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For every constant $d geq 3$ and $epsilon > 0$, we give a deterministic $mathrm{poly}(n)$-time algorithm that outputs a $d$-regular graph on $Theta(n)$ vertices that is $epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2sqrt{d-1} + epsilon$ (excluding the single trivial eigenvalue of~$d$).
Let $p(Y_1, dots, Y_d, Z_1, dots, Z_e)$ be a self-adjoint noncommutative polynomial, with coefficients from $mathbb{C}^{r times r}$, in the indeterminates $Y_1, dots, Y_d$ (considered to be self-adjoint), the indeterminates $Z_1, dots, Z_e$, and thei
We show that for every prime $d$ and $alphain (0,1/6)$, there is an infinite sequence of $(d+1)$-regular graphs $G=(V,E)$ with girth at least $2alpha log_{d}(|V|)(1-o_d(1))$, second adjacency matrix eigenvalue bounded by $(3/sqrt{2})sqrt{d}$, and man
Let $X$ be an infinite graph of bounded degree; e.g., the Cayley graph of a free product of finite groups. If $G$ is a finite graph covered by $X$, it is said to be $X$-Ramanujan if its second-largest eigenvalue $lambda_2(G)$ is at most the spectral
Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved tr
A graph $G = (V,E)$ is a double-threshold graph if there exist a vertex-weight function $w colon V to mathbb{R}$ and two real numbers $mathtt{lb}, mathtt{ub} in mathbb{R}$ such that $uv in E$ if and only if $mathtt{lb} le mathtt{w}(u) + mathtt{w}(v)