On Strongly Regular Graphs and the Friendship Theorem

TL;DR

This paper offers a new proof of the friendship theorem using a closed-form Lovász theta function for strongly regular graphs, and explores extensions and conditions related to subgraphs and graph energies.

Contribution

The paper introduces an alternative proof of the friendship theorem based on a novel closed-form expression for the Lovász theta function of strongly regular graphs.

Findings

Derived new necessary conditions for strongly regular graphs to be subgraphs of others.

Extended results to regular graphs and their subgraphs.

Provided insights into the structure and properties of strongly regular graphs.

Abstract

This paper presents an alternative proof of the celebrated friendship theorem, originally established by Erd\H{o}s, R\'{e}nyi, and S\'{o}s (1966). The proof relies on a closed-form expression for the Lov\'{a}sz $\vartheta$-function of strongly regular graphs, recently derived by the author. Additionally, the paper considers some known extensions of the theorem, offering discussions that provide insights into the friendship theorem, one of its extensions, and the proposed proof. Leveraging the closed-form expression for the Lov\'{a}sz $\vartheta$-function of strongly regular graphs, the paper further establishes new necessary conditions for a strongly regular graph to be a spanning or induced subgraph of another strongly regular graph. In the case of induced subgraphs, the analysis also incorporates a property of graph energies. Some of these results are extended to regular graphs and their subgraphs.