Computations regarding certain graphs associated to finite polar spaces

TL;DR

This paper investigates regular graphs derived from finite polar spaces, proposing a conjecture that graphs of different ranks have distinct degrees, and proves it under certain conditions.

Contribution

It introduces a conjecture about the degrees of graphs associated with finite polar spaces and provides a proof for cases with sufficiently large ranks.

Findings

Conjecture that no two graphs of different ranks share the same degree.

Proof of the conjecture for sufficiently large ranks.

Insights into the structure of graphs from finite polar spaces.

Abstract

We consider various regular graphs defined on the set of elements of given rank of a finite polar space. It is likely that no two such graphs, of the same kind but defined for different ranks, can have the same degree. We shall prove this conjecture under the hypothesis that the considered rank are not too small.