Fiedler vector analysis for particular cases of connected graphs

In this paper, some subclasses of block graphs are considered in order to analyze Fiedler vector of its members. Two families of block graphs with cliques of fixed size, the block-path and block-starlike graphs, are introduced. Cases A and B of classification for both families were considered, as well as the behavior of the algebraic connectivity for particular cases of block-path graphs.


Introduction
We consider G = (V, E) an undirected, unweighted and simple graph. The sizes of its sets of vertices and edges are |V | = n, |E| = m. The Laplacian matrix of a graph G, L(G), is the symmetric and semidefinite positive matrix D(G)− A(G), where D(G) is the diagonal matrix with the degrees of the vertices of G and A(G) is the adjacency matrix of G. The eigenvalues of L(G), the roots of the characteristic polynomial of L(G), are n non-negative real numbers, and zero is always an eigenvalue of L(G). We denote them as 0 = λ 1 ≤ λ 2 ≤ .... ≤ λ n . The second smallest eigenvalue of L(G), λ 2 (G), is called the algebraic connectivity of G. An eigenvector associated with the algebraic connectivity is called a Fiedler vector. The algebraic connectivity gives some measure of how connected the graph is. Fiedler ([3]), showed that λ 2 (G) > 0 if and only if G is connected. Moreover, the algebraic connectivity of G does not decrease if an edge is added to G and the complete graph has λ 2 (K n ) = n. Also, Fiedler proved that λ 2 (G) ≤ κ(G), where κ(G) is the vertex connectivity of G. For a connected graph G and a Fiedler vector y, the i − th entry of y, y i , gives a valuation of vertex i.
When the graph is a tree (acyclic and connected graph), the entries of a Fiedler vector of such tree provide extra information. A tree T is called a Type 1 tree if there is a vertex z such that y z = 0 and is adjacent to a vertex with non zero valuation. In this case z is called characteristic vertex. When there is no entry of y equal to zero, the tree has an edge (u, w) with y u > 0 and y w < 0, and in that case T is called a Type 2 tree and the edge (u, w) the characteristic edge ( [5]).
Fiedler vectors and algebraic connectivity of graphs have been studied by several authors. For instance some results about trees are presented in [7] and [10], others works such as [8] and [9] deal with connected graphs. The surveys in [1] and [6] provide overviews of the literature on algebraic connectivity.
Even for trees, to identify classes of connected graphs for which its elements are of the same type is not an easy task. For instance, path graphs P n are classified by type according to their parity. Also, in [10] Patra shows conditions for a broom tree by of Type 2 and claims that all the broom trees are of Type 2, he also says that will be nice to get a proof of this claim.
This paper consider two families of connected graphs, both of them belonging to the block graph class, the block-path and the block-starlike graphs. Some properties of Fiedler vectors for graphs in those families are analyzed and conditions for classifying by type such graphs are given.
The paper is organized as follows. In Section 2, we review some basic concepts and results of the literature. Section 3 presents the families block-path and block-starlike graphs, as well as some properties about them. Finally, the conclusions of the work are presented in Section 4 and the references in the last section.

Preliminaries
In this section, some concepts and results about Graph Theory and Spectral Graph Theory are introduced.
First, let G = (V, E) be a connected graph. The neighborhood of a vertex v ∈ V is denoted by N (v) = {w ∈ V : (v, w) ∈ E} and its closed neighborhood by The vertex connectivity of G, denoted κ(G), is the minimum number of vertices whose removal from G leaves a disconnected or trivial graph.
A vertex v ∈ V is a point of articulation or cutpoint if G\v is disconnected, and in this case κ(G) = 1. A maximal connected subgraph without any points of articulation is called a block. The distance, d G (u, v), in G between two vertices is the length of a shortest path between u, v in G. The eccentricity, e G (v), of a vertex v in G is its greatest distance from any other vertex. The center of G, denoted center(G), is the set of vertices with minimum eccentricity. For any A tree is an acyclic and connected graph. We denote by P j the path on j vertices and K 1,q the star on q + 1 vertices.
A tree T is called starlike if T is homeomorphic to a star K 1,m . For m ≥ 3, T has a unique vertex v of degree m and T \v is a union of m paths ( [11]).
A graph is a block graph if every block is a clique. A proper interval graph is an interval graph that has an intersection model in which no interval properly contains another. Proper interval graphs are also known as indifference graphs. A graph is a block indifference graph when it belongs to the intersection of both classes, indifference and block graphs [2].
Next, some concepts and properties on graph spectra are introduced. The following results consider G = (V, E) a weighted graph with w(e) ≥ 0 for all e ∈ E (the weight of edge), and define the Laplacian matrix of G, L(G) whose In [4] Fiedler proved the following theorem which describes some structure of any Fiedler vector of a connected graph.

Theorem 1 ([4]) Let G be a connected graph and y a Fiedler vector of G.
Then exactly one of the following cases occurs: Case A: There is a single block C in G which contains both positively and negatively valuated vertices. Each other block has either vertices with positive valuation only, vertices with negative valuation only, or vertices with zero valuation only. Every path P which contains at most two points of articulation in each block, which starts in C and contains just one vertex k in C has the property that the valuations at points of articulation contained in P form either an increasing, or decreasing or a zero sequence among this path according to whether y k > 0, y k < 0 or y k = 0; in the last case all vertices in P have valuation zero.
Case In other words, Theorem 2 can be rewritten as in Corollary 2.1.

Corollary 2.1 ([8]) Let G = (V, E) be a weighted graph. Case A of Theorem 1 holds if and only if there is a unique Perron component at every vertex of G. Case B of Theorem 1 holds if and only if there is a unique vertex at which there are two or more Perron components.
Observe that for unweighted graphs, we consider each of its edges weight equals 1 in order to apply Theorem 2 and Corollary 2.1.

Block-path and block-starlike classes
The block-path class, denoted B, is a subclass of block-indifference graphs, with elements denoted G k,p , that contain all cliques of size k ≥ 2 and p ≥ 1 points of articulation. Moreover, the number of vertices in G k,p is n = k(p + 1) − p. For p = 0, G k,0 = K k . The block-path graph G 4,3 is presented in Figure 1. Let G = S r,k,p1,...,pr ∈ A with r ≥ 3 and v its central vertex such that the connected components of G\v are denoted by G k,p1 \v, G k,p2 \v, . . ., G k,pr \v, with p 1 ≥ p 2 ≥ . . . ≥ p r . G is a block-starlike. The class of block-starlike graphs is denoted A. Figure 2 shows S 3,4,1,2,3 ∈ A with v its central vertex and S 3,4,1,2,3 \v = G 4,1 \v ∪ G 4,2 \v ∪ G 4,3 \v. Observe that in this example, the central vertex v do not belong to center (S 3,4,1,2,3 ). Proof. Let |V | = n, a, b ∈ V be true twin vertices and the Laplacian matrix of G denoted L = (l i,j ). As λ 2 (G) is a Laplacian eigenvalue of G associated with y, Ly = λ 2 (G)y and λ 2 (G)(y a − y b ) = n s=1 l a,s y s − n s=1 l b,s y s Vertices a and b are true twins then, for all s ∈ V \{a, b} l a,s = l b,s . So, Since l a,a = l b,b , However, by hypothesis the graph G has a point of articulation and then λ 2 (G) ≤ 1. So it is not possible to be y a = y b . Then, the result holds.
Theorem 4 Let G k,p ∈ B. Then p is an odd number if and only if Case B of Theorem 1 holds for G k,p .

Proof.
Label the vertices of G k,p from 1 to n = k(p + 1) − p in the following way: from one block with only 1 point of articulation, all the points of articulation are labelled by k, 2k − 1, 3k − 2, ..., (p + 1)k − p, and inside each block, the true twin vertices are labeled in order by block with numbers between the labels of the corresponding points of articulation of the block. Then vertex v * labeled ( p+1 2 k − p+1 2 + 1) is the unique vertex in center(G k,p ) with odd p ≥ 1. So, by definition of the block-path class, the graph G k,p − v * has two Perron components and by Corollary 2.1 it holds if and only if Case B of Theorem 1 is true for G k,p .

Remark 1 By Theorem 4 we can conclude that G k,p ∈ B with p an even number if and only if Case A of Theorem 1 holds for G k,p .
Remark 2 If G = S r,k,p1,...,pr ∈ A with r ≥ 2 and p 1 = p 2 = ... = p r , then by Corollary 2.1 Case B of Theorem 1 holds for G and the center of the graph equals the central vertex. Figure 3 shows S 3,4,1,1,1 ∈ A with v its central vertex.

Proof.
We will prove that for every vertex in G there is only one Perron component. Then Corollary 2.1 holds for G in the condition of the hypothesis. Observe that is sufficient to consider points of articulations of G. Let L = L(G) and v be the central vertex of G, then the connected components of G\v are G k,p1 \v, G k,p2 \v, . . . , G k,pr \v and for each 2 ≤ i ≤ r, G k,pi \v is isomorphic to a proper subgraph of G k,p1 \v, and ρ(L(G k,p1 \v) −1 ) > ρ(L(G k,pi \v) −1 ). So, ρ(L −1 v ) = ρ(L(G k,p1 \v) −1 ) and G k,p1 \v is the unique Perron component of G\v. Let w ∈ V , w = v. We consider two cases: 1. G\w = K k−1 ∪ G ′ \w with G ′ a block starlike, and 2. G\w = G k,a \w ∪ H\w, where H is a block starlike containing as subgraph a block path graph with p 2 + p 3 + 1 articulation points and exists 1 ≤ i ≤ 3 such that 1 < a ≤ p i − 1. When case 1 holds there is only one Perron component of G in w.
On the other hand, if case 2 holds, consider the following situations: If i = 1, p 1 − 1 ≥ a. Since by hypothesis p 2 + p 3 + 1 ≥ p 1 , we can conclude that p 2 + p 3 ≥ a. Moreover, as H contains at least 2 connected components of G\w and p 2 + p 3 ≥ a then G k,a is isomorphic to a subgraph of H.
If i = 1, since p i − 1 ≥ a then a ≤ p i − 1 < p 1 and then G k,a is isomorphic to a subgraph of H.
Thus, by Corollary 2.1 the theorem is true.
Grone and Merris ( [5]) showed that the algebraic connectivity of a Type 1 tree T does not change when a degree one vertex adjacent to the characteristic vertex is added to T . The next theorem generalizes that result for graphs G k,p ∈ B with p an odd number.
Observe that if p is odd, then center(G k,p ) = u where u is the point of articulation labeled ( p+1 2 k − p+1 2 + 1) as in proof of Theorem 4.
v w u Figure 4: G ′ , obtained from G = G 4,3 and G 4,0 , with λ 2 (G) = λ 2 (G ′ ) = 0.32938 Theorem 6 Let G = G k,p ∈ B with n vertices and p an odd number. Let y be a Fiedler vector of G and u ∈ V (G) the unique vertex in center(G). Let G ′ be the graph obtained as the coalescence between vertex u of G with a vertex of G k,0 . Let y ′ be a (n + k − 1)-dimensional vector such that y ′ v = y v , ∀v ∈ V (G) and y ′ v = 0 otherwise. Then G ′ ∈ A, y ′ is a Fiedler vector of G ′ and λ 2 (G ′ ) = λ 2 (G).
Proof. Let v, w be a pair of vertices different from u in the new block K k of G ′ (as in Figure 4). Since v, w are twin vertices, by Lemma 3 y v = y w . Then n s=1 l v,s y s = (k + 1)y v − ky v − y u = y v − y u = λ 2 (G ′ )y v y v = 1 1−λ2(G ′ ) y u ( * ) As p ≥ 1, G ′ \u has 3 connected components, 2 of them are isomorphic and contain at least one block K k and the other one is isomorphic to K k−1 . Then, by Theorem 2 case B holds for G ′ and y u = 0. From equation ( * ), we have that y ′ v = 0 for all twin vertices in the new block K k of G ′ , and the result follows.

Conclusions
Two subclasses of block graphs were considered in this paper: block-path and block-starlike graphs. First, we showed some properties of Fiedler vectors associated with those graphs, such as identifying whether Cases A or B of Theorem 1 holds for block-path graphs. Then we exhibited particular conditions to have block-starlike graphs for which Case A of the same theorem holds. Next, we obtained the necessary conditions to mantain the value of the algebraic connectivity of a block-path graph when some vertices and edges are added. We plan on exploring other subclasses of block graphs in the future.