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1. The Second Eigenvalue of some Normal Cayley Graphs of Highly Transitive Groups

   https://doi.org/10.37236/8054  

   Huang, Xueyi; Huang, Qiongxiang; Cioabă, Sebastian M. (April 2019, The Electronic Journal of Combinatorics)

   Let $$G$$ be a finite group acting transitively on $$[n]=\{1,2,\ldots,n\}$$, and  let $$\Gamma=\mathrm{Cay}(G,T)$$ be a Cayley graph of $$G$$. The graph $$\Gamma$$ is called  normal if $$T$$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $$\Gamma$$ in terms of the second eigenvalues of certain subgraphs of $$\Gamma$$. Using this result, we develop a recursive method to  determine the second eigenvalues of certain  Cayley graphs of $$S_n$$, and we determine the second eigenvalues  of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $$S_n$$  with  $$\max_{\tau\in T}|\mathrm{supp}(\tau)|\leqslant 5$$, where $$\mathrm{supp}(\tau)$$ is the set of points in $[n]$ non-fixed by $$\tau$$. 

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2. The second eigenvalue of some normal Cayley graphs of highly transitive groups

   Huang, Xueyi; Huang, Qiongxiang; Cioaba, Sebastian M (June 2019, The Electronic journal of combinatorics)

   Let $$\Gamma$$ be a finite group acting transitively on $$[n]=\{1,2,\ldots,n\}$$, and let $$G=\mathrm{Cay}(\Gamma,T)$$ be a Cayley graph of $$\Gamma$$. The graph $$G$$ is called normal if $$T$$ is closed under conjugation. In this paper, we obtain an upper bound for \textcolor[rgb]{0,0,1}{the second (largest) eigenvalue} of the adjacency matrix of the graph $$G$$ in terms of the second eigenvalues of certain subgraphs of $$G$$ (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $$S_n$$ and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $$S_n$$ with $$\max_{\tau\in T}|\mathrm{supp}(\tau)|\leq 5$$, where $$\mathrm{supp}(\tau)$$ is the set of points in $[n]$ non-fixed by $$\tau$$. 

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3. Maximal Chains in Bond Lattices

   https://doi.org/10.37236/10983  

   Ahirwar, Shreya; Fishel, Susanna; Gya, Parikshita; Harris, Pamela; Pham, Nguyen; Vindas Meléndez, Andrés; Vo, Dan Khanh (July 2022, The Electronic Journal of Combinatorics)

   Let $$G$$ be a graph with vertex set $$\{1,2,\ldots,n\}$$. Its bond lattice, $BL(G)$, is a sublattice of the set partition lattice. The elements of $BL(G)$ are the set partitions whose blocks induce connected subgraphs of $$G$$. In this article, we consider graphs $$G$$ whose bond lattice consists only of noncrossing partitions. We define a family of graphs, called triangulation graphs, with this property and show that any two produce isomorphic bond lattices. We then look at the enumeration of the maximal chains in the bond lattices of triangulation graphs. Stanley's map from maximal chains in the noncrossing partition lattice to parking functions was our motivation. We find the restriction of his map to the bond lattice of certain subgraphs of triangulation graphs. Finally, we show the number of maximal chains in the bond lattice of a triangulation graph is the number of ordered cycle decompositions. 

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4. Random Algebraic Graphs and Their Convergence to ErdőS–Rényi

   https://doi.org/10.1002/rsa.21276  

   Bangachev, Kiril; Bresler, Guy (January 2025, Random Structures & Algorithms)

   ABSTRACT A random algebraic graph is defined by a group with a uniform distribution over it and a connection with expectation satisfying . The random graph with vertex set is formed as follows. First, independent variables are sampled uniformly from . Then, vertices are connected with probability . This model captures random geometric graphs over the sphere, torus, and hypercube; certain instances of the stochastic block model; and random subgraphs of Cayley graphs. The main question of interest to the current paper is: when is a random algebraic graph statistically and/or computationally distinguishable from ? Our results fall into two categories. (1) Geometric. We focus on the case and use Fourier‐analytic tools. We match and extend the following results from the prior literature: For hard threshold connections, we match for , and for ‐Lipschitz connections we extend the results of when to the non‐monotone setting. (2) Algebraic. We provide evidence for an exponential statistical‐computational gap. Consider any finite group and let be a set of elements formed by including each set of the form independently with probability Let be the distribution of random graphs formed by taking a uniformly random induced subgraph of size of the Cayley graph . Then, and are statistically indistinguishable with high probability over if and only if . However, low‐degree polynomial tests fail to distinguish and with high probability over when 

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5. Local Access to Random Walks , January 2022

   Biswas, A.S.; Pyne, E.; Rubinfeld, R. (January 2022, Innovations in Theoretical Computer Science (ITCS 2022))

   For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work Ω(t). We desire local access algorithms supporting positionG(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/ poly(n) close to those of a uniformly random walk in ℓ1 distance. We first give an algorithm for local access to random walks on a given undirected d-regular graph with eO( 1 1−λ √ n) runtime per query, where λ is the second-largest eigenvalue of the random walk matrix of the graph in absolute value. Since random d-regular graphs G(n, d) are expanders with high probability, this gives an eO(√ n) algorithm for a graph drawn from G(n, d) whp, which improves on the naive method for small numbers of queries. We then prove that no algorithm with subconstant error given probe access to an input d-regular graph can have runtime better than Ω(√ n/ log(n)) per query in expectation when the input graph is drawn from G(n, d), obtaining a nearly matching lower bound. We further show an Ω(n1/4) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree nϵ graphs for any ϵ ∈ (0, 1]) and Cartesian product. 

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