2025/05/06 by Martínez-Martínez, C. T., Aguilar-Sánchez, R., Méndez-Bermúdez, J. A.
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.04008
We perform a detailed statistical study of the distribution of topological and spectral indices on random graphs G=(V,E) in a wide range of connectivity regimes. First, we consider degree-based topological indices (TIs), and focus on two classes of them: XΣ(G) = ∑uv ∈ E f(du,dv) and XΠ(G) = ∏uv ∈ E g(du,dv), where uv denotes the edge of G connecting the vertices u and v, du is the degree of the vertex u, and f(x,y) and g(x,y) are functions of the vertex degrees. Specifically, we apply XΣ(G) and XΠ(G) on Erdös-Rényi graphs and random geometric graphs along the full transition from almost isolated vertices to mostly connected graphs. While we verify that P(XΣ(G)) converges to a standard normal distribution, we show that P( XΠ(G)) converges to a log-normal distribution. In addition we also analyze Revan-degree-based indices and spectral indices (those defined from the eigenvalues and eigenvectors of the graph adjacency matrix). Indeed, for Revan-degree indices, we obtain results equivalent to those for standard degree-based TIs. Instead, for spectral indices, we report two distinct patterns: the distribution of indices defined only from eigenvalues approaches a normal distribution, while the distribution of those indices involving both eigenvalues and eigenvectors approaches a log-normal distribution.