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On the Mutual Visibility of Some Moore Graphs

Tonny K B, Shikhi M

Abstract

The concept of mutual visibility in a graph encodes combinatorial information about vertex subsets with prescribed visibility properties and serves as a useful algebraic invariant. In this paper, we study the visibility properties of some Moore graphs of diameter 2, with emphasis on the Petersen and Hoffman Singleton graphs. We first compute the visibility polynomial of the Petersen graph explicitly by analyzing its structural features in relation to the strong regularity of the graph. For the Hoffman Singleton graph, we establish an upper bound of 20 on its visibility number, and subsequently employ an integer programming approach to prove that this bound is tight. As a corollary, we deduce that the maximum size of an induced matching in the Hoffman Singleton graph is 10. These results demonstrate how visibility polynomials can be computed for extremal graphs of high symmetry, and how algebraic and optimization techniques can be combined to determine precise visibility parameters in complex graph families.

On the Mutual Visibility of Some Moore Graphs

Abstract

The concept of mutual visibility in a graph encodes combinatorial information about vertex subsets with prescribed visibility properties and serves as a useful algebraic invariant. In this paper, we study the visibility properties of some Moore graphs of diameter 2, with emphasis on the Petersen and Hoffman Singleton graphs. We first compute the visibility polynomial of the Petersen graph explicitly by analyzing its structural features in relation to the strong regularity of the graph. For the Hoffman Singleton graph, we establish an upper bound of 20 on its visibility number, and subsequently employ an integer programming approach to prove that this bound is tight. As a corollary, we deduce that the maximum size of an induced matching in the Hoffman Singleton graph is 10. These results demonstrate how visibility polynomials can be computed for extremal graphs of high symmetry, and how algebraic and optimization techniques can be combined to determine precise visibility parameters in complex graph families.

Paper Structure

This paper contains 4 sections, 18 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Mutual-visibility sets of some Moore graphs of diameter two
  4. Conclusion

Key Result

Lemma 1

Let $G$ be a graph of diameter $2$. A subset $S \subseteq V(G)$ is a mutual-visibility set of $G$ if and only if, for every pair of non-adjacent vertices $\{u,v\} \subseteq S$, there exists a common neighbour $w \in V(G)\setminus S$.

Theorems & Definitions (36)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • ...and 26 more