Eigenvalues of subgraphs of the cube

We describe the eigenvalues and the eigenspaces of the adjacency matrices of subgraphs of the Hamming cube induced by Hamming balls, and more generally, by a union of adjacent concentric Hamming spheres. As a corollary, we extend the range of cardinalities of subsets of the Hamming cube for which Hamming balls have essentially the largest maximal eigenvalue (among all subsets of the same size). We show that this holds even when the sets in question are large, with cardinali...

We consider a sparse random subraph of the $n$-cube where each edge appears independently with small probability $p(n) =O(n^{-1+o(1)})$. In the most interesting regime when $p(n)$ is not exponentially small we prove that the largest eigenvalue of the graph is asymtotically equal to the square root of the maximum degree.

Let G be a random subgraph of the n-cube where each edge appears randomly and independently with probability p. We prove that the largest eigenvalue of the adjacency matrix of G is almost surely \lambda_1(G)= (1+o(1)) max(\Delta^{1/2}(G),np), where \Delta(G) is the maximum degree of G and o(1) term tends to zero as max (\Delta^{1/2}(G), np) tends to infinity.

A neutral network is a subgraph of a Hamming graph, and its principal eigenvalue determines its robustness: the ability of a population evolving on it to withstand errors. Here we consider the most robust small neutral networks: the graphs that interpolate pointwise between hypercube graphs of consecutive dimension (the point, line, line and point in the square, square, square and point in the cube, and so on). We prove that the principal eigenvalue of the adjacency matrix of...

The hypercube Q_n is the graph whose vertex set is {0,1}^n and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph H of the cube, let ex(Q_n, H) be the maximum number of edges in a subgraph of Q_n which does not contain a copy of H. We find a wide class of subgraphs H, including all previously known examples, for which ex(Q_n, H) = o(e(Q_n)). In particular, our method gives a unified approach to proving that ex(Q_n, C_{2t}) = o(e(Q_n)) f...

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. These results are obtained as corollaries of a new family of graphs, which we construct by pi...

In this paper we study random induced subgraphs of the binary $n$-cube, $Q_2^n$. This random graph is obtained by selecting each $Q_2^n$-vertex with independent probability $\lambda_n$. Using a novel construction of subcomponents we study the largest component for $\lambda_n=\frac{1+\chi_n}{n}$, where $\epsilon\ge \chi_n\ge n^{-{1/3}+ \delta}$, $\delta>0$. We prove that there exists a.s. a unique largest component $C_n^{(1)}$. We furthermore show that $\chi_n=\epsilon$, $| C_...

Let $Q_n$ denote the graph of the $n$-dimensional cube with vertex set $\{0,1\}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. Suppose $G$ is a subgraph of $Q_n$ with average degree at least $d$. How long a path can we guarantee to find in $G$? Our aim in this paper is to show that $G$ must contain an exponentially long path. In fact, we show that if $G$ has minimum degree at least $d$ then $G$ must contain a path of length $2^d-1$. Note tha...

This paper investigates the asymptotic nature of graph spectra when some edges of a graph are subdivided sufficiently many times. In the special case where all edges of a graph are subdivided, we find the exact limits of the $k$-th largest and $k$-th smallest eigenvalues for any fixed $k$. It is expected that after subdivision, most eigenvalues of the new graph will lie in the interval $[-2,2]$. We examine the eigenvalues of the new graph outside this interval, and we prove s...

Let $d \geq 1$ and $s \leq 2^d$ be nonnegative integers. For a subset $A$ of vertices of the hypercube $Q_n$ and $n\geq d$, let $\lambda(n,d,s,A)$ denote the fraction of subcubes $Q_d$ of $Q_n$ that contain exactly $s$ vertices of $A$. Let $\lambda(n,d,s)$ denote the maximum possible value of $\lambda(n,d,s,A)$ as $A$ ranges over all subsets of vertices of $Q_n$, and let $\lambda(d,s)$ denote the limit of this quantity as $n$ tends to infinity. We prove several lower and uppe...