Related papers: Using Expander Graphs to test whether samples are …

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random…

An $(n,d,\lambda)$-graph is a $d$ regular graph on $n$ vertices in which the absolute value of any nontrivial eigenvalue is at most $\lambda$. For any constant $d \geq 3$, $\epsilon>0$ and all sufficiently large $n$ we show that there is a…

Consider two independent Erd\H{o}s-R\'enyi $G(N,1/2)$ graphs. We show that with probability tending to $1$ as $N\to\infty$, the largest induced isomorphic subgraph has size either $\lfloor x_N-\varepsilon_N\rfloor$ or $\lfloor…

We develop a new approach for approximating large independent sets when the input graph is a one-sided spectral expander - that is, the uniform random walk matrix of the graph has its second eigenvalue bounded away from 1. Consequently, we…

It is shown that there exists a sequence of 3-regular graphs $\{G_n\}_{n=1}^\infty$ and a Hadamard space $X$ such that $\{G_n\}_{n=1}^\infty$ forms an expander sequence with respect to $X$, yet random regular graphs are not expanders with…

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,…

An explicit algorithm is presented for testing whether two non-directed graphs are isomorphic or not. It is shown that for a graph of n vertices, the number of n independent operations needed for the test is polynomial in n. A proof that…

Testing for independence between graphs is a problem that arises naturally in social network analysis and neuroscience. In this paper, we address independence testing for inhomogeneous Erd\H{o}s-R\'{e}nyi random graphs on the same vertex…

In this thesis, which is supervised by Dr. David Penman, we examine random interval graphs. Recall that such a graph is defined by letting $X_{1},\ldots X_{n},Y_{1},\ldots Y_{n}$ be $2n$ independent random variables, with uniform…

We initiate the study of property testing in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time, improving on the previous bound of $\tilde{O}(\sqrt{n})$ for graphs on $n$ vertices. The constant-time…

In the regime of two-sample comparison, tests based on a graph constructed on observations by utilizing similarity information among them is gaining attention due to their flexibility and good performances for high-dimensional/non-Euclidean…

We consider numbers and sizes of independent sets in graphs with minimum degree at least $d$, when the number $n$ of vertices is large. In particular we investigate which of these graphs yield the maximum numbers of independent sets of…

The second largest eigenvalue of a graph is an important algebraic parameter which is related with the expansion, connectivity and randomness properties of a graph. Expanders are highly connected sparse graphs. In coding theory, Expander…

Consider a random geometric 2-dimensional simplicial complex $X$ sampled as follows: first, sample $n$ vectors $\boldsymbol{u_1},\ldots,\boldsymbol{u_n}$ uniformly at random on $\mathbb{S}^{d-1}$; then, for each triple $i,j,k \in [n]$, add…

We begin with a new analysis of formal words. Let w be a formal word in letters g_1,...,g_k. The word map associated with w maps the permutations s_1,...,s_k in S_n to the permutation obtained by replacing for each i, every occurrence of…

A graph $G$ is called self-ordered (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from $G$ to any graph that is isomorphic to $G$. We say that $G=(V,E)$ is robustly…

We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method…

This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. First, we prove that,…

Let $G$ be a simple graph of order $n$. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of $G$ is the polynomial $I(G,x)=\sum_{k=0}^{n} s(G,k) x^{k}$, where $s(G,k)$ is the number of…

This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. More precisely, in the family of of $d$-regular graphs, which graph $G$ maximizes/minimizes the…