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We study the problem of constructing explicit sparse graphs that exhibit strong vertex expansion. Our main result is the first two-sided construction of imbalanced unique-neighbor expanders, meaning bipartite graphs where small sets contained in both the left and right bipartitions exhibit unique-neighbor expansion, along with algebraic properties relevant to constructing quantum codes. Our constructions are obtained from instantiations of the tripartite line product of a large tripartite spectral expander and a sufficiently good constant-sized unique-neighbor expander, a new graph product we defined that generalizes the line product in the work of Alon and Capalbo and the routed product in the work of Asherov and Dinur. To analyze the vertex expansion of graphs arising from the tripartite line product, we develop a sharp characterization of subgraphs that can arise in bipartite spectral expanders, generalizing results of Kahale, which may be of independent interest. By picking appropriate graphs to apply our product to, we give a strongly explicit construction of an infinite family of $(d_1,d_2)$-biregular graphs $(G_n)_{n\ge 1}$ (for large enough $d_1$ and $d_2$) where all sets $S$ with fewer than a small constant fraction of vertices have $\Omega(d_1\cdot |S|)$ unique-neighbors (assuming $d_1 \leq d_2$). Additionally, we can also guarantee that subsets of vertices of size up to $\exp(\Omega(\sqrt{\log |V(G_n)|}))$ expand losslessly., Comment: New version contains stronger result, and many new technical ingredients. 45 pages, 2 figures

For an abelian group $H$ acting on the set $[\ell]$, an $(H,\ell)$-lift of a graph $G_0$ is a graph obtained by replacing each vertex by $\ell$ copies, and each edge by a matching corresponding to the action of an element of $H$. In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group $H \leqslant \text{Sym}(\ell)$, constant degree $d \ge 3$ and $\epsilon > 0$, we construct explicit $d$-regular expander graphs $G$ obtained from an $(H,\ell)$-lift of a (suitable) base $n$-vertex expander $G_0$ with the following parameters: (i) $\lambda(G) \le 2\sqrt{d-1} + \epsilon$, for any lift size $\ell \le 2^{n^{\delta}}$ where $\delta=\delta(d,\epsilon)$, (ii) $\lambda(G) \le \epsilon \cdot d$, for any lift size $\ell \le 2^{n^{\delta_0}}$ for a fixed $\delta_0 > 0$, when $d \ge d_0(\epsilon)$, or (iii) $\lambda(G) \le \widetilde{O}(\sqrt{d})$, for lift size ``exactly'' $\ell = 2^{\Theta(n)}$. As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes. Items $(i)$ and $(ii)$ above are obtained by extending the techniques of Mohanty, O'Donnell and Paredes [STOC 2020] for $2$-lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing'" depth-first search traversals. Result $(iii)$ is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion., Comment: 31 pages

We describe a new method to remove short cycles on regular graphs while maintaining spectral bounds (the nontrivial eigenvalues of the adjacency matrix), as long as the graphs have certain combinatorial properties. These combinatorial properties are related to the number and distance between short cycles and are known to happen with high probability in uniformly random regular graphs. Using this method we can show two results involving high girth spectral expander graphs. First, we show that given $d \geq 3$ and $n$, there exists an explicit distribution of $d$-regular $\Theta(n)$-vertex graphs where with high probability its samples have girth $\Omega(\log_{d - 1} n)$ and are $\epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2\sqrt{d - 1} + \epsilon$ (excluding the single trivial eigenvalue of $d$). Then, for every constant $d \geq 3$ and $\epsilon > 0$, we give a deterministic poly$(n)$-time algorithm that outputs a $d$-regular graph on $\Theta(n)$-vertices that is $\epsilon$-near-Ramanujan and has girth $\Omega(\sqrt{\log n})$, based on the work of arXiv:1909.06988 ., Comment: 21 pages

Computing subgraph frequencies is a fundamental task that lies at the core of several network analysis methodologies, such as network motifs and graphlet-based metrics, which have been widely used to categorize and compare networks from multiple domains. Counting subgraphs is however computationally very expensive and there has been a large body of work on efficient algorithms and strategies to make subgraph counting feasible for larger subgraphs and networks. This survey aims precisely to provide a comprehensive overview of the existing methods for subgraph counting. Our main contribution is a general and structured review of existing algorithms, classifying them on a set of key characteristics, highlighting their main similarities and differences. We identify and describe the main conceptual approaches, giving insight on their advantages and limitations, and provide pointers to existing implementations. We initially focus on exact sequential algorithms, but we also do a thorough survey on approximate methodologies (with a trade-off between accuracy and execution time) and parallel strategies (that need to deal with an unbalanced search space)., Comment: 35 pages

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 $2\sqrt{d-1} + \epsilon$ (excluding the single trivial eigenvalue of~$d$)., Comment: 26 pages

We precisely determine the SDP value (equivalently, quantum value) of large random instances of certain kinds of constraint satisfaction problems, ``two-eigenvalue 2CSPs''. We show this SDP value coincides with the spectral relaxation value, possibly indicating a computational threshold. Our analysis extends the previously resolved cases of random regular $\mathsf{2XOR}$ and $\textsf{NAE-3SAT}$, and includes new cases such as random $\mathsf{Sort}_4$ (equivalently, $\mathsf{CHSH}$) and $\mathsf{Forrelation}$ CSPs. Our techniques include new generalizations of the nonbacktracking operator, the Ihara--Bass Formula, and the Friedman/Bordenave proof of Alon's Conjecture., Comment: 50 pages excluding title page and table of contents