Inspired by the swarming behaviors of biological systems and great promises in numerous applications, the field of controllability of multi-agent systems has been studied extensively in recent years [1, 2, 3].By introducing the concept of matching, Liu et al.’s paper  printed in Nature in 2011 gives a method to find the minimum leaders set for directed networks. However, as pointed out by Ji in , when the topological structure of the systems is undirected, how to locate the leaders and what is the minimum number of leaders to insure the controllability are still difficult and largely unknown problems.
1.1 Literature Review
The neighbor-based controllability of undirected graph under a single leader was first formulated by Tanner in 
and a necessary and sufficient condition expressed in terms of eigenvalue and eigenvector was derived. In case of multiple leaders, some other algebraic conditions were developed in[6, 7, 8, 9] etc. These algebraic conditions lay the foundation for understanding interaction between topological structures of undirected graph and its controllability. And they are also serve as the theoretical basis of this paper. The research efforts on characterizing the controllability from a graphical point of view was also motivated by  to build controllable topologies. Many kinds of uncontrollable topologies were characterized, such as a symmetric graph with respect to the anchored nodes , quotient graphs  , nodes with the same number of neighbors , controllability destructive nodes  etc. Useful tools and methods were developed to study the controllability of undirected graph, such as downer branch for tree graphs , Zero forcing set[3, 13], equitable partitions [6, 14, 15, 16, 17], leader and follower subgraphs , -core vertex [18, 19]
, Distance-to-Leaders (DL) Vector, etc. Omnicontrollable systems are defined by , in such systems, the choice of leader vertices that control the follower graph is arbitrary. Minimal controllability problem (MCP) that aims to determine the minimum number of state variables that need to be actuated to ensure system s controllability was studied in [21, 22]. In study , two algorithms are established for selecting the fewest leaders to preserve the controllability and the algorithm for leaders locations to maximize non-fragility is also designed. Necessary and sufficient conditions to characterize all and only the nodes from which the path or cycle network systerm is controllability were provided in [24, 25].
Although many scholars have devoted themselves to the research in the controllability of undirected graph and achieved many remarkably strong and elegant results, this problem has not been solved yet. As it is well known that any undirected simple connected graph on vertices is always -omnicontrollable. To insure the minimal controllability, which vertices should be selected as leaders is important. Therefore, Our aim is to find a method for giving a direct interpretation of the leader vertices from a graph-theoretic vantage point. In this sense, we provide a new concept, minimal perfect critical vertex set, to identified the potential leader vertices. This provides a new direction for the study of controllability of undirected systems.
1.2 Notations and Preliminary Results
Let be an undirected and unweighted simple graph, where is a vertex set and is an edge set, with an edge is an unordered pair of distinct vertices in . If , then and are said to be adjacent, or neighbors. represents the neighboring set in of , where . The cardinality of is denoted by . is the induced subgraph, whose vertex set is and edge set is . The valency matrix of graph is a diagonal matrix with rows and columns indexed by , in which the -entry is the degree of vertex , e.g. . Any undirected simple graph can be represented by its adjacency matrix, , which is a symmetric matrix with 0-1 elements. The element in position in is 1 if vertices and are adjacent and 0 otherwise. The symmetric matrix defined as:
is the Laplacian of .The Laplacian is always symmetric and positive semidefinite, and the algebraic multiplicity of its zero eigenvalue is equal to the number of connected components in the graph. For a connected graph, the dimensional eigenvector associated with the single zero eigenvalue is the vector of ones, .
Throughout this paper, it is assumed without loss of generality that denotes follower vertex set and its vertices play followers role, and the vertices in are leaders(driver nodes), where denotes the complement set of . Let be a vector, denote the vector obtained from after deleting the elements in . Let denote the matrix obtained from after deleting the rows in and columns in . The system described by undirected graph is said to be controllable (for convenience, is controllable )if it can be driven from any initial state to any desired state in finite time. If the followers’ dynamics is (see (4) in )
where captures the state of a system which is the stack vector of all corresponding to follower vertex and is the external control inputs vector which is imposed by the controller and is injected to only some of the vertices, namely the leaders, the system is controllable with the follower vertex set if and only if the controllability matrix
For example, if the vertex is selected as leader, the system is controllable(see fig.1).But, if plays the leaders role, it is NOT controllable. This paper will address the graphical characterization of leaders to insure the systems’s controllability.
In most real systems such as multi-agent systems or complex networks, we are particularly interested in identifying the minimum number of leaders, whose control is sufficient and fully control the system’s dynamics.
The Proposition 2 gives the algebraic characteristics of leader vertex set. It is worth noting that the eigenvector y in Proposition 2 has the characteristic of arbitrary. Therefore, when L has multiple eigenvalues, it is not possible to draw a conclusion only by examining all the linearly independent eigenvectors, but also by further verifying all the eigenvectors with zero components. From the point of view of numerical calculation, this verification is too computational and difficult to implement. It is clear that the topology of the interconnection graph completely determines its controllability properties. So, this paper will focus on the graph theoretic characterization of the leader vertices.
The remainder of this paper is organized as follows. In Section 2, we provide three new concepts: critical vertex set, perfect critical vertex set and minimal perfect critical vertex set. Necessary and sufficient conditions for to be a minimal perfect critical 2 vertex set is presented. An interesting result that there never exist minimal perfect critical 3 vertex set is also proved in Section 2. Section 3 is the main part of this paper. In this section, we provide a algorithm to locate all leader vertices of path by finding out its all minimal perfect critical vertex set. Graphs constructed by adding paths incident to one vertex are investiaged in Section 4. Finally, our conclusions are summarized in Section 5.
2 Minimal Perfect Critical Vertex Set
According to proposition 2, for any and , if there exist an eigenvector y of Laplace matrix L such that , then cannot be used as a leader vertex set. So, in order to locate the leaders of graph , the following concepts are proposed.
2.1 Three Definitions
(critical vertex set) Let be a nonempty subset of , if there exist an eigenvector y such that , then is called a critical vertex set(CVS) and y is a inducing eigenvector. is called a critical vertex set, if .
(perfect critical vertex set) Let be a critical vertex set, if there exist a eigenvector y satisfy that , then is called a perfect critical vertex set(PCVS). And is called a perfect critical vertex set, if .
(minimal perfect critical vertex set) A perfect critical vertex set is called a minimal perfect critical vertex set (MPCVS) if its any proper subset is no longer a perfect critical vertex set. And is called a minimal perfect critical vertex set, if .
By the Definitions, is a trivially CVS and a PCVS induced by the eigenvector . is a MPCVS if and only if is controllable under any single vertex selected as leader, e.g. is omnicontrollable.
Proposition 2 can be restated as: The undirected graph is controllable under the leader vertex set if and only if for each MPCVS , .
It is Remark 2 that inspired us to study MPCVS. Because, from Remark 2 , there is a close relationship between MPCVS and the minimum leaders set. In other words, when we find all MPCVS of , we find the minimum leader set and hence the minimum number of leader vertices. For example, by Remark 2 and all its 4 MPCVS above, graph in fig.2 is not controllable under any single leader because , or any two vertices. So, the minimum leaders set are , where comes from and from . Therefor, the minmum number of leaders is 3.
2.2 Sufficient Conditions for Critical Vertex Set
For undirected graph, Laplacian matrix L is symmetric, all the eigenvectors are orthogonal to each other, so knowing is an eigenvector of L, it is immediate that all the other eigenvectors of L are orthogonal to , that is, for all eigenvector ,
The equality in (1) is useful throughout the paper.
If is a CVS, then
In fact, Suppose , without loss of generality, . Let be the inducing eigenvector associated with eigenvalue , then and . By and (1), , e.g. , this is in contradiction with the fact that y is an eigenvector. Further, since any subset with isn’t a CVS, by Remark 2, is controllable with the leader vertex set when .
Now, we are going to investigate the properties of critical vertex set. Firstly, a sufficient conditions for to be a CVS is provided in the following Proposition 3, which describes a special case of the symmetry-based uncontrollability results.
Let be an undirected connected graph of order , and , if for any , either or , then is a critical vertex set.
Proof Let , then
is Laplacian of subgraph , where denotes the
dimensional identity matrix. Considered (1), there exist an eigenvector of the Laplacian such that .
Set vector y as and . It can be seen that
Noticing that the rows in matrix are either ones or zeros, the conclusion is proved by . ∎
The condition provided in Proposition 3 implies that some critical vertex sets are closely related to equitable partitions. For example, let be the followers and be the leaders. From earlier results in the literature , we know that in the case of Proposition 3 the maximal relaxed equitable partition would put all the leaders into a single cell, hence the system is uncontrollable. But, some other critical vertex sets have nothing to do with equitable partitions or almost equitable partitions(AEP, see ). For example, let be the perfect critical vertex set in fig.4(a). The partitions obtained by putting all the vertices in into a single cell is not a AEP.
2.3 Minimal Perfect Critical 2 and 3 Vertex Set
Armed with the above properties, critical vertex set with can be determined directly from their graphical characterization. This is achieved via a detailed analysis of the inducing eigenvector.
Let be an undirected connected graph and be a perfect critical vertices set, then for any , and .
Proof Let be a perfect critical vertices set and be the inducing eigenvector. since is a perfect critical vertices set.
, suppose the , without loss of generality, say, and , then . On the other hand, and , so , this is a contradiction.
Together with (1), can be proved similarly.∎
Let be an undirected connected graph, and , then is a minimal perfect critical 2 vertex set if and only if , either or .∎
FromTheorem 1, we know that the perfect critical 2 vertex set is what named twins nodes by  and also it is double controllability destructive (DCD) node tuple given by  and . Hence, one can see that the perfect critical vertex set is the extension and generalization of twins nodes and controllability destructive nodes.
But the minimal perfect critical 3 vertex set is not the same as the triple controllability destructive nodes TCD nodes named by , because we will prove that there do not exist a minimal perfect critical 3 vertex set. That is the following Theorem 2.
Let be an undirected connected graph, and , then is NOT a minimal perfect critical vertex set.
Proof Suppose is a minimal perfect critical vertex set.Consider the subgraph , all 4 possible topology structures of are depicted in fig. 3.
For each topology of in fig.3, let be the vertex set of white vertices and be the black vertex, one can have either or .
By Lemma 1, , either or .
Noticing that , by Proposition 3, is a critical vertex set. This contradicts the assumption.∎
Although there does not exist a minimal perfect critical 3 vertex set, minimal perfect critical 4 vertex set does exist, see fig.4(a). From Theorem 1, perfect critical 2 vertex set is completely determined by the relationship between and , and have nothing to do with the interconnection topology of subgraph . But, unlike the perfect critical 2 vertex set, the topology structure of will have an effect on whether is a minimal perfect critical 4 vertex set or not, see fig.4. The virtue that perfect critical vertex set should have was needed to be characterized from both algebraic and graphical perspectives. Developing such a characterization is along the directions of our current research.
3 Minimal Perfect Critical Vertex Set of Path
In this section, we will solve the leaders location problem for path completely by means of MPCVS.
3.1 Spectral Propoerties
A path graph is a finite sequence of vertex starting with and ending with such that consecutive vertex are adjacent. A subset is said to be isolated vertex set where there are no edges among the verties in .
If be a perfect critical vertex set of path , by Lemma 1, must be isolated vertex set. So, without loss of generality, let be a isolated vertex set and . Let , , . Recall Lemma 1, we know that and , e.g.,
It is easy to see that the matrix is a block matrix with the following form
By rearrange the columns and rows we can always write the Laplacian of into the following form:
Since for path , only the consecutive vertices are adjacent, so there are exactly 2 elements are -1 while the other elements are 0 for every vector . That is
For path, the matrixes in right side of (5) have similar structure. Let and be the dimension of the following useful matrix and , respectively. These matrices play an important role to determine the locations of leaders under which the controllability of paths can be realized. Thus, the first submatrix in (5) can be written as . By symmetric permutation reversing all the components, the last submatrix in (5) can be written as . The other submatrices in (5) can be written as .
Naturally, We are going to investigate the spectral properties of and .
For convenient, we introduce some useful notations. For any , is called a angle associated with , where is defined as , and . If is a eigenvalue, then is called a eigenangle. Let and be the -th sequential principal minor of and , respectively, then we have the following useful lemmas.
Let be an eigenvector of .
(i) is a eigenvalue of if and only if is associated with .
Proof (i) By applying the Laplace expansion to the last row of , the following recurrence formula hold:
Then, from Gersgorin disk theorem, it follows that the eigenvalue , that means . Solving this recurrence and taking into consideration, we have
Thus concluding the first part of the proof.
(ii) The claim can be verified via mathematical induction from the fact: ∎
Similarly, we have
Let be an eigenvector of .
(i) is a eigenvalue of if and only if is associated with .
If is a perfect critical vertex set of , then all in (5) have at least one common eigenvalue.
Proof If is a perfect critical vertex set , then there exist an eigenvector y such that and . From (6), we have Now, consider (5), all of have at least one common eigenvalue because are the corresponding eigenvectors.∎
If is a perfect critical vertex set of , for all in (5), the following equalities holds:
Since and is the -th sequential principal minor of , by Proposition 4(ii), the -th element of the eigenvector is zero. This contradicts the fact that is a perfect critical vertex set.
The proof of (ii) can be carried out in the same manner as (i).∎
If is a perfect critical vertex set of , for vertex set and in (5), then either is empty or
Proof If is empty, the proof is trivial; thus, let and .
From Lemma 2, and have a common eigenvalue . Let y be the induced eigenvector of the perfect critical vertex set . According to Proposition 4(i) and Proposition 5(i), there exist numbers , for , such that
We claim that and are coprime. Otherwise, recall the proof of Lemma 3, we know that there exist at least one entry of the eigenvector vanish. Similarly, and are coprime, too. That implies , e.g. . ∎
3.2 Equivalence Characterization of MPCVSs of Path Graphs
The following Theorem 3 provided a equivalence characterization of MPCVS of path graph.
Let be a vertex set of path and is isolated. Let , . Then is a minimal perfect critical vertex set if and only if the following assertions hold:
(i) , .
(ii) or .
(iii) is a odd prime.
Sufficiency: Case 1 .
Since , all of the eigenangles in (10) are common eigenangles of all submatrices . None of the eigenangles in (10) is a eigenangle of any sequential principal minor of because of the condition (iii). So, from Proposition 4(ii) and Proposition 5(ii), we know any eigenvectors associated with the common eigenangles have zero elements. Therefor, we only need to proof that there exist a eigenvector y of L such that .
Arbitrarily selecta common eigenangle in (10) and a real number . By Proposition 4(ii), there exist a eigenvector of associated with the common eigenangle , say , such that . For the same reason, there exist a eigenvector of associated with , say , such that and
This means the vector y is the eigenvector of L.
Case 2 .
The proof is similar as Case 1 and trivial by noticing that . ∎
If is a perfect critical vertex set of path , then by Theorem 3, we have . That is must be an odd. Therefor, a straightforward consequence of Theorem 3 is that there exist a perfect critical vertex set of a path graph if and only if is odd. The following corollary follows straight from Theorem 3, which have been proved in  by using different mathematical tools.
Let for some , then the path is controllable with any vertex selected as leader, e.g. is omnicontrollable. ∎
3.3 Algorithm and Examples
In fact, Theorem 3 described all perfect critical vertex set of path graph . Next, we provide a method to locate the leader vertices. That is the following Algorithm I.
|1: input: .|
|2: initialize: , , .|
|3: while , for some do|
|4: , ,|
|5: for do|
|8: end for|
|11: end while|
For a path , obtained by Algorithm I is the set of leaders. That is is controllable with the vertex located in and only with those vertices.
For example, let is even. Since and only 3 is a odd prime factor, by Algorithm I, , . So, any vertex in can be select as leader.
Let is also even but with 3 being a multiple factor. By Algorithm I, there is only one follower vertex set needed to be considered. let , . So, any vertex in can be select as leader.
In the case of multiple factor, Algorithm I is a much more efficient algorithm to locate the leader vertex than the method provided in . What’s more, Algorithm I can be easily applied to much lager path graph. For example, let , since , there are only three follower vertex set being calculate, that is
and leaders are located in the vertex set :
That is any one of and only of all these 56 vertices can be selected as leader.
4 Minimal Perfect Critical Vertex Set of Graphs Based on Path
Path graphs are simplest and basic graph structures. Some graphs can be constructed by adding paths. The minimal perfect critical vertex set of these graphs will be investigated as what follows.
Let be a graph and . We use to denote the graph by adding to incident to , as shown in fig.5.