Distributed Bounds on the Algebraic Connectivity of Graphs With Application to Agent Networks

This paper establishes several bounds on the algebraic connectivity and spectral radius of graphs. Before deriving these bounds, a directed graph with a leader node is first investigated, for which some bounds on the spectral radius and the smallest real part of all the eigenvalues of M = L + D are obtained using the properties of M-matrix and non-negative matrix under a mild assumption, where L is the Laplacian matrix of the graph and D = diag{d
₁
, d
₂
, ..., d
_N
} with d
_i
> 0 if node i can access the information of the leader node and 0 otherwise. Subsequently, by virtue of the results on directed graphs, the bounds on the algebraic connectivity and spectral radius of an undirected connected graph are provided. Besides establishing these bounds, another important feature is that all these bounds are distributed in the sense of only knowing the information of edge weights' bounds and the number of nodes in a graph, without using any information of inherent structures of the graph. Therefore, these bounds can be in some sense applied to agent networks for reducing the conservatism where control gains in control protocols depend on the eigenvalues of matrices M or L, which are global information. Also some examples are provided for corroborating the feasibility of the theoretical results.