Structural and Spectral properties of Corona Graphs

TL;DR

This paper introduces Corona graphs, a new class of product graphs modeled by iteratively applying the corona product to a seed graph, and analyzes their spectral and degree distribution properties.

Contribution

It defines Corona graphs based on the corona product, derives explicit formulas for their eigenvalues, and characterizes their degree and betweenness distributions.

Findings

Degree distribution decays exponentially for regular seed graphs

Betweenness distribution follows a power law for seed graphs that are cliques

Explicit eigenvalue formulas are provided for regular and star seed graphs

Abstract

Product graphs have been gainfully used in literature to generate mathematical models of complex networks which inherit properties of real networks. Realizing the duplication phenomena imbibed in the definition of corona product of two graphs, we define Corona graphs. Given a small simple connected graph which we call seed graph, Corona graphs are defined by taking corona product of a seed graph iteratively. We show that the cumulative degree distribution of Corona graphs decay exponentially when the seed graph is regular and cumulative betweenness distribution follows power law when seed graph is a clique. We determine explicit formulae of eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues of Corona graphs when the seed graph is regular. Computable expressions of eigenvalues and signless Laplacian eigenvalues of Corona graphs are also obtained when the seed graph is a star graph.