On Strongly Regular Graphs and the Friendship Theorem

This paper presents an alternative proof of the celebrated friendship theorem, originally established by Erdős, Rényi, and Sós in 1966. The proof relies on a closed-form expression for the Lovász ϑ-function of strongly regular graphs, recently derived by the author. Additionally, this 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 Lovász ϑ-function of strongly regular graphs, this 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.