Ramanujan polar graphs

TL;DR

This paper demonstrates that certain tangent graphs of finite classical polar spaces satisfy Ramanujan's eigenvalue condition, expanding the class of known Ramanujan graphs with applications in coding theory.

Contribution

It proves that specific families of tangent graphs of finite classical polar spaces are Ramanujan, including cases with unitary and orthogonal polarities over binary fields.

Findings

Some tangent graphs are Ramanujan graphs.

Explicit spectra are computed for strongly regular cases.

Identification of new Ramanujan graph families.

Abstract

Recently, a construction of minimal codes arising from a family of almost Ramanujan graphs was shown. Ramanujan graphs are examples of expander graphs that minimize the second-largest eigenvalue of their adjacency matrix. We call such graphs Ramanujan, since all known non-trivial constructions imply the Ramanujan conjecture on arithmetical functions. In this paper, we prove that some families of tangent graphs of finite classical polar spaces satisfy Ramanujan's condition. If the polarity is unitary, or it is orthogonal and the quadric is over the binary field, the tangent graphs are strongly regular, and we know their spectrum. By direct computation, it is possible to show which families of tangent graphs are Ramanujan.