*2.2. Graph Theory*

A communication topology G(N , *ξ*, A) contains three elements: N ∈{1, ... , *N*} is the node set of MASs, *<sup>ξ</sup>* ⊂N ×N represents an edge set for nodes in MASs, and A ∈ *<sup>R</sup>N*×*<sup>N</sup>* is an adjacent matrix of the graph. For <sup>A</sup> = [*aij*] <sup>∈</sup> *<sup>R</sup>N*×*N*, *aji* is the weight from node *<sup>i</sup>* to node *j*. In this paper, the graph has no self loops, i.e., *aii* = 0, *i* ∈ N . *aji* = 0 if and only if (*i*, *j*) ∈ *ξ*. For node *i*, its neighborhood is denoted as N*<sup>i</sup>* = {*j*|*j* ∈ N , *aji* = 0, *j* = *i*}. For G, it is defined as a strongly connected graph if and only if any node in it is mutually reachable. For adjacent matrix A, the corresponding Laplacian matrix is defined as L = [*lij*]*N*×*N*, where *lii* = Σ*<sup>N</sup> <sup>j</sup>*=1*aij* and *lij* = −*aij*, *i* = *j*. If a topology is call as a balanced topology, the edges of it must be balanced, which means that edge (*i*, *j*) belongs to set *ξ* if edge (*j*, *i*) belongs to set *ξ*.

**Lemma 1** ([34])**.** *for a Laplacian matrix* L*, zero is one of the eigenvalues and it has a fixed right eigenvector* 1*N. The other nonzero eigenvalues are all positive. If there exists a directed spanning tree in* G*, zero is a simple eigenvalue of* L*.*

**Lemma 2** ([35])**.** *considering* G *as a strongly connected graph, suppose that r* = [*r*1, ... ,*rN*] *is the left eigenvector connected with the eigenvalue zero. Then, we have <sup>R</sup>*<sup>L</sup> <sup>+</sup> <sup>L</sup>*TR* <sup>≥</sup> <sup>0</sup>*, where R* = *diag*{*r*1,...,*rN*}*. For a balanced graph, we have r*<sup>1</sup> = ··· = *rN.*

**Lemma 3** ([35])**.** *If* G *is a strongly connected graph; its generalized algebraic connectivity is define as <sup>a</sup>*(L) = *min <sup>r</sup><sup>T</sup> <sup>x</sup>*=0,*<sup>x</sup>*<sup>=</sup><sup>0</sup> *<sup>x</sup>T*(*R*L+L*TR*)*<sup>x</sup>* <sup>2</sup>*xTRx . Due to the topology in this paper being defined as balanced, matrix R can be converted into R* = *r*<sup>1</sup> *IN. The generalized algebraic connectivity is equal to a*(L) = *<sup>λ</sup>min*( *He*(L) <sup>2</sup> )*.*
