Eigenvalues and eigenfunctions of a Hamming ball

We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube $Q_d$ of a given order. We believe that in most cases, Hamming balls are maximisers, and our results support this belief. We show that the Hamming balls of radius $o(d)$ have largest eigenvalue that is within $1 + o(1)$ of the maximum value. We also prove that Hamming balls with fixed radius maximise the largest eigenvalue exactly, rather than asymptotically, when $d$ is sufficiently la...

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...

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.

The Hamming graph $H(n,q)$ is the graph whose vertices are the words of length $n$ over the alphabet $\{0,1,\ldots,q-1\}$, where two vertices are adjacent if they differ in exactly one coordinate. The adjacency matrix of $H(n,q)$ has $n+1$ distinct eigenvalues $n(q-1)-q\cdot i$ with corresponding eigenspaces $U_{i}(n,q)$ for $0\leq i\leq n$. In this work we study functions belonging to a direct sum $U_i(n,q)\oplus U_{i+1}(n,q)\oplus\ldots\oplus U_j(n,q)$ for $0\leq i\leq j\le...

The Laplacian matrix of the $n$-dimensional hypercube has $n+1$ distinct eigenvalues $2i$, where $0\leq i\leq n$. In 2004, B\i y\i ko\u{g}lu, Hordijk, Leydold, Pisanski and Stadler initiated the study of eigenfunctions of hypercubes with the minimum number of weak and strong nodal domains. In particular, they proved that for every $1\leq i\leq \frac{n}{2}$ there is an eigenfunction of the hypercube with eigenvalue $2i$ that have exactly two strong nodal domains. Based on comp...

In this work we present a survey of results on the problem of finding the minimum cardinality of the support of eigenfunctions of graphs.

We study functions defined on the vertices of the Hamming graphs $H(n,q)$. The adjacency matrix of $H(n,q)$ has $n+1$ distinct eigenvalues $n(q-1)-q\cdot i$ with corresponding eigenspaces $U_{i}(n,q)$ for $0\leq i\leq n$. In this work, we consider the problem of finding the minimum possible support (the number of nonzeros) of functions belonging to a direct sum $U_i(n,q)\oplus U_{i+1}(n,q)\oplus\ldots\oplus U_j(n,q)$ for $0\leq i\leq j\leq n$. For the case $n\geq i+j$ and $q\...

We find minimal supports of eigenfunctions of Hamming graphs for eigenvalue n(q-1)-q and describe eigenfunctions with minimal support.

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.

We prove a vertex isoperimetric inequality for the $n$-dimensional Hamming ball $\mathcal{B}_n(R)$ of radius $R$. The isoperimetric inequality is sharp up to a constant factor for sets that are comparable to $\mathcal{B}_n(R)$ in size. A key step in the proof is a local expansion phenomenon in hypercubes.