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تسجيل مستخدم جديدRandom growth on a Ramanujan graph
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The behavior of a certain random growth process is analyzed on arbitrary regular and non-regular graphs. Our argument is based on the Expander Mixing Lemma, which entails that the results are strongest for Ramanujan graphs, which asymptotically maximize the spectral gap. Further, we consider ErdH{o}s--Renyi random graphs and compare our theoretical results with computational experiments on flip graphs of point configurations. The latter is relevant for enumerating triangulations.
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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
radius $rho(X)$ of $X$, and more generally $k$-quasi-$X$-Ramanujan if $lambda_k(G)$ is at most $rho(X)$. In case $X$ is the infinite $Delta$-regular tree, this reduces to the well known notion of a finite $Delta$-regular graph being Ramanujan. Inspired by the Interlacing Polynomials method of Marcus, Spielman, and Srivastava, we show the existence of infinitely many $k$-quasi-$X$-Ramanujan graphs for a variety of infinite $X$. In particular, $X$ need not be a tree; our analysis is applicable whenever $X$ is what we call an additive product graph. This additive product is a new construction of an infinite graph $mathsf{AddProd}(A_1, dots, A_c)$ from finite atom graphs $A_1, dots, A_c$ over a common vertex set. It generalizes the notion of the free product graph $A_1 * cdots * A_c$ when the atoms $A_j$ are vertex-transitive, and it generalizes the notion of the universal covering tree when the atoms $A_j$ are single-edge graphs. Key to our analysis is a new graph polynomial $alpha(A_1, dots, A_c;x)$ that we call the additive characteristic polynomial. It generalizes the well known matching polynomial $mu(G;x)$ in case the atoms $A_j$ are the single edges of $G$, and it generalizes the $r$-characteristic polynomial introduced in [Ravichandran16, Leake-Ravichandran18]. We show that $alpha(A_1, dots, A_c;x)$ is real-rooted, and all of its roots have magnitude at most $rho(mathsf{AddProd}(A_1, dots, A_c))$. This last fact is proven by generalizing Godsils notion of treelike walks on a graph $G$ to a notion of freelike walks on a collection of atoms $A_1, dots, A_c$.
We introduce the random graph $mathcal{P}(n,q)$ which results from taking the union of two paths of length $ngeq 1$, where the vertices of one of the paths have been relabelled according to a Mallows permutation with real parameter $0<q(n)leq 1$. Thi
s random graph model, the tangled path, goes through an evolution: if $q$ is close to $0$ the graph bears resemblance to a path and as $q$ tends to $1$ it becomes an expander. In an effort to understand the evolution of $mathcal{P}(n,q)$ we determine the treewidth and cutwidth of $mathcal{P}(n,q)$ up to log factors for all $q$. We also show that the property of having a separator of size one has a sharp threshold. In addition, we prove bounds on the diameter, and vertex isoperimetric number for specific values of $q$.
The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group $
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We study the distributional properties of horizontal visibility graphs associated with random restrictive growth sequences and random set partitions of size $n.$ Our main results are formulas expressing the expected degree of graph nodes in terms of
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For a binomial random variable $xi$ with parameters $n$ and $b/n$, it is well known that the median equals $b$ when $b$ is an integer. In 1968, Jogdeo and Samuels studied the behaviour of the relative difference between ${sf P}(xi=b)$ and $1/2-{sf P}
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