Exploring the "Middle Earth" of network spectra via a Gaussian matrix function

Area of Science:

• Network Science
• Graph Theory
• Quantum Chemistry

Background:

• Existing graph matrix functions may not fully capture network structural information.
• Dynamical processes in quantum systems can be modeled using network Hamiltonians.
• The Folded Spectrum Method is a precursor for analyzing network spectra.

Purpose of the Study:

• To introduce and analyze the Gaussian Estrada index for network characterization.
• To investigate the properties of this index in various graph structures.
• To connect network spectral properties to underlying structural patterns.

Main Methods:

• Application of a Gaussian matrix function to the adjacency matrix of networks.
• Mathematical derivation of bounds for the Gaussian Estrada index in simple graphs.
• Formulation of the index for Erdős-Rényi and Barabási-Albert random graphs.

Main Results:

• The Gaussian Estrada index is maximized in star graphs and complete bipartite graphs.
• Formulas for the index are derived for specific random graph models.
• A correlation is found between the index and the presence of bicliques in real-world networks.

Conclusions:

• The Gaussian Estrada index provides novel insights into network topology.
• It effectively highlights structural features like bicliques, often arising from network evolution.
• This matrix function offers a more comprehensive characterization of networks compared to previous methods.