Graphs with two trivial critical ideals
aa r X i v : . [ m a t h . C O ] A p r GRAPHS WITH TWO TRIVIAL CRITICAL IDEALS
CARLOS A. ALFARO AND CARLOS E. VALENCIA
Abstract.
The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrixassociated to a graph. A basic property of the critical ideals of graphs asserts that the graphs with atmost k trivial critical ideals, Γ ≤ k , are closed under induced subgraphs. In this article we find the set ofminimal forbidden subgraphs for Γ ≤ , and we use this forbidden subgraphs to get a classification of thegraphs in Γ ≤ . As a consequence we give a classification of the simple graphs whose critical group hastwo invariant factors equal to one. At the end of this article we give two infinite families of forbiddensubgraphs. Introduction
Given a connected graph G = ( V ( G ) , E ( G )) and a set of indeterminates X G = { x u | u ∈ V ( G ) } ,the generalized Laplacian matrix L ( G, X G ) of G is the matrix with rows and columns indexed by thevertices of G given by L ( G, X G ) uv = ( x u if u = v, − m uv otherwise , where m uv is the multiplicity of the edge uv , that is, the number of the edges between vertices u and v of G . For all 1 ≤ i ≤ n , the i - critical ideal of G is the determinantal ideal given by I i ( G, X G ) = h{ det( m ) | m is a square submatrix of L ( G, X G ) of size i }i ⊆ Z [ X G ] . We say that a critical ideal is trivial when it is equal to h i .Critical ideals are a generalization of the characteristic polynomials of the adjacency matrix and theLaplacian matrix associated to a graph. Also, critical ideals generalize the critical group of a graph asshown below: if d G ( u ) is the degree of a vertex u of G , then the Laplacian matrix of G , denoted by L ( G ), is the evaluation of L ( G, X G ) on x u = d G ( u ). Given a vertex s of G , the reduced Laplacian matrix of G , denoted by L ( G, s ), is the matrix obtained from L ( G ) by removing the row and column s . The critical group of a connected graph G , denoted by K ( G ), is the cokernel of L ( G, s ). That is, K ( G ) = Z e V / Im L ( G, s ) , where e V = V ( G ) \ s . Therefore the critical group of a graph can be obtained from their critical idealsas shows [3, theorems 3.6 and 3.7]. The critical group have been studied intensively on several contextsover the last 30 years. However, a well understanding of the combinatorial and algebraic nature of thecritical group still remains.Let assume that G is a connected graph with n vertices. A classical result (see [6, section 3.7]) assertsthat the reduced Laplacian matrix is equivalent to a integer diagonal matrix with entries d , d , ..., d n − Mathematics Subject Classification.
Primary 05C25; Secondary 05C50, 05E99.
Key words and phrases.
Critical ideal, Critical group, Generalized Laplacian matrix, Forbidden induced subgraph.Both authors were partially supported by CONACyT grant 166059, the first author was partially supported by CONA-CyT and the second author was partially supported by SNI. where d i > d i | d j if i ≤ j . The integers d , . . . , d n − are unique and are called invariant factors .With this in mind, the critical group is described in terms of the invariant factors as follows [8, theorem1.4]: K ( G ) ∼ = Z d ⊕ Z d ⊕ · · · ⊕ Z d n − . Given an integer number k , let f k ( G ) be the number of invariant factors of the Laplacian matrix of G equal to k . Let G i = { G | G is a simple connected graph with f ( G ) = i } . The study and characteri-zation of G i is of great interest. In particular, some results and conjectures on the graphs with cycliccritical group can be found in [10, section 4] and [13, conjectures 4.3 and 4.4]. On the other hand, DinoLorenzini, notice in [9] that G consists only of the complete graphs. More recently, Merino in [11] posedinterest on the characterization of G and G . In this sense, few attempts have done. For instance, in[12] it was characterized the graphs in G whose third invariant factor is equal to n , n − n −
2, or n −
3. In [2] the characterizations of the graphs in G with a cut vertex, and the graphs in G withnumber of independent cycles equal to n − ≤ i denotes the family of graphs with at most i trivial critical ideals, then it is not difficult tosee that G i ⊆ Γ ≤ i for all i ≥
0. At first glance, critical ideals behave better than critical ideals. Forinstance, by [3, proposition 3.3] we have that Γ ≤ i is closed under induced subgraphs at difference of G i .This property will play a crucial role in order to get a characterization of Γ ≤ on this paper. Also, ifΓ i is the family of graphs with exactly i trivial critical ideals, then we will shown on this paper that Γ has a more simple description that G .The main goals of this paper are three: to get a characterization of the graphs with at most twotrivial critical ideals, to get a characterization of the graphs with two invariant factors equal to one, andto give two infinite families of forbidden subgraphs for Γ ≤ i .This article is divided as follows: We begin by recalling some basic concepts on graph theory insection 2 and establishing some of basic properties of critical ideals in section 3. In section 4 we willcharacterize the graphs with at most two trivial critical ideals by finding their minimal set of forbiddengraphs. As consequence, we will get the characterization of the graphs with two invariant factors equalto one. Finally, in section 5 we give two infinite families of forbidden graphs for Γ ≤ i .2. Basic definitions
In this section, we give some basic definitions and notation of graph theory used in later sections. Itshould be pointed that we will consider the natural number as the the non-negative integers.Given a graph G = ( V, E ) and a subset U of V , the subgraph of G induced by U will be denoted by G [ U ]. If u is a vertex of G , let N G ( u ) be the set of neighbors of u in G . Here a clique of a graph G is amaximum complete subgraph, and its order is the clique number of G , denoted by ω ( G ). The path with n vertices is denoted by P n , a matching with k edges by M k , the complete graph with n vertices by K n and the trivial graph of n vertices by T n . The cone of a graph G is the graph obtained from G by addinga new vertex, called appex , which is adjacent to each vertex of G . The cone of a graph G is denoted by c ( G ). Thus, the star S k of k + 1 vertices is equal to c ( T k ). Given two graphs G and H , their union isdenoted by G ∪ H , and their disjoint union by G + H . The join of G and H , denoted by G ∨ H , is thegraph obtained from G + H when we add all the edges between vertices of G and H . For m, n, o ≥ K m,n,o be the complete tripartite graph . You can consult [4] for any unexplained concept of graphtheory.Let M ∈ M n ( Z ) be a n × n matrix with entries on Z , I = { i , . . . , i r } ⊆ { , . . . , n } , and J = { j , . . . , j s } ⊆ { , . . . , n } . The submatrix of M formed by rows i , . . . , i r and columns j , . . . , j s is RAPHS WITH TWO TRIVIAL CRITICAL IDEALS 3 denoted by M [ I ; J ]. If |I| = |J | = r , then M [ I ; J ] is called a r -square submatrix or a square submatrix of size r of M . A r -minor is the determinant of a r -square submatrix. The set of i -minors of a matrix M will be denoted by minors i ( M ). Finally, the identity matrix of size n is denoted by I n and the allones m × n matrix is denoted by J m,n . We say that M, N ∈ M n ( Z ) are equivalent , denoted by N ∼ M ,if there exist P, Q ∈ GL n ( Z ) such that N = P M Q . Note that if N ∼ M , then K ( M ) = Z n /M t Z n ∼ = Z n /N t Z n = K ( N ). 3. Graphs with few trivial critical ideals
In this section, we will introduce the critical ideals of a graph and the set of graphs with k or lesstrivial critical ideals, denoted by Γ ≤ k . After that, we define the set of minimal forbidden graphs of Γ ≤ k .We finish this section with the classification of G , that we already know that they are the completegraphs.Let G be a graph and X G = { x v | v ∈ V ( G ) } be the set of indeterminates indexed by the vertices of G . For all 1 ≤ i ≤ n , the i -critical ideal I i ( G, X G ) is defined as the ideal of Z [ X G ] given by I i ( G, X G ) = h{ det( m ) | m is a square matrix of L ( G, X G ) of size i }i . By convention I i ( G, X G ) = h i if i <
1, and I i ( G, X G ) = h i if i > n . The algebraic co-rank of G ,denoted by γ ( G ), is the number of critical ideals of G equal to h i . Definition 3.1.
For all k ∈ N , let Γ ≤ k = { G | G is a simple connected graph with γ ( G ) ≤ k } and Γ ≥ k = { G | G is a simple connected graph with γ ( G ) ≥ k } . Note that, Γ ≤ k and Γ ≥ k +1 are disjoint sets and that a characterization of one of them leads to acharacterization of the other one. Now, let us recall some basic properties about critical ideals, see[3] for details. It is known that if i ≤ j , then I j ( G, X G ) ⊆ I i ( G, X G ). Moreover, if H is an inducedsubgraph of G , then I i ( H, X H ) ⊆ I i ( G, X G ) for all i ≤ | V ( H ) | and therefore γ ( H ) ≤ γ ( G ). This impliesthat Γ ≤ k is closed under induced subgraphs, that is, if G ∈ Γ ≤ k and H is an induced subgraph of G ,then H ∈ Γ ≤ k . Definition 3.2.
Let f k ( G ) be the number of invariant factors of K ( G ) that are equal to k and G i = { G | G is a simple connected graph with f ( G ) = i } . Presumably Γ ≤ k behaves better than G k . It is not difficult to see that unlike of Γ ≤ k , G k is not closedunder induced subgraphs. For instance, considerer c ( S ), clearly it belongs to G , but S belongs to G .Similarly, K \ { P } belongs to G meanwhile K \ { P } belongs to G . Moreover, if H is an inducedsubgraph of G , it is not always true that K ( H ) E K ( G ). For example, K ( K ) ∼ = Z K ( K ) ∼ = Z .Finally, theorems 4.2 and 4.14 gives us additional evidence in the sense that Γ ≤ k behaves better than G k . Moreover, theorem 3.6 of [3] implies that γ ( G ) ≤ f ( G ) for any graph and therefore G k ⊆ Γ ≤ k forall k ≥ G is forbidden (or an obstruction ) for Γ ≤ k if and only if γ ( G ) ≥ k + 1. Let Forb (Γ ≤ k ) be theset of minimal (under induced subgraphs property) forbidden graphs for Γ ≤ k . Also, a graph G is called γ - critical if γ ( G \ v ) < γ ( G ) for all v ∈ V ( G ). That is, G ∈ Forb (Γ ≤ k ) if and only if G is γ -critical with γ ( G ) = k + 1.Given a family of graphs F , a graph G is called F -free if no induced subgraph of G is isomorphic toa member of F . Thus, G belongs to Γ ≤ k if and only if G is Forb (Γ ≤ k )-free, or equivalently, G belongsto Γ ≥ k +1 if and only if G contains a graph of Forb (Γ ≤ k ) as an induced subgraph. CARLOS A. ALFARO AND CARLOS E. VALENCIA
These ideas are useful to characterize Γ ≤ k . For instance, since γ ( P ) = 1 and no one of its inducedsubgraphs has γ ≥
1, then P ∈ Forb (Γ ≤ ). Moreover, it is easy to see that T is the only connectedgraph that is P -free. Thus, since I ( T , { x } ) = h i , we get that Forb (Γ ≤ ) = { P } , and Γ ≤ consistsof the graph with one vertex. Also, it is not difficult to prove that G = Γ ≤ and that the set of non-necessarily connected graphs with algebraic co-rank equal to zero consists only of the trivial graphs. Inthe next section we will use this kind of arguments in order to get Forb (Γ ≤ k ) and characterize Γ ≤ k for k equal to 1 and 2. Finally, section 5 will be devoted to explore in general the set Forb (Γ ≤ k ).Now, we obtain the characterization of Γ ≤ . Theorem 3.3. If G is a simple connected graph, then the following statements are equivalent:(i) G ∈ Γ ≤ ,(ii) G is P -free,(iii) G is a complete graph.Proof. ( i ) ⇒ ( ii ) Since γ ( P ) = 2, then clearly G must be P -free.( ii ) ⇒ ( iii ) If G is not a complete graph, then it has two vertices not adjacent, say u and v . Let P bethe smallest path between u and v . Thus, the length of P is greater or equal to 3. So, P is an inducedsubgraph of P and hence of G . Therefore, G is a complete graph.( iii ) ⇒ ( i ) It is easy to see that for any non-trivial simple connected graph, its first critical ideal istrivial, meanwhile I ( K , { x } ) = h x i . On the other hand, the 2-minors of a complete graphs are of theforms: − x i x j and 1 + x i . Since − x i x j ∈ h x , ..., x n i , then(3.1) I ( K n , X K n ) = ( h− x x i if n = 2 , and, h x , ..., x n i if n ≥ . Therefore γ ( K n ) ≤
1. In fact, the set { x , ..., x n } is a reduced Gr¨obner basis for I ( K n , X K n ),see [3, theorem 3.14]. (cid:3) In light of theorem 3.3, the characterization of G is as follows: Clearly, G ⊆ Γ ≤ \ G . Now, let G ∈ Γ ≤ \ { K } , that is, G = K n with n ≥ f ( G ) ≥
1. It is easy to verify from equation 3.1 thatthe second invariant factor of K ( G ) is equal to I ( K n , X K n ) | { x v = n − | v ∈ K n } which is different to 1. Corollary 3.4. [9] If G is a simple connected graph with n ≥ vertices, then f ( G ) = 1 if and only if G is a complete graph. A crucial fact in the proof of theorem 3.3 was that P belongs to Forb (Γ ≤ ) and the fact that anyother connected simple graph belonging to Γ ≥ contains P . This leads to the following corollary. Corollary 3.5. Forb (Γ ≤ ) = { P } . Next corollary give us the non-connected version of theorem 3.3.
Corollary 3.6. If G is a simple non-necessary connected graph, then the following statements areequivalent:(i) γ ( G ) ≤ ,(ii) G is { P , P } -free,(iii) G is a disjoint union of a complete graph and a trivial graph. RAPHS WITH TWO TRIVIAL CRITICAL IDEALS 5
Before to proceed with the proof of corollary 3.6 present a lemma that help us to calculate the criticalideal of a non connected graph. It may be useful to recall that the product of the ideals I and J of acommutative ring R , which we denote by IJ , is the ideal generated by all the products ab where a ∈ I and b ∈ J . Lemma 3.7. [3, Proposition 3.4] If G and H are vertex disjoint graphs, then I i ( G + H, { X G , Y H } ) = (cid:10) ∪ ij =0 I j ( G, X G ) I i − j ( H, Y H ) (cid:11) for all ≤ i ≤ | V ( G + H ) | . By this lemma we have that γ ( G + H ) = γ ( G ) + γ ( H ) when G and H are vertex disjoint. Proof of corollary 3.6. ( i ) ⇒ ( ii ) It follows since γ (2 P ) = 2 and γ ( P ) = 2.( ii ) ⇒ ( iii ) Let G , . . . , G s be the connected components of G . Then by theorem 3.3 and lemma 3.7, G i is a complete graph for all 1 ≤ i ≤ s . Since 2 P must not be an induced subgraph of G , then at mostone of the G i has order greater than 1.( iii ) ⇒ ( i ) If G = K n + T m , then it is not difficult to see that I ( T m , Y T m ) = h y , ..., y m i and I ( T m , Y T m ) = DQ i = j y i y j E . Thus by lemma 3.7, I ( G, { X K n , Y T m } ) = h I ( K n , X K n ) , I ( K n , X K n ) I ( T m , Y T m ) , I ( T m , Y T m ) i 6 = h i . (cid:3) Graphs with algebraic co-rank equal to two
The main goal of this section is to classify the simple graphs on Γ ≤ . After that, using the fact that G ⊆ Γ ≤ we will classify the simple graphs whose critical group has two invariant factors equal to 1.As in the case of Γ ≤ , the characterization of Γ ≤ relies heavely in the fact that Γ ≤ is closed underinduced subgraphs and the fact that we have a good guessing about Forb (Γ ≤ ). We begin with theintroduction of a set of graphs in Forb (Γ ≤ ). Proposition 4.1.
Let F be the set of graphs consisting of P , K \ S , K \ M , cricket and dart ; seefigure 1. Then F ⊆ Forb (Γ ≤ ) . P K \ S K \ M cricket dart Figure 1.
The set F of graphs. Proof.
It is not difficult to see that the generalized Laplacian matrix of the graphs on F are given by: L ( P ) = x − − x − − x −
10 0 − x , L ( K \ S ) = x − − x − − − − − x − − − − − x − − − − x ,L ( cricket ) = x − x − − − x − − − − x −
10 0 0 − x , L ( dart ) = x − − − x − − − x − − − − x −
10 0 0 − x , CARLOS A. ALFARO AND CARLOS E. VALENCIA L ( K \ M ) = x − − − − x − − − − − − x − − − − − x − − − − − − x − − − − − x . In this matrices we marked with gray some 3 × ± γ ( G ) ≥ G ∈ F . Finally, using any algebraic system, for instance Macaulay 2 , one cannote that the graphs in F has algebraic co-rank equal to 3. Moreover, it can be checked that any ofhis induced subgraphs has algebraic co-rank less or equal to 2. (cid:3) One of the main results of this article is the following:
Theorem 4.2.
Let G be a simple connected graph. Then, G ∈ Γ ≤ if and only if G is an inducedsubgraph of K m,n,o or T n ∨ ( K m + K o ) . We divide the proof of theorem 4.2 in two steps. First we classify the connected graphs that are F -free. After that, we check that all these graphs have algebraic co-rank less or equal than two. Theorem 4.3.
A simple connected graph is F -free if and only if it is an induced subgraph of K m,n,o or T n ∨ ( K m + K o ) .Proof. First, one implication is clear because K m,n,o and T n ∨ ( K m + K o ) are F -free. The another partof the proof is divided in three cases: when ω ( G ) = 2, ω ( G ) = 3, and ω ( G ) ≥ ω ( G ) = 2 is very simple. Since ω ( G ) = 2, there exist a, b ∈ V ( G ) such that ab ∈ E ( G ).Clearly, N G ( a ) ∩ N G ( b ) = ∅ . Moreover, if x ∈ { a, b } , then uv / ∈ E ( G ) for all u, v ∈ N G ( x ). On the otherhand, since G is P -free, then uv ∈ E ( G ) for all u ∈ N G ( a ) and v ∈ N G ( b ). Therefore G is the completebipartite graph.Now, assume that ω ( G ) = 3. Let a , b and c be vertices of G that induce a complete graph. For all X ⊆ { a, b, c } let V X = { v ∈ V ( G ) | N G ( v ) ∩ { a, b, c } = X } . Clearly V { a,b,c } = ∅ because ω ( G ) = 3. In asimilar way, if X ⊆ { a, b, c } has size two, then set V X induce a trivial graph. Also, since G is cricket -free, V x induces a complete graph for all x ∈ { a, b, c } . Thus V x has at most two vertices.Now, given U, V ∈ V ( G ), let E ( U, V ) = { uv ∈ E ( G ) | u ∈ U and v ∈ V } . Let x = y ∈ { a, b, c } and z ∈ { a, b, c } such that { x, y, z } = { a, b, c } Assume that V x , V y and V { x,y } are not empty. Let u ∈ V x and v ∈ V y . If uv ∈ E ( G ), then { u, v, y, z } induced a P . Therefore E ( V x , V y ) = ∅ . In a similar way, since G is P -free, we get E ( V x , V { x,y } ) = ∅ . Claim 4.4.
At least two of the sets V a , V b or V c are empty. Furthermore, if V a = ∅ , then G is aninduced subgraph of T l ∨ ( K + K ) , where l = | V { b,c } | + 1 .Proof. First, assume that V x and V y are non empty. Let u ∈ V y , v ∈ V x . Since u and v are not adjacent,the vertices { u, x, y, v } induces a P . Therefore at least one of V x or V y is empty.Without loss of generality we can assume that V a is not empty. Since there is no edge between V a and V { a,b } , then V { a,b } = ∅ . Otherwise, if u ∈ V { a,b } and v ∈ V a , then the vertices { u, v, a, b, c } induces a dart . In a similar way V { a,c } = ∅ . On the other hand, if V { b,c } is not empty and there exist u ∈ V { b,c } and v ∈ V a such that uv / ∈ E ( G ), then the vertices { u, b, a, v } induces a P . Therefore, either E ( V a , V { b,c } ) = { uv | u ∈ V a and v ∈ V { b,c } } or the set V { b,c } is empty. Finally, since V a is a completegraph with at most two vertices, the result follows. (cid:3) RAPHS WITH TWO TRIVIAL CRITICAL IDEALS 7
Now, we can assume that V x = ∅ for all x ∈ { a, b, c } . Let { x, y, z } = { a, b, c } . If uv / ∈ E ( G ) forsome u ∈ V { x,y } , v ∈ V { x,z } , then { u, y, z, v } induces a P . Therefore uv ∈ E ( G ) for all u ∈ V { x,y } and u ∈ V { x,z } , and G is an induced subgraph of the complete tripartite graph.We finish with case when ω ( G ) ≥
4. Let W = { a, b, c, d } be a complete subgraph of G of size fourand let V i = { v ∈ V ( G ) \ W (cid:12)(cid:12) | N G ( V ) ∩ W | = i } for all i = 0 , , , , . Since G is K \ S -free, V = ∅ . Claim 4.5.
The graph induced by V is a complete graph.Proof. Let u, u ′ ∈ V and suppose there is no edge between u and u ′ . Let x, y ∈ W be the verticesadjacent to u and u ′ , respectively. If x = y , then { u, x, y, u ′ } induces a P ; a contradiction. On theother hand, if x = y , let z = w ∈ W \ x . Since u and u ′ are not adjacent to both z and w , then { x, z, w, u, u ′ } induces a cricket ; a contradiction. (cid:3) Let v, v ′ ∈ V and assume that are adjacent. Let x, y ∈ W such that x / ∈ N G ( v ) and y / ∈ N G ( v ′ ). If x = y , then { v, v ′ } ∪ W induces a K \ M ; a contradiction. On the other hand, if x = y , then { v, v ′ } ∪ W contains a K \ S as induced graph; a contradiction. Therefore V induces a trivial graph.Now, let u ∈ V , v ∈ V , x, y ∈ W such that xu ∈ E ( G ), yv / ∈ E ( G ). Assume that uv / ∈ E ( G ). Let z ∈ W \ { x, y } . If x = y , then { v, z, x, u } induces a P ; a contradiction. On the other hand, if x = y , then G must contains a dart as induced subgraph; a contradiction. Therefore E ( V , V ) contains all the possibleedges. Since uv ∈ E ( G ), then x = y . Otherwise, if x = y , then { y, z, v, u } induces a P ; a contradiction.Therefore we can assume without loss of generality that { a } = N G ( V ) ∩ W = ( N G ( V ) ∩ W ) c .Now, let w ∈ V , u ∈ V , and v ∈ V . If uw ∈ E ( G ), then { u, w, a, b, c } induces a K \ S . Therefore, E ( V , V ) = ∅ . In a similar way, if vw / ∈ E ( G ), then { v, a, w, b, c } induces a K \ S . Therefore, E ( V , V ) = { vw | v ∈ V and w ∈ V } .Since G is { K \ S , K \ M } -free, then it is not difficult to see that the graph induced by V is { K + T , C } -free. Thus V induces either a trivial graph, a complete graph, or a complete graph minusan edge. Moreover, if ww ′ / ∈ E ( G ) for some w = w ′ ∈ V and v ∈ V , then { w, w ′ , a, v, b, c } induces a K \ M . Thus, if V = ∅ , then V induces a complete graph.Clearly, if V , V , V = ∅ , then G is a complete graph. Claim 4.6. If V , V = ∅ and V = ∅ , then G is an induced subgraph of T ∨ ( K m + K n ) for some m, n ∈ N .Proof. If | V | = | V | = 1, then the result is clear. Therefore we can assume that | V | ≥ | V | ≥ V , when it induces a trivial graph, a complete graph, ora complete graph minus an edge. Assume that V induces a trivial graph. If | V | ≥
2, let o ∈ V and w, w ′ ∈ V . If ow ∈ E ( G ) and ow ′ / ∈ E ( G ), then { o, w, w ′ , a } induces a P ; a contradiction. Thus, either E ( o, V ) = { ow | w ∈ V } or E ( o, V ) is empty. Therefore, since G is connected, we get the result when | V | = 1.Now, assume that | V | ≥
2. Since G is connected, there exist o ∈ V such that ow ∈ E ( G ) for some w ∈ V . Let o ′ ∈ V such that E ( o ′ , V ) is empty. Since G is connected, there exist a path from o ′ to o .Let P be a minimum path between o ′ and o . In this case, { V ( P ) , w, a } induces a path with more thatfour vertices; a contradiction. Therefore, E ( V , V ) = { ow | o ∈ V and w ∈ V } . Moreover, since G is K \ M -free, then V induces a trivial graph and we get the result. CARLOS A. ALFARO AND CARLOS E. VALENCIA
Now, assume that V induces a complete graph. Since G is K \ S -free, o is adjacent to at mostone vertex in V . Moreover, all the vertices in V are adjacent to a unique vertex in V . Otherwise,let o, o ′ ∈ V and w, w ′ ∈ V such that ow, o ′ w ′ ∈ E ( G ) and ow ′ , o ′ w / ∈ E ( G ). If oo ′ ∈ E ( G ), then { a, w, o, o ′ } induces a P ; a contradiction. Also, if oo ′ / ∈ E ( G ) and ww ′ ∈ E ( G ), then { w, w ′ , o, o ′ } induces a P ; a contradiction. Let w ∈ V such that all the vertices in V are adjacent to w . Then V induces a complete graph. Otherwise, { a, b, w, o, o ′ } induces a cricket ; a contradiction. Therefore G isan induced subgraph of T ∨ ( K m + K n ) for some m, n ∈ N .Finally, when V induces a complete graph minus an edge, following similar arguments to those givenin the case when V induces a complete graph we get that G is an induced subgraph of T ∨ ( K m + K n )for some m, n ∈ N . (cid:3) Therefore we can assume that V ∪ V = ∅ . Let u ∈ V ∪ V , o ∈ V , and x = y ∈ W such that x / ∈ N G ( u ) and y ∈ N G ( u ). If uo ∈ E ( G ), then { x, y, u, o } induces a P ; a contradiction. Thus, thereare no edges between V and V ∪ V . Moreover, let w ∈ V . If ow ∈ E ( G ), then { a, b, u, w, o } induces a dart when u ∈ V and { u, a, w, o } induces a P when u ∈ V . Therefore there are no edges between V and V . Since G is connected, V = ∅ and therefore G is an induced subgraph of T n ∨ ( K m + K o ). (cid:3) To finish the proof of theorem 4.2 we need to prove that the third critical ideal of the graphs K m,n,o and T n ∨ ( K m + K o ) is not trivial. If m + n + o ≤
2, then the third critical ideal is equal to zero. Also,if m + n + o = 3, then the third critical ideal is equal to the determinant of the generalized Laplacianmatrix. Moreover, [3, theorem 3.16] proves that the algebraic co-rank of the complete graph is equal to1. Theorem 4.7. If K m,n,o is connected with m ≥ n ≥ o and m + n + o ≥ , then (4.1) I ( K m,n,o , { X, Y, Z } ) = h , S mi =1 x i , S ni =1 y i , S oi =1 z i i if m, n, o ≥ , h S mi =1 x i , S ni =1 y i , z + 2 i if m ≥ , n ≥ , o = 1 , h S mi =1 x i , y + z + 2 i if m ≥ , n = 1 , o = 1 , h x x + x + x , x z + x , x z + x , y + z + 2 i if m = 2 , n = 1 , o = 1 , h S mi =1 x i , S ni =1 y i i if m ≥ , n ≥ , o = 0 , h S mi =1 x i , y + y i if m ≥ , n = 2 , o = 0 , h x y , x + x , y + y i if m = 2 , n = 2 , o = 0 , h S mi =1 x i i if m ≥ , n = 1 , o = 0 . Theorem 4.8. If T n ∨ ( K m + K o ) is connected with m ≥ o , m + n + o ≥ , and such that T n ∨ ( K m + K o ) is not the complete graph or the complete bipartite graph, then (4.2) I ( T n ∨ ( K m + K o ) , { X, Y, Z } ) = h , S mi =1 ( x i + 1) , S ni =1 y i , S oi =1 ( z i + 1) i if m, n, o ≥ , h S mi =1 ( x i + 1) , y + 2 , S oi =1 ( z i + 1) i if m ≥ , n = 1 , o ≥ , h S mi =1 ( x i + 1) , S ni =1 y i , z − i if m ≥ , n ≥ , o = 1 , h x + z , S ni =1 y i i if m = 1 , n ≥ , o = 1 , h x + z , y + y , y z , i if m = 1 , n = 2 , o = 1 , h S mi =1 ( x i + 1) , z y + z − i if m ≥ , n = 1 , o = 1 , h S mi =1 ( x i + 1) , S ni =1 y i i if m ≥ , n ≥ , o = 0 , h x + x + 2 , S ni =1 y i i if m = 2 , n ≥ , o = 0 , h S mi =1 ( x i + 1) , y y + y + y i if m ≥ , n = 2 , o = 0 , h x + x + 2 , x y + y , x y + y , y y + y + y i if m = 2 , n = 2 , o = 0 , RAPHS WITH TWO TRIVIAL CRITICAL IDEALS 9
The proofs of theorems 4.7 and 4.8 relies on the description of the 3-minors of the generalized Laplacianmatrices of K m,n,o and T n ∨ ( K m + K o ). Proof of theorem 4.7.
In order to simplify the arguments in the proof we separate it in two parts. Webegin by finding the 3-minors of the generalized Laplacian matrix of the complete bipartite graph andusing it to calculate their third critical ideal. An after that, we do the same for the general case of thecomplete tripartite graph.
Lemma 4.9.
For m, n ≥ , let L m,n be the generalized Laplacian matrix of the complete bipartite graph K m,n . That is, L m,n = L ( K m,n , { X T m , Y T n } ) = (cid:20) L ( T m , X T m ) − J m,n − J n,m L ( T n , Y T n ) (cid:21) . Then -minors (with positive leading coefficient) of L m,n are the following: • y j , y j y j , and y j y j y j when n ≥ , • x i , x i x i , and x i x i x i when m ≥ , • y j y j x i − y j − y j when n ≥ , • x i x i y j − x i − x i when m ≥ , • x i + x i , y j + y j and x i y j when m ≥ and n ≥ , where ≤ i < i < i ≤ n and ≤ j < j < j ≤ n .Proof. Before to proceed with the proof we establish some notation corresponding to row and columnindices. Let I = { i , i , i } such that 1 ≤ i < i < i ≤ m + n , and J = { j , j , j } such that1 ≤ j < j < j ≤ m + n . Let I = I ∩ [ m ], I = I c , J = J ∩ [ m ], and J = J c . Also in the following i ′ t = i t − m and j ′ t = j t − m , for all 1 ≤ t ≤ L m,n we need to calculate the determinants of all non-singularmatrices of the form L m,n [ I , J ]. Since the generalized Laplacian matrix is symmetric, we can assumewithout loss of generalization that |I | ≤ |J | . Let L = L m,n [ I ; J ] be non-singular. First, consider thecase when I is empty. Since the determinant of L is equal to zero when |J | ≥
2, only remains toconsider the cases when |J | = 0 or |J | = 1. If |J | = 0, then m ≥ L is a submatrix of L ( T m , X T m ),and the determinant of L is equal to x i x i x i . In a similar way, if |J | = 1, then m ≥ n ≥
1, and L is equal to (up to row permutation) x j − x j −
10 0 − whose determinant is equal to − x j x j .Now, consider the case when |I | = 1. In a similar way, L has determinant different from zero when |J | = 1 or |J | = 2. If |J | = 1, then there are essentially only four 3 × L m,n : x i − A − − − B , where A is equal to 0 (when m ≥
3) and x i , and B is equal to 0 (when n ≥
2) and y i ′ . Clearlydet( L ) = ABx i − A − x i . Thus we have the following minors: x i x i y i ′ − x i − x i , − x i − x i , − x i .If |J | = 2, then m ≥ n ≥
2, and L has determinant equal todet x j − − − − − y i ′ = − x j y i ′ . When |I | = 2 we have two cases, when either |J | = 2 or |J | = 3. If |J | = 2, then L is equal to: A − − − y i ′ − B where A is equal to 0 (when m ≥
2) or x i and B is equal to 0 (when n ≥
3) or y i ′ . It is easy to see thatdet( L ) = ABy i ′ − A − y i ′ . Thus we have the following minors: x i y i ′ y i ′ − y i ′ − y i ′ , − y i ′ − y i ′ , − y i ′ . If |J | = 3, then m ≥ n ≥ − − − y i ′ y i ′ = − y i ′ y i ′ . Finally, if |I | = 3, then n ≥ L is a submatrix of L ( T m , Y T m ), and therefore its determinant is equalto y i ′ y i ′ y i ′ . (cid:3) We can use lemma 4.9 to get the third critical ideal of the complete bipartite graph. For instance, itis not difficult to see that I ( K m,n , { X, Y } ) = h S mi =1 x i , S ni =1 y i i when m ≥ n ≥
3. In a similarway, since x i + x i , x i y j , y j y j x i − y j − y j , x i x i , x i x i x i ∈ h S mi =1 x i , y + y i , I ( K m,n , { X, Y } ) = h S mi =1 x i , y + y i when m ≥ n = 2. The other cases follow in a similar way.Therefore in order to calculate the third critical ideal of the complete tripartite graph we need tocalculate their 3-minors as below. Theorem 4.10.
For m, n, o ≥ , let L m,n,o be the generalized Laplacian matrix of the tripartite completegraph K m,n,o . That is, L m,n,o = L ( K m,n,o , { X T m , Y T n , Z T o } ) = L ( T m , X T m ) − J m,n − J m,o − J n,m L ( T n , Y T n ) − J n,o − J o,m − J o,n L ( T o , Z T o ) . Then the -minors (with positive leading coefficient) of L m,n,o are the following: • x i , x i x i , and x i x i x i when m ≥ , • when m ≥ , n ≥ and o ≥ , • y j , y j y j , and y j y j y j when n ≥ , • − − x i − y j − z k + x i y j z k , • z k , z k z k , and z k z k z k when o ≥ , • x i , y j , x i + 2 , y j + 2 , x i + x i , y j + y j , and x i y j when m ≥ and n ≥ , • x i , z k , x i + 2 , z k + 2 , x i + x i , z k + z k , and x i z k when m ≥ and o ≥ , • y j , z k , y j + 2 , z k + 2 , y j + y j , z k + z k , and y j z k , when n ≥ and o ≥ , • y j + z k + 2 , x i ( y j + 1) , x i ( z k + 1) , x i x i + x i + x i , x i x i y j − x i − x i , and x i x i z k − x i − x i when m ≥ , • x i + z k + 2 , y j ( x i + 1) , y j ( z k + 1) , y j y j + y j + y j , y j y j x i − y j − y j , and y j y j z k − y j − y j when n ≥ , • x i + y j + 2 , z k ( x i + 1) , z k ( y j + 1) , z k z k + z k + z k , z k z k x i − z k − z k , and z k z k y j − z k − z k when o ≥ , where ≤ i < i < i ≤ m , ≤ j < j < j ≤ n , and ≤ k < k < k ≤ o .Proof. We will follow a similar proof to the proof given for lemma 4.9. Let I = { i , i , i } with 1 ≤ i < i < i ≤ m + n + o and J = { j , j , j } with 1 ≤ j < j < j ≤ m + n + o . Moreover, let I = I ∩ [ m ], I = I ∩ { m + 1 , . . . , m + n } , I = I ∩ { m + n + 1 , . . . , m + n + o } , J = J ∩ [ m ], J = J ∩ { m + 1 , . . . , m + n } , J = J ∩ { m + n + 1 , . . . , m + n + o } . Also, in the following i ′ t = i t − m , i ′′ t = i t − m − n , j ′ t = j t − m and j ′′ t = j t − m − n , for t ∈ [3].Let L = L m,n,o [ I ; J ]. First, in the same way that in the proof of lemma 4.9 we can assume that L is non-singular. Several of the 3-minor of L m,n,o can be calculated using lemma 4.9. For instance, if I i = J i = ∅ for some i = 1 , ,
3, then L is a submatrix of L ( K m,n , { X T m , Y T n } ) and the corresponding RAPHS WITH TWO TRIVIAL CRITICAL IDEALS 11 I i = ∅ , then J i = ∅ forall i = 1 , ,
3. Moreover, if I i = ∅ , then |J i | = 1 for all i = 1 , ,
3. Because otherwise either L will havetwo identical columns; a contradiction to the fact that L is non-singular. In a similar way, if J i = ∅ ,then |I i | = 1 for all i = 1 , ,
3. If |I i | = 3 for some i = 1 , ,
3, then L is a submatrix of the generalizedLaplacian matrix of a complete bipartite graph. Therefore we can assume that |I i | ≤ |J i | ≤ i = 1 , , I i = ∅ 6 = J i for all 1 ≤ i ≤
3, that is, |I i | = |J i | = 1for all 1 ≤ i ≤
3. In this case we have that L = A − − − B − − − C , where A is equal to 0 (when m ≥
2) or x i , B is equal to 0 (when n ≥
2) or y i ′ , and C is equal to 0(when n ≥
2) or z i ′′ . Since det L = ABC − A − B − C − L m,n,o . Since |I i | ≤ |J i | ≤
2) for all i = 1 , ,
3, then there are no two I ’s ( J ’s) empty. Therefore only remains thecases: when only one of the I ’s is empty and the case when one of the I ’s is empty and one of the J ′ s is empty.Consider the case when only one of the sets I ’s is empty, that is, |J i | = 1 for all i = 1 , ,
3. Assumethat I = ∅ . Then we need to consider the following two matrices (when |I | = 1 and |I | = 2): L = A − − − − − B − and L = A − − − − − B − , where A is equal to 0 (when m ≥ m ≥
3, respectively) or x i ′ and B is equal to 0 (when n ≥ n ≥
2, respectively) or y i ′ . It is not difficult to see that det( L ) = AB − B and det( L ) = AB − A .Thus, we get the minors x i y i ′ − y i ′ (when n ≥ x i y i ′ − x i (when m ≥ − y i ′ and − x i (when m ≥ n ≥ I = ∅ or I = ∅ .Finally, consider the case when one of the I ’s is empty and one of the J ′ s is empty. Assume that I = ∅ and J = ∅ . Then |I | = 1 and L is equal to: A − A ′ − − − − , where A is equal to 0 or x i ′ and A ′ is equal to 0 or x i . Clearly det L = − AA ′ − A − A ′ . Thus we getthe 3-minors x i x i + x i + x i (when m ≥
2) and x i and x i (when m ≥ J = ∅ and the other cases. (cid:3) The rest it follows by similar arguments to those in the case of the bipartite complete graph. (cid:3)
Proof of theorem 4.8.
Following the proof of theorem 4.7 we need to find the 3-minors of the generalizedLaplacian matrix of T n ∨ ( K m + K o ). We begin with K m ∨ T n and after that we do the same for T n ∨ ( K m + K o ). We omit the proofs of lemma 4.11 and theorem 4.12 because are rutinary and bothfollows by using similar arguments to those in lemma 4.9 and in theorem 4.10, respectively. Lemma 4.11.
For m, n ≥ , let L ′ m,n be the generalized Laplacian matrix of K m ∨ T n . That is, L ′ m,n = L ( K m ∨ T n , { X K n , Y T m } ) = (cid:20) L ( K m , X K m ) − J m,n − J n,m L ( T n , Y T n ) (cid:21) . Then the -minors (with positive leading coefficient) of L ′ m,n are the following: • y j , y j y j , and y j y j y j when n ≥ , • ( x i +1)( x i +1) , ( x i +1)( y i +1) , and x i x i x i − x i − x i − x i − when m ≥ , • x i y j y j − y j − y j when n ≥ , • x i x i y j − x i − x i − y j − when m ≥ , • y j when m ≥ and n ≥ , • x i + 1 when m ≥ and n ≥ , • x i + x i + 2 , x i + y j , x i y j y j and y j y j + y j + y j when m ≥ and n ≥ , where ≤ i < i < i ≤ m and ≤ j < j < j ≤ n . Theorem 4.12.
For m, n, o ≥ , let L ′ m,n,o be the the generalized Laplacian matrix of T n ∨ ( K m + K o ) .That is, L ′ m,n,o = L ( T n ∨ ( K m + K o ) , { X K m , Y T n , Z K o } ) = " L ( K m , X K m ) − J m,n m,o − J n,m L ( T n , Y T n ) − J n,o o,m − J o,n L ( K o , Z K o ) . Then the -minors (with positive leading coefficient) of L ′ m,n,o are the following: • x i + 1 when m ≥ and ( o ≥ or n ≥ ), • z k + 1 when o ≥ and ( m ≥ or n ≥ ), • y j when n ≥ and ( m ≥ or o ≥ ), • when m ≥ , n ≥ , and o ≥ , • y j , y j y j , and y j y j y j when n ≥ , • x i y j z k − x i − z k , • x i + 1 , z k ( x i + 1) , ( x i + 1)( x i + 1) , ( x i + 1)( y j + 1) , and x i x i x i − x i − x i − x i − , when m ≥ , • z k + 1 , x i ( z k + 1) , ( z k + 1)( z k + 1) , ( z k + 1)( y j + 1) , and z k z k z k − z k − z k − z k − when o ≥ , • x i + z k , y j + y j , x i y j , y j z k , x i y j y j − y j − y j , and y j y j z k − y j − y j when n ≥ , • x i + 1 , x i x i − , y j z k + z k − , z k ( x i + 1) , x i x i z k − z k , and x i x i y j − x i − x i − y j − , when m ≥ , • z k + 1 , x i ( z k z k − , z k z k − , x i y j + x i − , x i ( z k + 1) , and z k z k y j − z k − z k − y j − , when o ≥ , • x i + 1 , y j + 2 , z k + 1 , x i x i − , and z k z k − when m ≥ and o ≥ , • x i + 1 , y j , z k − , x i + y j , x i + x i + 2 , y j ( x i + 1) , and y j y j + y j + y j when m ≥ and n ≥ , • x i − , y j , z k + 1 , z k + y j , z k + z k + 2 , y j ( z k + 1) , and y j y j + y j + y j , when n ≥ and o ≥ , where ≤ i < i < i ≤ m , ≤ j < j < j ≤ n , and ≤ k < k < k ≤ o . (cid:3) Theorems 4.7 and 4.8 implies that
Forb (Γ ≤ ) = F . Now, we present the non-conected version oftheorem 4.2. Corollary 4.13.
A simple graph has algebraic co-rank equal to two if and only if is the disjoint unionof a trivial graph with one of the following graphs: • K m,n,o , where m ≥ , n + o ≥ , • T n ∨ ( K m + K o ) , where m, o ≥ , m, n, o ≥ , or n ≥ and m + o ≥ .Proof. It is not difficult to see that in the non-connected case we need to add the graphs P + P and3 P to the set of forbidden graphs. The rest follows directly from theorem 4.2. (cid:3) We finish this section with the classification of the graphs having critical group with 2 invariantfactors equal to one.
Theorem 4.14.
The critical group of a connected simple graph has exactly two invariant factor equalto if and only if is one of the following graphs: • K m,n,o , where m ≥ n ≥ o satisfy one of the following conditions: ∗ m, n, o ≥ with the same parity, ∗ m, n ≥ , o = 1 , and gcd( m + 1 , n + 1) = 1 , ∗ m ≥ , n = o = 1 , ∗ m, n ≥ , o = 0 and gcd( m, n ) = 1 , ∗ m ≥ , n = 2 , and o = 0 , or RAPHS WITH TWO TRIVIAL CRITICAL IDEALS 13 ∗ m = 2 and n = 1 . • T n ∨ ( K m + K o ) , where m ≥ o and n satisfy one of the following conditions: ∗ m, n, o ≥ with the same parity, ∗ m, o ≥ , n = 1 , and gcd( m + 1 , o + 1) = 1 , ∗ m, n ≥ , o = 1 , and gcd( m + 1 , n − = 1 , ∗ m ≥ , n = o = 1 , ∗ n ≥ , m = o = 1 , ∗ m, n ≥ , o = 0 , and gcd( m, n ) = 1 , ∗ m ≥ , n = 2 , o = 0 , or ∗ m = 2 , n ≥ , o = 0 .Proof. It turns out from theorems 4.7 and 4.8. (cid:3) The set
Forb (Γ ≤ k ) . The characterization of the γ -critical graphs with a given algebraic co-rank, Forb (Γ ≤ k ), is veryimportant. For instance, we were able to characterize Γ ≤ k for k equal to 1 and 2 because we got afinite set of γ -critical graphs with algebraic co-rank equal to k + 1 (for k equal to 1 and 2), and afterthat we proved that all the graphs that do not contain a graph from this set as an induced subgraphhas algebraic co-rank less or equal to k . In this section we give two infinite families of forbidden simplegraphs. This will prove that Forb (Γ ≤ k ) is not empty for all k ≥
0. Moreover, we conjecture that
Forb (Γ ≤ k ) is finite for all k ≥
0. To finish we present an example of a simple graph G with algebraicco-rank equal to 5 but with no 5-minor equal to 1. That is, the 1 can be obtained uniquely from a nontrivial algebraic combination of 5-minors of L ( G, X ).We begin by proving that the path with n + 2 vertices is γ -critical with algebraic co-rank equal to n + 1. Theorem 5.1. If n ≥ , then P n +2 ∈ Forb (Γ ≤ n ) .Proof. It is not difficult to prove γ ( P n +2 ) = n + 1, see corollary 4.10 of [3]. On the other hand, if H = P n +2 \ v for some v ∈ V ( P n +2 ), then H is a disjoint union of at most two paths. Let H = P n + · · · + P n s with 1 ≤ s ≤ P si = n i = n + 1, then by lemma 3.7 we get that γ ( H ) = s X i =1 γ ( P n i ) = s X i =1 ( n i −
1) = s X i =1 n i − s = n + 1 − s < n + 1 . Therefore P n +2 ∈ Forb (Γ ≤ n ). (cid:3) Now, we present another infinite family of graph that are γ -critical. Let K n be the complete graphwith n vertices and M k a matching of K n with k edges. We begin by finding the critical group of K n \ M k . Proposition 5.2. If K n be the complete graph with n vertices and M k is a matching of k edges, then K ( K n \ M k ) ∼ = ( Z n − k − n ⊕ Z kn ( n − if n ≥ k + 2 , Z n − ⊕ Z k − n ( n − if n = 2 k + 1 . Proof. If n = 2 k + 1 the result follows by [7, Theorem 1]. Therefore we can assume that n ≥ k + 2.Given a ∈ Z k , let N k +1 ( a ) be the matrix given by (cid:20) a0 t I k (cid:21) . If M k = { v v , . . . , v k − v k } , then L ( K n \ M k , v n ) = (cid:20) [( n − I + J ] ⊗ I k − J k − J k,n − k − − J n − k − , k nI n − k − − J n − k − (cid:21) , where ⊗ is the tensor product of matrices. Now, since det( N n − ( a )) = 1 for all a , then L ( K n \ M k , v n ) ∼ N n − ( ) t N n − ( ) L ( K n \ M k , v n ) N n − ( − )= I ⊕ nI n − k − k M i =1 (cid:20) n − n − (cid:21) . On the other hand (cid:20) n − n − (cid:21) ∼ (cid:20) − n − (cid:21) (cid:20) n − n − (cid:21) (cid:20) − ( n − (cid:21) = (cid:20) n ( n − (cid:21) . Therefore L ( K n \ M k , v n ) ∼ I k +1 ⊕ nI n − k − ⊕ n ( n − I k . (cid:3) Corollary 5.3. If n = 2 k + 2 , then K n \ M k ∈ Forb (Γ ≤ k ) .Proof. First, by proposition 5.2 we have that γ ( K n \ M k ) ≤ ( k + 1 if n ≥ k + 2 ,k if n = 2 k + 1 . Now, let n ≥ k + 2, M k = { v v , . . . , v k − v k } , and M = L ( K n \ M k , X )[ { , . . . , k + 1 } , { , . . . , k + 2 } ]be a square submatrix of generalized Laplacian matrix of K n \ M k . Then M = − − − − − −
1. . . − − − − − − − − − − − . By [3, theorem 3.13], det( M ) = det( L ( K k , X K k )) | { x =0 ,...,x k − =0 ,x k = − } . = − γ ( K n \ M k ) = k + 1 for all n ≥ k + 2. Finally, if n = 2 k + 2 and v ∈ V ( K n \ M k ), then ( K n \ M k ) \ v is equalto K n − \ M k or K n − \ M k − . Thus, γ (( K n \ M k ) \ v ) ≤ k and therefore K n \ M k ∈ Forb (Γ ≤ k ). (cid:3) This results proves that
Forb (Γ ≤ k ) is not empty for all k ≥ Corollary 5.4. If k ≥ , then Forb (Γ ≤ k ) is not empty. For i ≥
3, the set
Forb (Γ ≤ i ) is more complex than Forb (Γ ≤ ) and Forb (Γ ≤ ). For instance, in [1]was proved that Forb (Γ ≤ ) has 49 graphs. Moreover, we conjecture that Forb (Γ ≤ k ) is finite. Conjecture 5.5.
For all k ∈ N the set Forb (Γ ≤ k ) is finite. RAPHS WITH TWO TRIVIAL CRITICAL IDEALS 15
Until now, all the graphs that were presented has algebraic co-rank equal to k because its generalizedLaplacian matrix has a k -minor equal to one. Next example shows a graph G with γ Z ( G ) = 5 havingno a 5-minor equal to 1. Example 5.6.
Let G be the graph on figure 2 and f = det( L ( G, X )[ { , , , , } ; { , , , , } ]) = v v v v v v G L ( G, X ) = x − − − − − x − − − − x − − −
10 0 − x − − −
10 0 0 − x − − − − − − x − − − − − − − x Figure 2.
A graph G with seven vertices and its generalized Laplacian matrix. x + x + x x , and f = det( L ( G, X )[ { , , , , } ; { , , , , } ]) = − (1 + x + x + x x ) . Then h f , f i = 1 and therefore γ Z ( G ) = 5 . However, it is not difficult to check that L ( G, X ) has no -minoris equal to one. References [1] C. A. Alfaro and C. E. Valencia, Graphs with three trivial critical ideals, in preparation.[2] W. H. Chan, Y. Hou and W.C. Shiu, Graphs whose critical groups have larger rank, Acta Math. Sinica 27 (2011)1663–1670.[3] H. Corrales and C. Valencia, On the critical ideals of graphs, preprint, arXiv:1205.3105 [math.AC][4] R. Diestel, Graph Theory, Fourth Edition, Springer, 2010.[5] C. Godsil and G. Royle, Algebraic Graph Theory, GTM 207, Springer-Verlag, New York, 2001.[6] N. Jacobson, Basic Algebra I, Second Edition, W. H. Freeman and Company, New York, 1985.[7] B. Jacobson, A. Niedermaier and V. Reiner, Critical groups for complete multipartite graphs and Cartesian productsof complete graphs, J. Graph Theory 44 (2003) 231–250.[8] D. J. Lorenzini, Arithmetical Graphs, Math. Ann. 285 (1989) 481–501.[9] D. J. Lorenzini, A finite group attached to the laplacian of a graph, Discrete Mathematics 91 (1991) 277–282.[10] D. J. Lorenzini, Smith normal form and Laplacians, J. Combin. Theory B 98 (2008), 1271-1300.[11] C. Merino, The chip-firing game, Discrete Mathematics 302 (2005) 188–210.[12] Y. Pan and J. Wang, A note on the third invariant factor of the Laplacian matrix of a graph, preprint,arXiv:0912.3608v1 [math.CO][13] D. G. Wagner, The critical group of a directed graph, preprint, arXiv:math/0010241v1 [math.CO]
Departamento de Matem´aticas, Centro de Investigaci´on y de Estudios Avanzados del IPN, ApartadoPostal 14–740, 07000 Mexico City, D.F.
E-mail address ::