Fiedler vector analysis for particular cases of connected graphs
aa r X i v : . [ c s . D M ] N ov Fiedler vector analysis for particular cases ofconnected graphs
Daniel Felisberto Tracin´a FilhoP´os-gradua¸c˜ao em Sistemas e Computa¸c˜ao - Instituto Militar de [email protected] Marcela JustelP´os-gradua¸c˜ao em Sistemas e Computa¸c˜ao - Instituto Militar de [email protected]
Abstract
In this paper, some subclasses of block graphs are considered in orderto analyze Fiedler vector of its members. Two families of block graphswith cliques of fixed size, the block-path and block-starlike graphs, areintroduced. Cases A and B of classification for both families were consid-ered, as well as the behavior of the algebraic connectivity for particularcases of block-path graphs.
Keyword: block graphs, spectral graph theory, Fiedler vector, algebraic con-nectivity.
We consider G = ( V, E ) an undirected, unweighted and simple graph. The sizesof its sets of vertices and edges are | V | = n , | E | = m . The Laplacian matrix of agraph 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 thecharacteristic polynomial of L ( G ), are n non-negative real numbers, and zero isalways an eigenvalue of L ( G ). We denote them as 0 = λ ≤ λ ≤ .... ≤ λ n . Thesecond smallest eigenvalue of L ( G ), λ ( G ), is called the algebraic connectivityof G . An eigenvector associated to the algebraic connectivity is called a Fiedlervector. The algebraic connectivity gives some measure of how connected thegraph is. Fiedler ([3]), showed that λ ( G ) > 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 ( K n ) = n . Also, Fiedler proved that λ ( 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 .1hen the graph is a tree (a particular case of connected graphs), the entriesof a Fiedler vector of such tree provide extra information. A tree T is called aType 1 tree if there is a vertex z such that y z = 0 and is adjacent to a vertexwith non zero valuation. In this case z is called characteristic vertex. Whenthere is no entry of y equal to zero, the tree has an edge ( u, w ) with y u > y w <
0, and in that case T is called a Type 2 tree and the edge ( u, w ) thecharacteristic edge ([5]).Fiedler vectors and algebraic connectivity of graphs have been studied byseveral authors. For instance some results about trees are presented in [7] and[10], others works such as [8] and [9] deal with connected graphs, and there aresome surveys about these issues in [1] and [6].In this work we introduce two families of connected graphs, both of thembelonging to the class block graphs, the block-path and the block-starlikegraphs. Some properties of Fiedler vectors for graphs in these families areanalyzed.The paper is organized as follows. In Section 2, we review some basic con-cepts and results of the literature. Section 3 presents the families block-pathand block-starlike graphs, as well as some properties about them. Finally, theconclusions of the work are presented in Section 4 and the references in the lastsection. In this section, some concepts and results about Graph Theory and SpectralGraph 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 neighborhoodby N [ v ] = N ( v ) ∪ { v } . Two vertices u, v ∈ V are true twins if N [ u ] = N [ v ].The vertex connectivity of G , denoted κ ( G ), is the minimum number ofvertices whose removal from G leaves a non-connected or trivial graph.A vertex v ∈ V is a point of articulation or cutpoint if G \ v is non-connected,and in this case κ ( G ) = 1. A maximal connected subgraph without any points ofarticulation is called a block. The distance, d G ( u, v ), in G between two verticesis the length of a shortest path between u, v in G . The eccentricity, e G ( v ), of avertex 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 S ⊆ V , the subgraph of G induced by S is denoted G [ S ]. If G [ S ] is a completesubgraph then S is a clique in G .A tree is an acyclic and connected graph. We denote by P j the path on j vertices and K ,q the star on q + 1 vertices.A tree T is called starlike if T is homeomorphic to a star K ,m . For m ≥ T has a unique vertex v of degree m and T \ v is a union of m paths ([11]).2 graph is a block graph if every block is a clique. A proper interval graphis an interval graph that has an intersection model in which no interval properlycontains another. Proper interval graphs are also known as indifference graphs.A graph is a block indifference graph when it belongs to the intersection ofboth classes, indifference and block graphs [2].Next, some concepts and properties on graph spectra are introduced.In [4] Fiedler proved the following theorem which describes some structureof 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 andnegatively valuated vertices. Each other block has either vertices with positivevaluation only, vertices with negative valuation only, or vertices with zero val-uation only. Every path P which contains at most two points of articulationin each block, which starts in C and contains just one vertex k in C has theproperty that the valuations at points of articulation contained in P form ei-ther an increasing, or decreasing or a zero sequence among this path accordingto whether y k > , y k < or y k = 0 ; in the last case all vertices in P havevaluation zero.Case B: No block of G contains both positively and negatively valuatedvertices. There exists a unique vertex z which has valuation zero and isadjacent to a vertex with a non-zero valuation. This vertex z is a point ofarticulation. Each block contains either vertices with positive valuation only,with negative valuation only, or with zero valuation only. Every path containingboth positively and negatively valuated vertices passes through z . Kirkland et al ([8]) presented another characterization of graphs for whichCase B of Theorem 1 holds. This characterization is based on the concept ofPerron components at points of articulation.For a positive n × n matrix M , the Perron value of M , ρ ( M ) is definedas the maximum eigenvalue of the matrix. Let G be a connected weightedgraph with Laplacian matrix L ( G ). For a vertex v , we denote the connectedcomponents of G \ v by C , ..., C p . If p ≥ v is a point of articulation. Foreach connected component C i , 1 ≤ i ≤ p let L ( C i ) be the principal submatrixof L ( G ) corresponding to the vertices of C i . The Perron value of C i is thePerron value of the positive matrix L ( C i ) − , denoted by ρ ( L ( C i ) − ). Wesay that C j is the Perron component at v if ρ ( L ( C j ) − ) = max ≤ i ≤ p ρ ( L ( C i ) − ). Theorem 2 ([8]) Let G = ( V, E ) be a weighted graph. Case B of Theorem 1holds if and only if there are two or more Perron components at z . Further, in hat case, the algebraic connectivity λ ( G ) is given by ρ ( L ( C ) − ) for any Perroncomponent C at z . If the Perron components at z are C , ..., C m , for ≤ i ≤ m ,let the Perron vector of L ( C i ) − be x i , normalized so that its entries sum to 1.For each ≤ i ≤ m , let b i − be the Fiedler vector which valuates the vertices of C by x , the vertices of C i by − x i and all other vertices zero; then b , ..., b m − is a basis for the eigenspace corresponding to λ ( G ) . In particular, if there are m Perron components at z , then the multiplicity of λ ( G ) is m − , and everyFiedler vector has zeros in the positions corresponding to z , and to the verticesof the non-Perron components at z . Finally, for any vertex v = z , the uniquePerron component at v is the component containing z . 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 Theorem1 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 thereare two or more Perron components. 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 ≥ p ≥ G k,p is n = k ( p + 1) − p . The block-path graph G , is presented in Figure 1.Figure 1: G , ∈ B Let r ≥ G k,p , ...., G k,p r ∈ B . The graph S r,k,p ,...,p r obtained by thecoalescence of G k,p , ...., G k,p r by identifying a vertex of degree k − G k,p i , ≤ i ≤ r is a block-starlike .The class of block-starlike graphs is denoted A . The vertex of the coalescenceis called central vertex in G .Figure 2 shows S , , , , ∈ A obtained by the coalescence of G , , G , and G , with v its central vertex. 4 Figure 2: S , , , , ∈ A Next, we present some new results about classes A and B . Lemma 3
Let G be a block graph with a point of articulation. If a, b are truetwins in G and y is a Fiedler vector, then y a = y b .Proof. Let | V | = n , a, b ∈ V be true twin vertices and the Laplacian matrixof G denoted L = ( l i,j ). As λ ( G ) is a Laplacian eigenvalue of G associated to y , L. y = λ ( G ) y and λ ( G )( y a − y b ) = P ns =1 l a,s y s − P ns =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, λ ( G )( y a − y b ) = l a,a y a + l a,b y b − l b,a y a − l b,b y b λ ( G )( y a − y b ) = ( l a,a − l b,a ) y a + ( l a,b − l b,b ) y b Since l a,a = l b,b , λ ( G )( y a − y b ) = λ ( G )( y a − y b ) = q ( y a − y b ), with q = l a,a − l b,a = deg G ( a ) + 1 > ∈ N .However, by hypothesis the graph G has a point of articulation and then λ ( 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 ofTheorem 1 holds for G k,p .Proof. Label the vertices of G k,p from 1 to n = k ( p + 1) − p in thefollowing way: from one block with only 1 point of articulation, all the pointsof articulation are labelled by k, k − , k − , ..., ( p + 1) k − p , and inside eachblock, the true twin vertices are labeled in order by block with numbers betweenthe labels of the corresponding points of articulation of the block. Then vertex v ∗ labeled ( p +12 k − p +12 + 1) is the unique vertex in center ( G k,p ) with odd p ≥ G k,p − v ∗ has two Perroncomponents and by Corollary 2.1 it holds if and only if Case B of Theorem 1 istrue for G k,p . 5 emark 1 By Theorem 4 we can conclude that G k,p ∈ B with p an even numberif and only if Case A of Theorem 1 holds for G k,p . Remark 2 If G = S r,k,p ,...,p r ∈ A with r ≥ and p = p = ... = p r , thenby Corollary 2.1 Case B of Theorem 1 holds for G and the center of the graphequals the central vertex. Figure 3 shows S , , , , ∈ A with v its central vertex. v Figure 3: S , , , , ∈ A Theorem 5
Let G = S r,k,p ,...,p r ∈ A with r ≥ and v its central vertex suchthat the connected components of G \ v are denoted by G k,p \ v , G k,p \ v , . . . , G k,p r \ v , with p ≥ p ≥ . . . ≥ p r . If p + p + 1 ≥ p and p > p then Case Aof Theorem 1 holds for G .Proof. We will prove that for every vertex in G there is only onePerron component. Then Corollary 2.1 holds for G in the condition of thehypothesis. Let L = L ( G ) and v be the central vertex of G , then the con-nected components of G \ v are G k,p \ v , G k,p \ v , . . . , G k,p r \ v and for each2 ≤ i ≤ r , G k,p i \ v is isomporphic to a proper subgraph of G k,p \ v , and ρ ( L ( G k,p \ v ) − ) > ρ ( L ( G k,p i \ v ) − ). So, ρ ( L − v ) = ρ ( L ( G k,p \ v ) − ) and G k,p \ v is the unique Perron component of G \ v .Let w ∈ V , w = v . We consider two cases:1. G \ w = K k − ∪ G ′ with G ′ a block starlike, and2. G \ w = G k,a ∪ P p i − a for some i and 1 < a ≤ p − G in w . On the otherhand, if case 2 holds, p i − a ≥ p − ≥ a . Since by hypothesis p + p + 1 ≥ p , we can conclude that p + p ≥ a . Moreover, as G k,a containsat least 2 connected components of G \ w and p + p ≥ a then there is a sub-graph of G k,a isomorphic to P p i − a . Thus, by Corollary 2.1 the theorem is true.Grone and Merris ([5]) showed that the algebraic connectivity of a Type 1tree T does not change when a degree one vertex adjacent to the characteristicvertex is added to T . The next theorem generalizes that result for graphs6 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 ofarticulation labeled ( p +12 k − p +12 + 1) as in proof of Theorem 4. v wu Figure 4: G ′ , obtained from G = G , and K , with λ ( G ) = λ ( G ′ ) = 0 . Theorem 6
Let G = G k,p ∈ B with n vertices and p an odd number. Let y be aFiedler vector of G and u ∈ V ( G ) the unique vertex in center ( G ) . Let G ′ be thegraph obtained as the coalescence between vertex u of G with a vertex of K k . Let y ′ be a ( n + k − -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 ( G ′ ) = λ ( 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 P ns =1 l v,s y s = ( k + 1) y v − ky v − y u = y v − y u = λ ( G ′ ) y v y v = − λ ( G ′ ) y u ( ∗ )As p ≥ G ′ \ u has 3 connected components, 2 of them are isomorphic andcontain at least one block K k and the other one is isomorphic to K k − . 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. Two subclasses of block graphs were considered in this paper: block-path andblock-starlike graphs. First, we showed some properties of Fiedler vectors asso-ciated to those graphs, such as identifying whether Cases A or B of Theorem1 holds for block-path graphs. Then we exhibited particular conditions to haveblock-starlike graphs for which Case A of the same theorem holds. Next, weobtained the necessary conditions to mantain the value of the algebraic connec-tivity of a block-path graph when some vertices and edges are added. We planon exploring other subclasses of block graphs in the future.7 cknowledgments
This work was supported by CAPES, Coordena¸c˜ao de Aperfei¸coamento de Pes-soal do N´ıvel Superior - C´odigo de Fianciamento 001.
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