Mutual visibility in Moore graphs and $(d,2)$-graphs with defect

TL;DR

This paper investigates the mutual visibility number in specific classes of graphs, deriving algebraic conditions, exact values for certain cases, and bounds for others, with a focus on Moore graphs like the Hoffman-Singleton graph.

Contribution

It provides new algebraic conditions for mutual visibility in $(d,2)$-graphs, determines this parameter for specific cases, and employs integer programming to establish tight bounds.

Findings

Mutual-visibility number for $(d,2,-2)$-graphs with $d=3,4$

Upper bound of 20 for the Hoffman-Singleton graph's mutual-visibility number

Maximum induced matching size in Hoffman-Singleton graph is 10

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 derive algebraic conditions for the mutual-visibility number of $(d,2)$-graphs with non-negative defect. We then determine this parameter for $(d,2,-2)$-graphs for $d=3$ and $4$, and establish an upper bound for $d=5$. In the case $\delta=0$, that is, for Moore graphs of diameter $2$, we focus on the Hoffman-Singleton graph. We establish an upper bound of $20$ for its mutual-visibility number and subsequently employ an integer programming approach to show that this bound is tight. As a corollary, we deduce that the maximum size of an induced matching in the Hoffman--Singleton graph is $10$.