Normalized Sombor indices as complexity measures of random graphs

TL;DR

This paper investigates the behavior of normalized Sombor indices across various random graph models, revealing their scaling properties and correlation with graph entropy, thus proposing them as effective complexity measures.

Contribution

It introduces the application of normalized Sombor indices to different random graph models and explores their correlation with graph entropy as a complexity metric.

Findings

Normalized Sombor indices scale with the average degree of graphs.

Selected indices are highly correlated with Shannon entropy of eigenvectors.

Normalized indices serve as effective complexity measures.

Abstract

We perform a detailed computational study of the recently introduced Sombor indices on random graphs. Specifically, we apply Sombor indices on three models of random graphs: Erd\"os-R\'enyi graphs, random geometric graphs, and bipartite random graphs. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the graph, scale with the graph average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random graphs and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the graph adjacency matrix.