Topological and spectral properties of random digraphs.

Area of Science:

• Graph Theory
• Network Science
• Random Graphs

Background:

• Erdős-Rényi (ER) random digraphs D(n,p) are fundamental models in network science.
• Understanding their topological and spectral properties is crucial for network analysis.

Purpose of the Study:

• To investigate topological properties like nonisolated vertices, Randić index, and sum-connectivity index in ER random digraphs.
• To analyze spectral properties, including eigenvalue distribution and related invariants.
• To establish scaling relationships and refine bounds for these properties.

Main Methods:

• Scaling analysis to identify key parameters (average degree, sqrt[np(1-p)]).
• Derivation of expressions relating graph properties to digraph parameters (n,p).
• Computation and analysis of six eigenvalue-related invariants.

Main Results:

• Average degree 〈k〉 scales topological indices (V_{x}(D), R(D), χ(D)).
• Eigenvalue distribution converges to a circle of radius sqrt[np(1-p)].
• Six eigenvalue invariants scale with sqrt[np(1-p)], with refined bounds established.

Conclusions:

• The average degree and sqrt[np(1-p)] are critical scaling parameters for ER random digraphs.
• Established relationships and extended bounds for topological and spectral invariants.
• Provides a deeper understanding of the structure and behavior of random digraphs.