Ramanujan polar graphs
Valentino Smaldore
TL;DR
This work identifies Ramanujan tangent graphs arising from finite classical polar spaces by explicitly computing spectra for several polar-family tangent graphs and comparing them to the Ramanujan bound $\lambda \le 2\sqrt{d-1}$. Through detailed spectral analysis of the families $NO^{\varepsilon}(2m,2)$, $NO(2m+1,2)$, and $NU(m,4)$, the authors establish Ramanujan properties for most parameter ranges (with small-$m$ exceptions) and show asymptotic non-Ramanujan behavior for $q>2$. The approach leverages the unitary and orthogonal-binary cases to obtain strongly regular structures with known eigenvalues, enabling direct verification. The results connect spectral graph theory with finite geometry and have potential implications for coding-theory constructions, such as minimal codes and LDPC-type schemes, via Ramanujan polar graphs. Overall, the paper broadens the catalog of explicit Ramanujan graphs arising from polar-space tangencies and highlights practical applications in combinatorial designs and communications.
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.