31 problems

Let $G$ be a finite regular graph, and let $H$ be a 2-lift of $G$. An eigenvalue of $H$ not inherited from $G$ is called new, and let $\rho(\mathsf{UCT}(G))$ denote the relevant un…

Let $n$ be fixed and let $p$ be large. Consider Cayley graphs of $SL_2(\mathbb Z/p^n\mathbb Z)$ generated by random generating sets, together with the analogous graphs for the proj…

Let $G_{n,d}$ be a uniformly random simple $d$-regular graph, and write … for the maximum absolute value of its nontrivial adjacency eigenvalues. A $d$-regular graph is Ramanujan w…

View a graph from model (R1') as a random $n$-cover of a bouquet, the one-vertex multigraph with $d$ self-loops. Hall–Puder–Sawin's conjecture. Their proof should work in this sett…

Folklore conjecture. The Ramanujan graphs constructed by Lubotzky, Phillips, and Sarnak should be lossless expanders.

Level-packet invariant-vector conjecture. If the non-tempered representation in the A-packet has no $K_p'$-invariant vectors, then neither does the supercuspidal representation fro…

Let $p$ and $q$ be distinct odd primes with $q>2\sqrt p$ and Legendre symbol $\left(\frac pq\right)=1$, let $G_q=\operatorname{PSL}_2(\mathbb{F}_q)$, and let $S_{p,q}\subset G_q$ b…

Let $X$ be a sequence of Ramanujan graphs, let $n$ denote the number of vertices, let $d$ be their common degree, and set $p=d-1$. Let $d_2(t)$ be the average squared $\ell^2$ dist…

Let $q$ be fixed and large, and let $X_n$ be the family of $(q+1)$-regular Ramanujan graphs constructed by Lubotzky, Phillips, and Sarnak. For a vertex set $Y\subset X_n$, write…

Camarero–Martínez conjecture. The graph $\mathcal{G}_p$ is a Ramanujan graph for every prime $p$ such that $p\equiv 3 \pmod{4}$.

Let $q$ be a prime power, let $\mathbb{F}_q$ be the finite field with $q$ elements, and let $g\in\mathbb{F}_q[t]$ satisfy $\gcd(g,t(t-1))=1$. Let $X^{q,g}$ be the associated $q+1$-…

Let $G$ be a $k$-regular Ramanujan graph, and let $\varepsilon>0$. Sarnak's diameter conjecture. The diameter of $G$ is bounded from above by … as $|G|\to\infty$. This is stronger…

Let $N,m,\lambda_1,\dots,\lambda_4\in\mathbb{Z}$ satisfy … Four-square strong approximation conjecture. If $N\gg m^{4+\varepsilon}$, then there exists…

Let $G$ range over LPS Ramanujan graphs, and let $|G|$ denote the number of vertices. Sardari's diameter conjecture. The diameter is asymptotically … The paper states that the lowe…

Fix an integer $k\geq 3$, and let $G$ be a $k$-regular connected Ramanujan graph, with $|G|$ vertices. Let $\operatorname{diam}(G)$ denote its diameter. Ramanujan graph diameter co…

Mohar's conjecture. For every degree matrix $D$, if $\mathbb{T}_D$ is weakly Ramanujan, then it is $k_D$-quasi-Ramanujan; moreover, if $k_D=1$, then $\mathbb{T}_D$ is Ramanujan. Th…

Let $G$ be a finite graph, and let $H$ be a lift of $G$. Write $\mathsf{spec}(G)$ for the spectrum of $G$, and let $\rho(\mathsf{UCT}(G))$ denote the relevant universal-cover spect…

Asymptotic Ramanujan conjecture. If $p\equiv 3\pmod 4$, then $G_p$ is asymptotically Ramanujan; equivalently,

Let $X^{p,q}$ be the LPS Ramanujan graph, whose vertices are represented by $7\mathbf a\in S^3(\mathbb Z/q\mathbb Z)/\pm$ when $p$ is a quadratic residue modulo $q$. The diameter c…

Let $p\geq 5$ be an odd prime, let $\delta,a,c\in\mathbb{F}_p^{\times}$, and let $X_p(\delta,a,c)$ be Allen's Cayley graph on the specified subgroup of $\operatorname{GL}(3,p)$. Le…

Let $\mathcal{D}$ be a random $k$-regular digraph on $n$ vertices, and let $\rho(\mathcal{D})$ denote its nontrivial spectral radius. Almost-Ramanujan conjecture for random regular…

Asymptotic distance conjecture. For almost all such diagonal vertices, the distance from $v$ to the identity vertex is

Let $p$ and $q$ be primes, and let $X_{p,q}$ be the associated LPS Ramanujan graph. Asymptotic diameter conjecture. The diameter of $X_{p,q}$ is asymptotic to … The preceding resul…

Optimal strong approximation conjecture. The same strong approximation result as for quadratic forms in five or more variables should hold for $F$ provided that

Proportion conjecture. As $n\to\infty$, the proportion of Ramanujan graphs among all $k$-regular graphs on $n$ vertices tends to a constant in $(0,1)$.