On the maximum spectral radius of connected graphs with a prescribed order and size

Key Points

• This research aims to maximize the spectral radius of connected simple graphs with a given number of vertices and edges.
• Considered connected simple graphs with n vertices and n-1 + e edges.
• Focused on cases where e ≤ 130 or n ≥ e + 2 + 13√e.
• Analyzed the eigenvalues of the adjacency matrix for these graphs.
• Identified optimal conditions for spectral radius in the specified graph sets.
• Provided new bounds for spectral radius based on the number of edges and vertices.