13 problems

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…

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…

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

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 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 $G$ 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…

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 $G$ be an infinite $d$-regular graph. Call $G$ Ramanujan if the spectral radius of its Markov operator on $l^2$ of the vertices equals … the spectral radius of the $d$-regular…

Let $G$ be the $p$-adic group and $I$ an Iwahori subgroup, and let $\Gamma$ be a cocompact lattice in $G$. Consider the space of $I$-invariant vectors … An irreducible unitary…

Let $D$ be a degree matrix of order $t\geq 2$, let $T_D$ be its universal cover, and let $\lambda_2(D)$ denote the second eigenvalue of $D$. A finite graph with degree matrix $D$ i…