Clique structure and other network properties of the tensor product of Erd\H{o}s-R\'enyi graphs

TL;DR

This paper investigates the structural properties of the tensor product of Erdős-Rényi graphs, focusing on clique counts, clustering coefficients, isolated vertices, and their implications for graph efficiency and maximum common subgraph problems.

Contribution

It provides new analytical insights into the clique structure, clustering, and isolated vertices of tensor products of Erdős-Rényi graphs, linking these properties to graph efficiency and modular products.

Findings

Derived formulas for clique counts in tensor products

Extended clustering coefficient analysis and its relation to local efficiency

Characterized the number of isolated vertices in tensor products

Abstract

We analyze the number of cliques of given size and the size of the largest clique in tensor product $G \times H$ of two Erd\H{o}s-R\'enyi graphs $G$ and $H$. Then an extended clustering coefficient is introduced and is studied for $G \times H$. Restriction to the standard clustering coefficient has a direct relation to the local efficiency of the graph, and the results are also interpreted in terms of the efficiency. As a last statistic of interest, the number of isolated vertices is analyzed for $G \times H$. The paper is concluded with a discussion of the modular product of random graphs, and the relation to the maximum common subgraph problem.