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On Strongly Regular Graphs and the Friendship Theorem

Igal Sason

TL;DR

This work addresses proving the friendship theorem through a novel, analytics-driven route based on the closed-form Lovász $\vartheta$-function for strongly regular graphs and extends these ideas to subgraph embedding questions. By exploiting the SRG theta formula $\vartheta(G) = \dfrac{n (t+\mu-\lambda)}{2d+t+\mu-\lambda}$ with $t = \sqrt{(\mu-\lambda)^2+4(d-\mu)}$ and the relation $\vartheta(G)\vartheta(\overline{G}) = n$, the paper provides an alternative proof of the theorem and derives new necessary conditions for a graph to be a spanning or induced subgraph of another SRG. It further extends the framework to regular graphs, incorporating graph energy for induced-subgraph results, and demonstrates concrete implications for families such as line graphs and triangular graphs. Overall, the work offers a powerful spectral-analytic toolkit linking $\vartheta$, eigenvalues, and energy to classical combinatorial structures, with potential applications in design, coding, and network theory.

Abstract

This paper presents an alternative proof of the celebrated friendship theorem, originally established by Erdős, Rényi, and Sós (1966). The proof relies on a closed-form expression for the Lovász $\vartheta$-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 Lovász $\vartheta$-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.

On Strongly Regular Graphs and the Friendship Theorem

TL;DR

This work addresses proving the friendship theorem through a novel, analytics-driven route based on the closed-form Lovász -function for strongly regular graphs and extends these ideas to subgraph embedding questions. By exploiting the SRG theta formula with and the relation , the paper provides an alternative proof of the theorem and derives new necessary conditions for a graph to be a spanning or induced subgraph of another SRG. It further extends the framework to regular graphs, incorporating graph energy for induced-subgraph results, and demonstrates concrete implications for families such as line graphs and triangular graphs. Overall, the work offers a powerful spectral-analytic toolkit linking , eigenvalues, and energy to classical combinatorial structures, with potential applications in design, coding, and network theory.

Abstract

This paper presents an alternative proof of the celebrated friendship theorem, originally established by Erdős, Rényi, and Sós (1966). The proof relies on a closed-form expression for the Lovász -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 Lovász -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.

Paper Structure

This paper contains 9 sections, 16 theorems, 34 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Friendship Theorem: An Alternative Proof and Further Extensions
  4. The friendship theorem
  5. A new alternative proof of the friendship theorem relying on the Lovász $\vartheta$-function
  6. Another proof of the friendship theorem relying on spectral properties of strongly regular graphs
  7. Further discussions and insights on extensions of the friendship theorem
  8. Spanning Subgraphs of Strongly Regular Graphs
  9. Induced Subgraphs of Strongly Regular Graphs

Key Result

Proposition 2.1

BrouwerM22 A graph $\mathsf{G}$ is strongly regular if and only if its complement $\overline{\mathsf{G}}$ is so. Furthermore, if $\mathsf{G}$ is a strongly regular graph in the family $\mathop{\mathrm{\mathsf{srg}}}\nolimits(n,d,\lambda,\mu)$, then $\overline{\mathsf{G}}$ is a strongly regular graph

Figures (2)

  • Figure 1: Windmill graph (also known as friendship graph) consisting of multiple triangles sharing a common central vertex $u$.
  • Figure 2: The Shrikhande graph $\mathsf{G}$ and two induced cycles of length 6 (left plot) and 8 (right plot), with their edges highlighted in red.

Theorems & Definitions (51)

  • Definition 2.1: Graph complement
  • Definition 2.2: Integer-valued graph invariants
  • Definition 2.3: Orthonormal representations of a graph
  • Definition 2.4: Lovász $\vartheta$-function, Lovasz79_IT
  • Definition 2.5: Strongly regular graphs
  • Proposition 2.1
  • Theorem 2.1: Bounds on Lovász function of regular graphs, Sason23
  • Theorem 2.2: The Lovász $\vartheta$-function of strongly regular graphs, Sason23
  • Remark 2.1
  • Corollary 2.1: Sason23
  • ...and 41 more