The vertex-isoperimetric number of the incidence andnon-incidence graphs of unitals

TL;DR

This paper establishes bounds and exact values for the vertex-isoperimetric number of incidence and non-incidence graphs of unitals, advancing understanding of their combinatorial properties.

Contribution

It provides the first precise calculations of the vertex-isoperimetric number for classical and certain BM-unitals, and bounds for cases with smaller arcs.

Findings

Exact value for classical unital graphs.

Bounds for unital graphs with small arcs.

Exact value for non-incidence graphs.

Abstract

We derive upper and lower bounds for the vertex-isoperimetric number of the incidence graphs of unitals and determine its order of magnitude. In the case when a unital contains sufficiently large arcs, these bounds agree and give rise to the precise value of this parameter. In particular, we obtain the exact value of the vertex-isoperimetric number of the incidence graphs of classical unitals and a certain subfamily of BM-unitals. In the case when the maximum size of arcs in the unital is relatively small, we obtain an upper bound for this parameter in terms of the vertex-isoperimetric number of the incidence graph. We also determine the exact value of the vertex-isoperimetric number of the non-incidence graph of any unital.