Eigenvalues of neutral networks: interpolating between hypercubes

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...

We give an upper bound on the smallest eigenvalue of the adjacency matrix of graphs with no p-cliques.

Let $\Gamma=(V,E)$ be a graph. The square graph $\Gamma^2$ of the graph $\Gamma$ is the graph with the vertex set $V(\Gamma^2)=V$ in which two vertices are adjacent if and only if their distance in $\Gamma$ is at most two. The square graph of the hypercube $Q_n$ has some interesting properties. For instance, it is highly symmetric and panconnected. In this paper, we investigate some algebraic properties of the graph ${Q^2_n}$. In particular, we show that the graph ${Q^2_n}$...

In this paper we study the maximum value of the largest eigenvalue for simple bipartite graphs, where the number of edges is given and the number of vertices on each side of the bipartition is given. We state a conjectured solution, which is an analog of the Brualdi- Hoffman conjecture for general graphs, and prove the conjecture in some special cases.

Let $\mathcal{H}$ be a uniform hypergraph. Let $\mathcal{A(H)}$ and $\mathcal{Q(H)}$ be the adjacency tensor and the signless Laplacian tensor of $\mathcal{H}$, respectively. In this note we prove several bounds for the spectral radius of $\mathcal{A(H)}$ and $\mathcal{Q(H)}$ in terms of the degrees of vertices of $\mathcal{H}.$

We derive bounds on the size of an independent set based on eigenvalues. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erd\H{o}s-R\'{e}nyi graphs. We investigate further properties of our bounds, and show how our results on the Erd\H{o}s-R\'{e}nyi graphs can be extended to other polarity graphs.

We study the relative importance of "top-speed" (long-term growth rate) and "acceleration" (how quickly the long-term growth rate can be reached) in the evolutionary race to increase population size. We observe that fitness alone does not capture growth rate: robustness, a property of neutral network shape, combines with fitness to include the effect of deleterious mutations, giving growth rate. Similarly, we show that growth rate alone does not capture population size: regul...

For real $\alpha\in [0,1)$ and a hypergraph $G$, the $\alpha$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ is the adjacency matrix of $G$, which is a symmetric matrix with zero diagonal such that for distinct vertices $u,v$ of $G$, the $(u,v)$-entry of $A(G)$ is exactly the number of edges containing both $u$ and $v$, and $D(G)$ is the diagonal matrix of row sums of $A(G)$. We study the $\alpha$-spectr...

Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these graphs have optimal diameter. We study twisted hypercubes in the setting where the instances can have general dependence, and also in the particular case where they are identical. We show that the resultant graph shares properties with random...

In this paper, we give a new proof on a Theorem of Tutte which says that the determinants of the adjacency matrices of two hypomorphic graphs are the same. We also study the lowest eigenvectors.