Spectral gap of a weighted 3-simplicial complex

In this paper, we construct a new framework that’s we call the weighted [Formula: see text]-simplicial complex and we define its spectral gap. An upper bound for our spectral gap is given by generalizing the Cheeger constant. The lower bound for our spectral gap is obtained from the first nonzero eigenvalue of the Laplacian acting on the functions of certain weighted [Formula: see text]-simplicial complexes.

Fast Consensus Seeking on Networks with Antagonistic Interactions

It is well known that all agents in a multiagent system can asymptotically converge to a common value based on consensus protocols. Besides, the associated convergence rate depends on the magnitude of the smallest nonzero eigenvalue of Laplacian matrix L. In this paper, we introduce a superposition system to superpose to the original system and study how to change the convergence rate without destroying the connectivity of undirected communication graphs. And we find the result if the eigenvector x of eigenvalue λ has two identical entries xi=xj, then the weight and existence of the edge eij do not affect the magnitude of λ, which is the argument of this paper. By taking advantage of the inequality of eigenvalues, conditions are derived to achieve the largest convergence rate with the largest delay margin, and, at the same time, the corresponding topology structure is characterized in detail. In addition, a method of constructing invalid algebraic connectivity weights is proposed to keep the convergence rate unchanged. Finally, simulations are given to demonstrate the effectiveness of the results.

On Consensus of Star-Composed Networks with an Application of Laplacian Spectrum

In this paper, we mainly study the performance of star-composed networks which can achieve consensus. Specifically, we investigate the convergence speed and robustness of the consensus of the networks, which can be measured by the smallest nonzero eigenvalue λ2 of the Laplacian matrix and the H2 norm of the graph, respectively. In particular, we introduce the notion of the corona of two graphs to construct star-composed networks and apply the Laplacian spectrum to discuss the convergence speed and robustness for the communication network. Finally, the performances of the star-composed networks have been compared, and we find that the network in which the centers construct a balanced complete bipartite graph has the most advantages of performance. Our research would provide a new insight into the combination between the field of consensus study and the theory of graph spectra.

On Optimality of Designs with Three Distinct Eigenvalues

Let ${\cal D}_{v,b,k}$ denote the family of all connected block designs with $v$ treatments and $b$ blocks of size $k$. Let $d\in{\cal D}_{v,b,k}$. The replication of a treatment is the number of times it appears in the blocks of $d$. The matrix $C(d)=R(d)-\frac{1}{k}N(d)N(d)^\top$ is called the information matrix of $d$ where $N(d)$ is the incidence matrix of $d$ and $R(d)$ is a diagonal matrix of the replications. Since $d$ is connected, $C(d)$ has $v-1$ nonzero eigenvalues $\mu_1(d),\ldots,\mu_{v-1}(d)$.Let ${\cal D}$ be the class of all binary designs of ${\cal D}_{v,b,k}$. We prove that if there is a design $d^*\in{\cal D}$ such that (i) $C(d^*)$ has three distinct eigenvalues, (ii) $d^*$ minimizes trace of $C(d)^2$ over $d\in{\cal D}$, (iii) $d^*$ maximizes the smallest nonzero eigenvalue and the product of the nonzero eigenvalues of $C(d)$ over $d\in{\cal D}$, then for all $p>0$, $d^*$ minimizes $\left(\sum_{i=1}^{v-1}\mu_i(d)^{-p}\right)^{1/p}$ over $d\in{\cal D}$. In the context of optimal design theory, this means that if there is a design $d^*\in{\cal D}$ such that its information matrix has three distinct eigenvalues satisfying the condition (ii) above and that $d^*$ is E- and D-optimal in ${\cal D}$, then $d^*$ is $\Phi_p$-optimal in ${\cal D}$ for all $p>0$. As an application, we demonstrate the $\Phi_p$-optimality of certain group divisible designs. Our proof is based on the method of KKT conditions in nonlinear programming.

ON A GENERALIZATION OF A THEOREM OF LICHNEROWICZ TO MANIFOLDS WITH BOUNDARY

A theorem due to Lichnerowicz which establishes a lower bound on the lowest nonzero eigenvalue of the Laplacian acting on functions on a compact, closed manifold is reviewed. It is shown how this theorem can be extended to the case of a manifold with nonempty boundary. Lower bounds for different boundary conditions, analogous to the empty boundary case, are formulated and some novel proofs are presented.