## 1 Introduction  
Leaders around the world obviously prefer the present era of connectivity as the Fourth Industrial Revolution [2]. Industry 4.0 has been significantly contributing to the transformation of many industries such as transportation, manufacturing, health-care, agriculture, etc. via enabling data transmission and integration between disciplines. Today, we are in the fourth one, a generation of connection between our physical, digital, social, and biological worlds. In particular, data and information from these different areas have been made available and have been connected in complex and dense networks. For better integration and utilization, control aspect of these complex networks, researchers from different communities have brought different contributions varying from topology inference to control strategy, deputizing for interacting systems, which are modeled by graphs, whose vertices represent the components of the system while edges stand for the interactions between these components. In the last decade there has been dramatic increasing number of publications in the cooperative control of multi-agent systems. In control of multi-agent systems, the performance of the whole system depends on both structure and the connections between individuals of the systems. Here, the total connections can be defined by the graph Laplacian matrix, and its spectrum is involved in some useful properties of the network system [3; 4]. For instance, the second smallest graph Laplacian eigenvalue, i.e., the so-called algebraic connectivity of the graph, has the main role in the convergence time of various distributed algorithms as well as the performance and robustness of dynamical systems [5]. Conceptually, agents share their information with each other to achieve common objectives, relative position information, or common control algorithms. This is called consensus problem [6; 7], where a group of agents' approaches average consensus in an undirected network under simple linear iteration scheme. It is well known that in a multi-agent system, consensus is achieved if and only if the network is connected (the algebraic connectivity) being strictly greater than zero [8]. On the other hand, the largest Laplacian eigenvalue is an important factor to decide the stability of the system. For example, minimizing the spectral radius in [9] leads to maximize the robustness of the network to time delays under a linear consensus protocol. Furthermore, to speed up consensus algorithms, the optimal Laplacian-based consensus matrix is obtained with a stepsize which is the inverse of the sum of the smallest and the largest non-zero graph Laplacian eigenvalue [10]. Moreover, the authors in [11; 12] have proved that the spectrum of the Laplacian matrix can be used to design consensus matrices to obtain average consensus in finite number of steps. In order to investigate network efficiency, structural robustness of a network which is related to its performance despite changes in the network topology [13] has been also studied. The concept of natural connectivity as a spectral measure of robustness was introduced in [14]. It is expressed in mathematical form as the average eigenvalue of the adjacency matrix of the graph representing the network topology. The Laplacian spectrum \(sp(\mathbf{L})=\{\lambda_{1}^{m_{1}},\ldots,\lambda_{i},\ldots\}\) can also be employed to compute the robustness indices, for instance, the number of spanning trees and the effective graph resistance (Kirchhoff index) [15]. The smaller (or greater) the Kirchhoff index (or the number of spanning trees) is, the more robust the network becomes. In addition, it has been pointed out that adding an edge strictly decreases the Kirchhoff index and hence increases the robustness. In [16], the authors have proposed a method to monitor collaboratively the robustness of the networks partitioned into sub-networks by Kirchhoff index \(\mathcal{R}_{\mathbf{L}}=N\sum_{i=2}^{D+1}\frac{m_{i}}{\lambda_{i}}\).