Integer Laplacian Eigenvalues of Chordal Graphs
Nair Maria Maia de Abreu, Claudia Marcela Justel, Lilian Markenzon
aa r X i v : . [ c s . D M ] J u l Integer Laplacian Eigenvaluesof Chordal Graphs
Nair Maria Maia de Abreu
PEP-COPPE - Universidade Federal do Rio de JaneiroCentro de Tecnologia, Bloco F, sala F105, Cidade Universit´aria, RJ, Brazil
Claudia Marcela Justel ∗ Departamento de Engenharia de Computa¸c˜ao - Instituto Militar de EngenhariaPra¸ca General Tib´urcio 80, Praia Vermelha, 22290-270, RJ, Brazil
Lilian Markenzon
NCE - Universidade Federal do Rio de JaneiroAv. Athos da Silveira, 274, Pr´edio do CCMN, Cidade Universit´aria, 21941-611, RJ, Brazil
Abstract
In this paper, structural properties of chordal graphs are analysed, in order toestablish a relationship between these structures and integer Laplacian eigen-values. We present the characterization of chordal graphs with equal vertex andalgebraic connectivities, by means of the vertices that compose the minimal ver-tex separators of the graph; we stablish a sufficient condition for the cardinalityof a maximal clique to appear as an integer Laplacian eigenvalue. Finally, wereview two subclasses of chordal graphs, showing for them some new properties.
Keywords: chordal graphs, structural properties, algebraic connectivity,integer Laplacian eigenvalues
1. Introduction “Is it possible, looking only at a graph structure, to determine its integer Lapla-cian eigenvalues?” In this paper we address this question for chordal graphs.Structural properties of chordal graphs allow the development of efficient so-lutions for many theoretical and algorithmic problems. In this context, its ∗ Corresponding author
Email addresses: [email protected] (Nair Maria Maia de Abreu), [email protected] (Claudia Marcela Justel), [email protected] (Lilian Markenzon)
Preprint submitted to Elsevier July 12, 2019 lique-based structure and the minimal vertex separators play a decisive role.Analysing these structures we present new results about integer Laplacian eigen-values of chordal graphs and some subclasses. Firstly, we characterize thechordal graphs with equal vertex and algebraic connectivities. Then we revisitan integral Laplacian subclass, the quasi-threshold graphs [16], and, based onresults from [2], we show the relation between its structural and spectral prop-erties. The tree representation of a quasi-threshold graph is analysed, allowingthe determination of its Laplacian eigenvalues expressed in terms of maximalcliques and simplicial vertices. Another interesting subclass is restudied: the( k, t )-split graphs [8, 11] and its properties raised. Finally, the importance ofsimplicial vertices in the existence of integer Laplacian eigenvalues is established.
2. Basic concepts
Let G = ( V, E ) (or G = ( V ( G ) , E ( G ))) be a connected graph, where | E | = m is its size and | V | = n is its order . The set of neighbors of a vertex v ∈ V isdenoted by N ( v ) = { w ∈ V ; { v, w } ∈ E } and its closed neighborhood by N G [ v ] = N G ( v ) ∪ { v } . Two vertices u and v are false twins in G if N G ( u ) = N G ( v ), and true twins in G if N G [ u ] = N G [ v ]. For any S ⊆ V , the subgraph of G inducedby S is denoted G [ S ]. If G [ S ] is a complete subgraph then S is a clique in G .The complete graph on n vertices is denoted by K n . A vertex v is said to be simplicial in G when N ( v ) is a clique in G .The Laplacian matrix of a graph G of order n is defined as L ( G ) = D ( G ) − A ( G ),where D ( G ) = diag ( d , ..., d n ) denote the diagonal degree matrix and A ( G ) theadjacency matrix of G . As L ( G ) is symmetric, there are n eigenvalues of L ( G ).We denote the eigenvalues of L ( G ), called the Laplacian eigenvalues of G , by µ ( G ) ≥ · · · ≥ µ n ( G ). Since L ( G ) is positive semidefinite, µ n ( G ) = 0 . Fiedler[7] showed that G is a connected graph if and only if µ n − ( G ) >
0; this eigenvalueis called algebraic connectivity and it is denoted by a ( G ). Moreover, Fidlerproved that for G = K n , a ( G ) ≤ κ ( G ). All different Laplacian eigenvalues of G together with their multiplicities form the Laplacian spectrum of G , denotedby SpecL ( G ). A graph is called Laplacian integral if its L -spectrum consists ofintegers. A chordal graph is a graph in which every cycle of length four and greater hasa cycle chord. Basic concepts about chordal graphs are assumed to be knownand can be found Blair and Peyton [4] and Golumbic [9]. Following the mostpertinent concepts are reviewed.A subset S ⊂ V is a separator of G if at least two vertices in the same connectedcomponent of G are in two distinct connected components of G [ V \ S ]. The set S is a minimal separator of G if S is a separator and no proper set of S separates2he graph. The vertex connectivity of G , κ ( G ), is defined as the minimumcardinality of a separator of G .Let G = ( V, E ) be a chordal graph and u, v ∈ V . A subset S ⊂ V is a vertex separator for non-adjacent vertices u and v (a uv -separator) if the removalof S from the graph separates u and v into distinct connected components.If no proper subset of S is a uv -separator then S is a minimal uv -separator .When the pair of vertices remains unspecified, we refer to S as a minimal vertexseparator ( mvs ). The set of minimal vertex separators is denoted by S . Aminimal separator is always a minimal vertex separator but the converse is nottrue.A clique-tree of G is defined as a tree T whose vertices are the maximal cliquesof G such that for every two maximal cliques Q and Q ′ each clique in the pathfrom Q to Q ′ in T contains Q ∩ Q ′ . The set of maximal cliques of G is denotedby Q . For a chordal graph G and a clique-tree T = ( Q , E T ), a set S ⊂ V is a minimal vertex separator of G if and only if S = Q ∩ Q ′ for some edge { Q, Q ′ } ∈ E T . Moreover, the multiset M of the minimal vertex separators of G is the same for every clique-tree of G . The multiplicity of the minimal vertexseparator S , denoted by µ ( S ), is the number of times that S appears in M . Theset of minimal vertex separators of G is denoted by S . The determination of theminimal vertex separators and their multiplicities can be performed in lineartime [14].If G = ( V , E ) and G = ( V , E ) are graphs on disjoint set of vertices, their graph sum is G + G = ( V ∪ V , E ∪ E ) . The join G ∇ G of G and G is agraph obtained from G + G by adding new edges from each vertex in G toall vertices of G .If A is a finite n -element set then a chain is a collection of subsets B , B , . . . , B k of A such that for all i, j ∈ { , , ..., k } where i = j we have that B i ⊂ B j or B j ⊂ B i . Some other graph classes will be mentionned in this paper. Their definitionsand characterizations can be found in [5].A graph G is a comparability graph if it transitively orientable, i.e. its edges canbe directed such that if a → b and b → c are directed edges, then a → c is adirected edge.A graph G is a cograph (short for complement-reducible graph) if one of thefollowing equivalent conditions hold: • G can be constructed from isolated vertices by disjoint union and joinoperations. • G is P -free. 3 split graph is a graph G = ( V, E ) if V can be partitioned as the disjoint unionof an independent set and a clique. Split graphs are chordal graphs.
3. Chordal graphs with κ ( G ) = a ( G ) In this section we characterize chordal graphs that have equal vertex and alge-braic conectivities. Our result particularizes an important result from Kirkland et al. [12], allowing us to recognize this property based on the graph structure.
Theorem 1. [12]
Let G be a non-complete connected graph on n vertices. Then κ ( G ) = a ( G ) if and only if G can be written as G ∇ G , where G is a discon-nected graph on n − κ ( G ) vertices and G is a graph on κ ( G ) vertices with a ( G ) ≥ κ ( G ) − n . Theorem 2.
Let G be a non-complete connected chordal graph. Then κ ( G ) = a ( G ) if and only if there is a minimal separator of G such thatall its elements are universal vertices. Proof.
The vertex connectivity of a non-complete connected chordal graph isgiven by the cardinality of its minimum separator.Graph G is connected, so, in order to obtain the disconnected graph G statedby Theorem 1, we must remove at least a minimal separator S of G . And more,as G results from a join operation between G and G , the vertices of S mustbe adjacent to all vertices of V ( G ). A minimal separator of chordal graph G isa minimal vertex separator of G . As a minimal vertex separator is a clique, allthe vertices of V ( G ) are pairwise adjacent. Hence, they are universal verticesand S is a minimum graph separator.As G is a clique, a ( G ) = | V ( G ) | = κ ( G ). Then κ ( G ) ≥ κ ( G ) − n = n ≥ κ ( G )is always true. (cid:3) It is interesting to analyse the structure of a graph that satisfies Theorem 2.It is not difficult to see that the minimal vertex separators play an importantrole in their characterization. As the minimal vertex separator S = V ( G ) iscomposed by universal vertices, it must be a subset of any other minimal vertexseparator of the graph and, there is only one mvs with this property.Figure 1 shows some examples. It is well known that cographs obey this prop-erty. Quasi-threshold graphs satisfy this property, since, as it will be reviewedin the next section, they are cographs and chordal graphs. However there arechordal graphs that satisfy Theorem 2 and are not cographs as it is the case ofthe graph of Figure 1(a).There is an efficient procedure to recognize if a graph G obeys Theorem 2. First,the set S = { S , . . . , S r } of minimal vertex separators must be determined; thisstep has O ( m ) complexity time [14]. Then we must determine ∩ ri =1 S i . As4a) (b) (c) Figure 1: Graphs with a ( G ) = κ ( G ) P i =1 ,r | S i | ≤ m this step has also O ( m ) complexity time. It is immediate that G has κ ( G ) = a ( G ) if and only if this intersection is for itself a minimal vertexseparator composed by universal vertices.
4. Quasi-threshold graphs
The quasi-threshold graphs are the object of our study in this section. Animportant subclass of chordal graphs, they were defined in 1962 by Wolk [15] as comparability graphs of a tree ; Golumbic [10] called them trivially perfect graphs .Ma et. al. [13] called them quasi-threshold graphs and studied algorithmicresults. Yan et al. [16] presented the following characterization theorem:
Theorem 3.
The following statements are equivalent for any graph G . G is a quasi-threshold graph. G is a cograph and is an interval graph. G is a cograph and is a chordal graph. G is P -free and C -free. For any edge uv in G , either N [ u ] ⊆ N [ v ] or N [ v ] ⊆ N [ u ] . If v , v , . . . , v n is a path with d ( v ) ≥ d ( v ) ≥ . . . ≥ d ( v n − ) , then { v , v , . . . , v n } is a clique. G is induced by a rooted forest.
It is immediate that, as a subclass of cographs, they are integral Laplaciangraphs.
It is interesting to review some subclasses of quasi-thresholds graphs alreadystudied; the results that will be presented for the quasi-threshold graphs will beobviously valid for all these classes. 5 the windmill graph
W d ( k, ℓ ), k ≥ ℓ ≥
2, is a graph constructedby joining ℓ copies of a complete graph K k at a shared universal vertex.Figure 2(a) presents W d (4 , • the split-complete graph , which is the join of a K k and a set of n − k independent vertices. Figure 2(b) presents an example with 7 vertices.(a) (b) (c) Figure 2: Examples
Recently, a class that generalises windmill and the split-complete graphs, the core-satellite graphs [6], were defined:Let c ≥ s ≥ η ≥
2. The core-satellite graph is Θ( c, s, η ) ∼ = K c ∇ ( ηK s ).That is, they are the graphs consisting of η copies of K s (the satellites) meetingin a K c (the core). Figure 2(c) presents a Θ(2 , ,
2) graph. In the same paper, the generalized core-satellite graphs were defined, relaxing the size of the satellitesin the previous definition; this class is equivalent to a chordal graph that hasonly one minimal vertex separator and, as so, it is a quasi-threshold graph.Windmills have all maximal cliques with the same cardinality and the minimalvertex separator of cardinality 1; split-complete graphs have all maximal cliqueswith cardinality k + 1 and the minimal vertex separator of cardinality k ; core-satellite graphs have all maximal cliques with the same cardinality.Another well-known subclass is composed by the threshold graphs: a graphis a threshold graph if it can constructed from the empty graph by repeatedlyadding either an isolated vertex or an universal vertex. This construction canbe expressed by a sequence of 0’s and 1’s, 0 representing the addition of anisolated vertex, 1 representing the addition of an universal vertex.Threshold graphs are known to be the intersection of split graphs and cographs[5]. In a connected threshold graph, the set of minimal vertex separators formsa chain (nested mvs ); the mvs with smaller cardinality gives κ ( G ) and, as this mvs is composed by universal vertices, it satisfies Theorem 2.6 .2. Structural × spectral propertiesQuasi-threshold graphs are defined recursively by the following operations:( ρ ) an isolated vertex is a quasi-threshold graph;( ρ ) adding a new vertex adjacent to all vertices of a quasi-threshold graphresults in a quasi-threshold graph;( ρ ) the disjoint union of two quasi-threshold graphs results in a quasi-thresholdgraph.Yan et al. [16] presented a recognition algorithm for the class of quasi-thresholdgraphs that, at the same time, build the rooted forest (or the tree, if the graphis connected) that induces the graph (item 7 of Theorem 3). This directed treeis called a tree representation of G . This algorithm is important in our questfor the integer Laplacian eigenvalues of the graph. Given the graph G = ( V, E )the algorithm consists of two steps: In Step 1 the directed graph G d = ( V, E d )is built. An edge { v, w } of E becomes an arc vw of E d if d G ( v ) ≥ d G ( w ). InStep 2 the directed tree T that induces G is built.Some structural properties of quasi-threshold graphs can be stated. Lemma 4.
Let G be a quasi-threshold graph. The following properties hold:(a) Every maximal clique of G has at least one simplicial vertex.(b) Every maximal clique of G contains at most one chain of minimal vertexseparators. Proof.
The proof is by construction.Operation ( ρ ) joins two quasi-threshold graphs G and G , resulting a newquasi-threshold graph G , not connected. The operation does not affect themaximal cliques, the minimal vertex separators or the simplicial vertices.Operation ( ρ ) establishes a new maximal clique composed by one simplicialvertex. It is an isolated vertex and it is also a quasi-threshold graph with onesimplicial vertex.Operation ( ρ ) adds a universal vertex v . Three cases must be analysed: • graph G has only one maximal clique Q : vertex v is added to Q and itbecomes a simplicial vertex. • graph G is connected and it has more than one maximal clique: vertex v is added to all maximal cliques. The simplicial vertices remain the sameand the minimal vertex separators have an increment of one vertex (vertex v ). 7 graph G is not connected: suppose that its components are C , C , . . . ,C k ; they are quasi-threshold graphs. In this case, vertex v establishes anew minimal vertex separator. Vertex v is added to all maximal cliques.It belongs to all disjoint graphs and, as so, it stablishes a new mvs thatseparates vertices belonging to different components.In all cases, a new maximal clique begins as a simplicial vertex that remainsas such. So, all maximal cliques have at least one simplicial vertex (item (a)).All vertices added by operation ( ρ ) are added to all minimal vertex separatorsalready existent. So, the new minimal vertex separator is contained in all theold ones and every maximal clique of G contains at most one chain of minimalvertex separators (item (b)). (cid:3) Lemma 5.
Let G be a connected quasi-threshold graph and let T be a tree rep-resentation of G with root r . Let P = h r = v , . . . , v p i be a maximal path in T .Then the vertices of P establish a maximal clique Q of G ; ∃ k, k ≥ , such that v k , v k +1 , . . . , v p are simplicial vertices of Q . Proof.
An edge { u, v } of G is an arc uv of G d if d G ( u ) ≥ d G ( v ). So, byTheorem 3, it is immediate that the vertices of P = h r = v , . . . , v p i establisha clique in G . As it is a maximal path, it is a maximal clique. In a maximalclique, the simplicial vertices have the smaller degree, since they are linked withonly the vertices of that maximal clique. So, all simplicial vertices apppear inthe end of the path. By Lemma 4 the non-simplicial vertices compose one chainof minimal vertex separators; they form the first part of the path.Observe that all leaves in T are simplicial vertices, since all maximal cliqueshave at least one simplicial vertex (Lemma 4). (cid:3) Bapat et al. [2] have presented the integer Laplacian eigenvalues of the classof weakly quasi-theshold, which generalises the quasi-threshold graphs. Theirresult provides the integer Laplacian eigenvalues of a quasi-threshold graph interms of the directed tree T : Theorem 6. (Corollary 2.2 [2])
Let G be a connected quasi-threshold graph.Suppose G is the comparability graph of a rooted tree T = ( V, E T ) with vertices u , u , . . . , u k . Then the nonzero Laplacian eigenvalues of G are (1) d G ( u i ) + 1 , repeated exactly once for each non-pendant vertex u i , and (2) m i + 1 , repeated c i − times for each non-pendant vertex u i ,being m i , ≤ i ≤ k , the distance of the vertex u i from the root vertex u and being c i = | child ( u i ) | where child ( u i ) is the set of vertices v such thatthere is an arc u i v ∈ E T . Theorem 7.
Let G be a connected quasi-threshold graph. Let Q be its set ofmaximal cliques, S its set of minimal vertex separators and Simp ( Q i ) the set ofsimplicial vertices of Q i , ∀ Q i ∈ Q . Then the nonzero Laplacian eigenvalues of G are (a) d G ( v ) + 1 , repeated exactly once for each non-simplicial vertex v , and (b) | Q i | , repeated Simp ( Q i ) − times, ∀ Q i ∈ Q , and (c) | S i | repeated µ ( S i ) times. Proof.
Let us consider the tree representation of G , which is the rooted tree T , as seen in Theorem 6. We must analyse each item of Theorem 6.(1) d G ( u i ) + 1, repeated exactly once for each non-pendant vertex u i .A non-pendant vertex of T can be a simplicial vertex or a vertex belonging toa minimal vertex separator. Firstly we are going to consider the non-simplicialvertices, rewriting partially item (1).(a) d G ( v ) + 1, repeated exactly once for each non-simplicial vertex v .We know, by Lemma 5, that all leaves of T are simplicial vertices and eachmaximal path of T corresponds to a maximal clique. So, if the maximal cliquehave only one simplicial, it is the leaf of the path; if there is more than onesimplicial, it is a non-pendant vertex. So, for each maximal clique Q i there is Simp ( Q i ) − | Q i | −
1. Item (1) can be completed:(b) | Q i | , repeated Simp ( Q i ) − ∀ Q i ∈ Q .We now consider item (2) of Theorem 6. The value m i , ≤ i ≤ k , correspondsto the distance of the vertex u i from the root vertex u of T . As it is repeated c i − T must be observed;only vertices belonging to minimal vertex separators obey this condition sincethis means that the vertex belongs to more than on maximal clique.Each mvs is represented by a path of the tree T . Let h r, v , . . . , v p i be a path suchthat child ( v p ) > v p is a non-pendant vertex). Suppose, without loss of gen-erality, that v p has two children, a and b . There is at least two maximal cliquesin the graph: { r, v , . . . , v p , a, . . . } and { r, v , . . . , v p , b, . . . } ; { r, v , . . . , v p } is a mvs S . By Theorem 6, |{ r, v , . . . , v p }| = | S | appears as an integer Laplacianeigenvalue, since it corresponds to the distance of v p from r plus 1. The numberof children of vertex v p corresponds to the multiplicity of the mvs plus 1. So,item (2) is equivalent to: 9c) | S i | repeated µ ( S i ) times. (cid:3) Theorem 8.
The determination of the integer Laplacian eigenvalues of a quasi-threshold graph has linear time complexity.
Proof.
The determination of the degrees of the vertices of any graph takes O ( m ) complexity time. For a chordal graph, the determination of maximalcliques and minimal vertex separators has linear time complexity [14]. Since asimplicial vertex is a vertex that appears in only maximal clique, their determi-nation also has O ( m ) complexity time. So, the results presented in Theorem 7can be determined in linear time complexity.
5. ( k, t )-split graphs
In this section we approach a subclass of split graphs, the ( k, t )-split graphs,which were studied by Freitas [8] and Kirkland et al. [11]. Our results establisha new approach of the results already presented. It is known that ( k, t )-splitgraphs have, at least, some integer Laplacian eigenvalues and we explicitly de-termine these values in terms of the graph structure.As already seen, G = ( V, E ) is a split graph if V can be partitioned in anindependent set and a clique. Such partition may not be unique and it is referredas a split partition of V . However, every connected bipartite split graph has asplit partition equals to the degree partition which is called split degree partition (or sdp ). A split-complete graph is a split graph such that each vertex of theindependent set is adjacent to every vertex of the clique.Let H be a bipartite graph were the degree partition equals the bipartite parti-tion. So, H is a regular or a biregular graph. In the last case, H will be called a genuine biregular graph . Observe that there are biregular bipartite graphs whichare not genuine biregular graphs; for instance, P . According to Kirkland et al. [11], every connected biregular split graph G has a split partition which is equalto its degree partition. Moreover, G has a unique regular or a unique genuinebiregular graph H as a spanning subgraph called the splitness of G . Note that H can be obtained from G taking off all edges in the clique determined by its sdp .Let G be a split graph, S = { S i : 1 ≤ i ≤ r } its set of minimal vertex separatorsand T - Simp its set of simplicial vertices. Graph G is a ( k, t ) -split graph if andonly if the following properties hold: • k = | S i | , ∀ S i ∈ S ; • S i ∩ S j = ∅ , ∀ S i , S j ∈ S ; • t = | A i | being A i the set of false twins adjacent to vertices of S i , 1 ≤ i ≤ r ; • S i =1 ,r A i = T - Simp . 10igure 3 displays a (2 , r = 3.If G is a ( k, t )-split graph, the following statements are valid:1. G is a biregular graph.2. n = ( k + t ) r .3. For r > G has a unique split partition. So the split partition equals thedegree partition.4. The splitness H of G can be obtained taking off all edges of the cliquedetermined by its non-simplicial vertices.5. The splitness H of G is isomorphic to r copies of K k,t , that is, H = rK k,t .6. G is a split-complete graph if and only if G is a ( k, t )-split graph with | S | = 1. 12 643 57 8 910 11 12 131415 Figure 3: A (2 , We are going to consider a particular labeling of the vertices of a ( k, t )-splitgraph, that will be used in the construction of its Laplacian matrix. We beginby labeling the vertices of each minimal vertex separator S i , ≤ i ≤ r . Thenwe label each set A i , ≤ i ≤ r . For instance, the graph depicted in Figure 3was already labeled in this way.The Laplacian matrix of a ( k, t )-split graph G has the form: L ( G ) = (cid:12)(cid:12)(cid:12)(cid:12) ( rk + t ) I rk − J rk − X rk × rt − X Trt × rk k I rt (cid:12)(cid:12)(cid:12)(cid:12) (1)where X is a block diagonal matrix with r blocks J k,t .The next theorem determines the integer Laplacian eigenvalues of a ( k, t )-splitgraph in terms of its structure. If r = 1, G is a split-complete graph and, as so,it is a Laplacian integral graph. When r = 1, the proof follows straightforwardfrom Theorem 2.2 in [11]. 11 heorem 9. Let G be a ( k, t ) -split graph and S = { S , . . . , S r } its set of min-imal vertex separators. Then G has, at least, n − r + 2 integer Laplacianeigenvalues. Proof. If r = 1, G is a split-complete graph with t = n − k vertices in itsindependent set. Then the Laplacian eigenvalues follow from Theorem 7. Thegraph has the following eigenvalues: 0 with multiplicity 1, k with multiplicity t − k + t with multiplicity k . If r = 1, the splitness of G is H = rK k,t .According to (1), the rank of block X of matrix L ( G ) is r , the number of copiesof K k,t . Then by Theorem 2.2 [11] the following integer Laplacian eigenvaluesof G are determined: 0 with multiplicity 1, k with multiplicity r ( t − rk + t with multiplicity r ( k −
1) and k + t with multiplicity 1.As n = ( k + t ) r , then r ( t −
1) + r ( k + 1) + 2 = n − r + 2. (cid:3) Form Theorem 9, it is immediate to see that a ( k, t )-split graph G has, at least,the following integer Laplacian eigenvalues:– 0 with multiplicity 1,– k with multiplicity r ( t − rk + t with multiplicity r ( k − k + t with multiplicity 1.Observe that ( k, t )-split graphs have an interesting property: for r = 1, thegraph is a split-complete graph and a ( G ) = κ ( G ); for r > a ( G ) is alwaysdifferent from κ ( G ) since, in this case, there is not a minimal vertex separatorcomposed by universal vertices (Theorem 2).
6. Simplicial vertices and Laplacian eigenvalues
For any block graph G , Bapat et al. [3] had shown that, under some conditions,the size of a maximal clique is a Laplacian eigenvalue of G . This fact wasalso observed in Abreu et al. [1] for the block-indifference graphs. For quasi-threshold graphs, we highlight (Theorem 7) the importance of the number ofsimplicial vertices in the determination of such eigenvalues. In this section wegeneralize these results proving that, for any chordal graph G , the cardinality ofall maximal cliques with at least two simplicial vertices is an integer Laplacianeigenvalue of G ; their multiplicities are also determined. The set of simplicialvertices of Q i , Simp ( Q i ), plays a decisive role. Theorem 10.
Let G = ( V, E ) be a connected chordal graph and let Q = { Q j , ≤ j ≤ ℓ } , be the set of maximal cliques of G . If | Simp ( Q i ) | ≥ , ≤ i ≤ ℓ , then | Q i | is an eigenvalue of L ( G ) with multiplicity at least | Simp ( Q i ) | − . igure 4: A chordal graph with ℓ = 6 Proof.
Let G be a non-complete chordal graph and Q i ∈ Q , | Q i | = n i . Let u, v be two simplicial vertices belonging Q i . Since G is a chordal graph and u and v are simplicial vertices in the same clique, they are true twins. So N [ u ] = N [ v ]and, as | Q i | = n i , then d G ( u ) = d G ( v ) = n i −
1. Considering the entries of L ( G ), we observe that: L ( G ) u,u = L ( G ) v,v = d G ( u ) = d G ( v ) and L ( G ) u,v = L ( G ) v,u = − y be the n -dimensional vector such that y ( u ) = 1, y ( v ) = − L ( G ) given by (1), weconclude that L ( G ) y = n i y . So, n i is an integer Laplacian eigenvalue of G .For every j , 1 ≤ j ≤ ℓ , let | Simp ( Q j ) | = { u j , ...., u jp j } . Suppose i ∈ { , . . . , ℓ } such that p i ≥
2. For all r , 2 ≤ r ≤ p i build p i − y ir such that y ir ( u i ) = 1, y ir ( u ir ) = − y ir , 2 ≤ r ≤ p i , are linearly independent. Besides, the vectors y ir ,2 ≤ r ≤ p i , satisfy L ( G ) y ir = n i y ir , so they are eigenvectors of L ( G ) correspond-ing to the eigenvalue n i . Then, the multiplicity of n i as Laplacian eigenvalue of G is p i − (cid:3) As an example, the Laplacian spectrum of G , presented in Figure 4, is: SpecL ( G ) = [ − , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , G are directlydetermined: 4 with multiplicity 3, 5 with multiplicity 1 and 6 with multiplicity1. 13 . Conclusions Several subclasses of chordal graphs were mentionned in this paper. Figure 5shows how they relate. chordal cographsplit graphs with a ( G ) = κ ( G ) quasi-threshold( k, t )-split threshold gen core-satsplit-complete windmill Figure 5: A hierarchy of chordal graphs
In all these subclasses we could notice the importance of the graph structure,its minimal vertex separators, its simplicial vertices and the maximal cliques.Other subclasses can be explored in the future.
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