Abstract:
This paper investigates some relationship between the algebraic connectivity and the clique number of graphs. We characterize all extremal graphs which have the maximum and minimum the algebraic connectivity among all graphs of order $n$ with the clique number $r$, respectively. In turn, an upper and lower bounds for the clique number of a graph in terms of the algebraic connectivity are obtained. Moreover, a spectral version of the Erd\H{o}s-Stone theorem in terms of the algebraic connectivity of graphs is presented.

Abstract:
We derive a new upper bound on the algebraic connectivity of a regular graph using the Higman-Sims technique. Together with a new result on the connectivity of the neighbourhood graph of strongly regular graphs, our result gives a characterization of a class of strongly regular graphs that maximize the algebraic connectivity amongst regular graphs.

Abstract:
We investigate the bounds on algebraic connectivity of graphs subject to constraints on the number of edges, vertices, and topology. We show that the algebraic connectivity for any tree on $n$ vertices and with maximum degree $d$ is bounded above by $2(d-2) \frac{1}{n}+O(\frac{\ln n}{n^{2}}) .$ We then investigate upper bounds on algebraic connectivity for cubic graphs. We show that algebraic connectivity of a cubic graph of girth $g$ is bounded above by $3-2^{3/2}\cos(\pi/\lfloor g/2\rfloor) ,$ which is an improvement over the bound found by Nilli [A. Nilli, Electron. J. Combin., 11(9), 2004]. Finally, we propose several conjectures and open questions.

Abstract:
We consider the problem of maximizing the algebraic connectivity of the communication graph in a network of mobile robots by moving them into appropriate positions. We define the Laplacian of the graph as dependent on the pairwise distance between the robots and we approximate the problem as a sequence of Semi-Definite Programs (SDP). We propose a distributed solution consisting of local SDP's which use information only from nearby neighboring robots. We show that the resulting distributed optimization framework leads to feasible subproblems and through its repeated execution, the algebraic connectivity increases monotonically. Moreover, we describe how to adjust the communication load of the robots based on locally computable measures. Numerical simulations show the performance of the algorithm with respect to the centralized solution.

Abstract:
In this paper we consider the following problem: Over the class of all simple connected graphs of order $n$ with $k$ pendant vertices ($n,k$ being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also discuss the algebraic connectivity of unicyclic graphs.

Abstract:
We prove that the algebraic connectivity a(G) of a graph embedded on a nonplanar surface satisfies a Heawood-type result. More precisely, it is shown that the algebraic connectivity of a surface S, defined as the supremum of a(G) over all graphs that can be embedded in S, is equal to the chromatic number of S. Furthermore, and with the possible exception of the Klein bottle, we prove that this bound is attained only in the case of the maximal complete graph that can be embedded in S. In the planar case, we show that, at least for some classes of graphs which include the set of regular graphs, a(G) is less than or equal to four. As an application of these results and techniques, we obtain a lower bound for the genus of Ramanujan graphs. We also present some bounds for the asymptotic behaviour of a(G) for certain classes of graphs as the number of vertices goes to infinity.

Abstract:
The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis and Bjorner. In this note, we provide a unifying approach which allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley--Reisner rings.

Abstract:
In this paper we characterize the unique graph whose algebraic connectivity is minimum among all connected graphs with given order and fixed matching number or edge covering number, and present two lower bounds for the algebraic connectivity in terms of the matching number or edge covering number.

Abstract:
For connectivity of \emph{random geometric graphs}, where there is no density for underlying distribution of the vertices, we consider $n$ i.i.d. \emph{Cantor} distributed points on $[0,1]$. We show that for this random geometric graph, the connectivity threshold $R_{n}$, converges almost surely to a constant $1-2\phi$ where $0 < \phi < 1/2$, which for the standard Cantor distribution is 1/3. We also show that $\| R_n - (1 - 2 \phi) \|_1 \sim 2 \, C(\phi) \, n^{-1/d_{\phi}}$ where $C(\phi) > 0$ is a constant and $d_{\phi} := - {\log 2}/{\log \phi}$ is the \emph{Hausdorff dimension} of the generalized Cantor set with parameter $\phi$.

Abstract:
Inspired by the concept of network algebraic connectivity, we adopt an extended notion named rigidity preservation index to characterize the rigidity property for a formation framework. A gradient based controller is proposed to ensure the rigidity preservation of multi-robot networks in an unknown environment, while the rigidity metric can be maximized over time during robots' motions. In order to implement the controller in a distributed manner, a distributed inverse power iteration algorithm is developed which allows each robot to estimate the global rigidity index information. Simulation results are provided to demonstrate the effectiveness of the estimation and control scheme.