Sorry for the bump, any ideas on this?
 

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