Vertex analysis of neural tree structures containing trichotomous nodes

Area of Science:

• Network theory
• Graph theory
• Computational biology

Background:

• Vertex analysis categorizes dichotomous networks using pendant (Vp) and nodal (Vd) vertices.
• Nodal vertices (Va, Vb, Vc) connect to 2, 1, or 0 pendant vertices, respectively.
• The vertex ratio (VR) of Va/Vb helps define network topology.

Purpose of the Study:

• To address the complexity introduced by trichotomous nodes (Vt) in vertex analysis.
• To adapt vertex ratio calculations for networks with both dichotomous and trichotomous nodes.
• To determine network growth modes using refined vertex analysis.

Main Methods:

• Introduced trichotomous nodes (Vt) with subclasses Va', Vb', Vc', and Vd'.
• Developed transformation rules for Vt subclasses into dichotomous Vd equivalents.
• Utilized computer growth simulation to calculate VR for various Vt frequencies.

Main Results:

• Established that each Vt is equivalent to 2Vd.
• Defined Vb' as equivalent to Va/3 + 4Vb/3 + Vc/3, assuming equal likelihood of Vd pair transformations.
• Demonstrated that VR can be computed for networks with varying Vt frequencies.

Conclusions:

• Refined vertex analysis accommodates trichotomous nodes, enhancing network topology studies.
• Computer simulation of network growth provides a method to ascertain the growth mode from VR.
• Accurate determination of network growth mode is possible if all trichotomous nodes are detectable.