Eigenvalues of neutral networks: interpolating between hypercubes

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.

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...

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.

This paper considers a kind of generalized measure $\lambda_s^{(h)}$ of fault tolerance in a hypercube-like graph $G_n$ which contain several well-known interconnection networks such as hypercubes, varietal hypercubes, twisted cubes, crossed cubes and M\"obius cubes, and proves $\lambda_s^{(h)}(G_n)= 2^h(n-h)$ for any $h$ with $0\leqslant h\leqslant n-1$ by the induction on $n$ and a new technique. This result shows that at least $2^h(n-h)$ edges of $G_n$ have to be removed t...

The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. This paper is primarily a survey of various aspects of the eigenvalues of the Laplacian matrix of a graph for the past teens. In a...

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...

This paper introduces a new variant of hypercubes, which we call Z-cubes. The n-dimensional Z-cube $H_n$ is obtained from two copies of the (n-1)-dimensional Z-cube $H_{n-1}$ by adding a special perfect matching between the vertices of these two copies of $H_{n-1}$. We prove that the n-dimensional Z-cubes $H_n$ has diameter $(1+o(1))n/\log_2 n$. This greatly improves on the previous known variants of hypercube of dimension n, whose diameters are all larger than n/3. Moreover,...

Lower bounds for the first and the second eigenvalue of uniform hypergraphs which are regular and linear are obtained. One of these bounds is a generalization of the Alon-Boppana Theorem to hypergraphs.

In this paper, we determine the set of all distinct eigenvalues of the line graph which is induced by the first and second layers of the hypercube $ Q_n $, $n>3$. We show that this graph has precisely five distinct eigenvalues and all of its eigenvalues are integers

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