Let Γ n,q be the point-hyperplane incidence graph of the projective space PG(n, q), where n ≥ 2 is an integer and q a prime power. We determine the order of magnitude of 1 − i V (Γ n,q ), where i V (Γ n,q ) is the vertex-isoperimetric number of Γ n,q . We also obtain the exact values of i V (Γ 2,q ) and the related incidence-free number of Γ 2,q for q ≤ 16.

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.

The w-wide diameter of a graph is a generalisation of its diameter and connectivity, and is useful for measuring the reliability of an interconnection network. The Gaussian networks, which are defined via quotient rings of the Gaussian integers, have been shown to satisfy many good properties of an interconnection network. In this paper, we will establish the wide diameters of the Gaussian networks and show that they are at most three more than the diameter.

No abstract

No abstract