- Email: [email protected]
Graphs with few trivial critical ideals
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 50 (2015) 391–396 www.elsevier.com/locate/endm
Graphs with few trivial critical ideals Carlos A. Alfaro 1,2 Banco de M´exico Calzada Legaria 691, m´ odulo IV Col. Irrigaci´ on 11500 Mexico City, D.F.
Carlos E. Valencia 3 Departamento de Matem´ aticas Centro de Investigaci´ on y de Estudios Avanzados del IPN Apartado Postal 14–740 07000 Mexico City, D.F.
Abstract The critical ideals of a graph are determinantal ideals of the generalized Laplacian matrix associated to a graph. Let Γ≤i denote the set of simple connected graphs with at most i trivial critical ideals. The main goal is to obtain a characterization of the graphs in Γ≤3 with clique number equal to 2, and the graphs in Γ≤3 with clique number equal to 3. This shows that there exists a strong connection between the structural properties of the graph (like the clique number and the stability number) with its critical ideals. Keywords: Critical ideal, generalized Laplacian matrix, forbidden induced subgraph.
http://dx.doi.org/10.1016/j.endm.2015.07.065 1571-0653/© 2015 Elsevier B.V. All rights reserved.
392
1
C.A. Alfaro, C.E. Valencia / Electronic Notes in Discrete Mathematics 50 (2015) 391–396
Introduction
Given a graph without loops G = (V, E), the Laplacian matrix L(G) of G is the matrix with rows and columns indexed by the vertices of G given by degG (u) if u = v, L(G)uv = −muv otherwise, where degG (u) denote the degree of u, and muv denote the number of edges from u to v. By considering the Laplacian matrix of a connected graph G as a linear operator on Zn , the critical group K(G) of G is the torsion part of the cokernel of L(G). The critical group has been studied intensively over the last 30 years on several contexts: the group of components [13], the Picard group [5,6], the Jacobian group [5,6], the sandpile group [1], chip-firing game [6,14], or Laplacian unimodular equivalence [11,15]. It is known (see [12, Theorem 3.9]) that the critical group of a connected graph G with n vertices can be described as follows: K(G) ∼ = Zf1 ⊕ Zf2 ⊕ · · · ⊕ Zfn−1 , where f1 , f2 , ..., fn−1 are positive integers with fi | fj for all i ≤ j. These integers are called invariant factors of the Laplacian matrix of G. If Δi (G) is the greatest common divisor of the i-minors of the Laplacian matrix L(G) of G, then the i-th invariant factor fi is equal to Δi (G)/Δi−1 (G), where Δ0 (G) = 1. Definition 1.1 Given an integer k, let fk (G) be the number of invariant factors of the Laplacian matrix of G equal to k. The computation of the invariant factors of the Laplacian matrix is an important technique used in the understanding of the critical group. For instance, several researchers addressed the question of how often the critical group is cyclic, that is, if f1 (G) denote the number of invariant factors equal to 1, then the question is how often f1 (G) is equal to n − 2 or n − 1? In [13] and [17] D. Lorenzini and D. G. Wagner, based on numerical data, suggest we could expect to find a substantial proportion of graphs having a cyclic critical group. Based on this, D. G. Wagner conjectured [17] that almost 1
Carlos A. Alfaro was supported by CONACyT grant 166059 and Carlos E. Valencia was supported by SNI and CONACyT grant 166059. 2 Email: [email protected] 3 Email: [email protected]
C.A. Alfaro, C.E. Valencia / Electronic Notes in Discrete Mathematics 50 (2015) 391–396
393
every connected simple graph has a cyclic critical group. A recent study [18] concluded that the probability that the critical group of a random graph is cyclic is asymptotically at most ζ(3)−1 ζ(5)−1 ζ(7)−1 ζ(9)−1 ζ(11)−1 · · · ≈ 0.7935212, where ζ is the Riemann zeta function, differing from Wagner’s conjecture. Besides, it is interesting [8] that for any given connected simple graph, there is an homeomorphic graph with cyclic critical group. The reader interested on this topic may consult [9,13,18] for more questions and results. A deeper study in the same direction is to characterize the graphs with a fixed number of invariant factors equal to 1. For instance, it follows from Kirchoff’s matrix-tree theorem that the order of K(G) is equal to the number κ(G) of spanning trees of G. Therefore, the only graphs with n vertices and n − 1 invariant factors equal to one are the trees. The counterpart of the characterization of the graphs with cyclic critical group is the characterization of the graphs with few invariant factors equal to one. In this sense, we define the following family of graphs. Definition 1.2 Let Gi = {G : G is a connected graph with f1 (G) = i}. The characterization of the family Gi of simple connected graphs has been of great interest. It is easy to see [15] that G1 consists only of the complete graphs. Several researchers (see [2,3,7,14,16]) have posed interest on the characterization of G2 and G3 . In this sense, several developments have been done. In [16] were characterized the graphs in G2 whose third invariant factor is equal to n, n − 1, n − 2, or n − 3. Later in [7], the characterizations of the graphs in G2 with a cut vertex, and the graphs in G2 with number of independent cycles equal to n − 2 were given. Recently, a complete characterization of G2 was obtained in [2]. This result was obtained by means of the critical ideals. The main result we present is the characterization of the graphs with clique number less or equal to 3 and at most 3 trivial critical ideals. The converse is more interesting, we find a family of graphs with at most 3 trivial critical ideals which the authors believe is the biggest family of the complete characterization.
2
Critical ideals
Critical ideals were defined in [10] as a generalization of the critical group and have been studied in [2,3,4,10]. Given a graph G = (V, E) and a set of indeterminates XG = {xu : u ∈ V (G)}, the generalized Laplacian matrix
394
C.A. Alfaro, C.E. Valencia / Electronic Notes in Discrete Mathematics 50 (2015) 391–396
L(G, XG ) of G is the matrix with rows and columns indexed by the vertices of G given by if u = v, xu L(G, XG )uv = −muv otherwise, where muv is the number of edges connecting u and v. Definition 2.1 For all 1 ≤ i ≤ |V (G)|, the i-th critical ideal of G is the determinantal ideal given by Ii (G, XG ) = {det(m) : m is an i × i submatrix of L(G, XG )} ⊆ Z[XG ]. We say that a critical ideal is trivial when it is equal to 1. The counterpart, in critical ideals, to the number f1 (G) of invariant factors equal to 1 is the algebraic co-rank defined as follows. Definition 2.2 The algebraic co-rank γ(G) of a graph G is the number of trivial critical ideals of G. Most of the basic properties of the critical ideals were obtained in [10]. For instance, it was proven that if H is an induced subgraph of G, then Ii (H, XH ) ⊆ Ii (G, XG ) for all i ≤ |V (H)|. Thus γ(H) ≤ γ(G). The algebraic co-rank allows to define the following graph family: Definition 2.3 Γ≤i = {G : G is a simple connected graph with γ(G) ≤ i}, We have that Γ≤i is closed under induced subgraphs. The following result serves as a bridge between the critical groups and critical ideals. Theorem 2.4 [10] If deg(G) = (degG (v1 ), ..., degG (vn )) is the degree vector of G, and f1 | · · · | fn−1 are the invariant factors of K(G), then i fj = Δi (G) for all 1 ≤ i ≤ n − 1. Ii (G, deg(G)) = j=1
Thus if the critical ideal Ii (G, XG ) is trivial, then Δi (G) and fi are equal to 1. Equivalently, if Δi (G) and fi are not equal to 1, then the critical ideal Ii (G, XG ) is not trivial. Hence, Gi ⊆ Γ≤i for all i ≥ 0. Therefore, after an analysis of the i-th invariant factor of the Laplacian matrix of the graphs in Γ≤i , the characterization of Gi can be obtained. In [2] these ideas were used to obtain a characterization of Γ≤1 and Γ≤2 . It was found that Γ≤1 consists
C.A. Alfaro, C.E. Valencia / Electronic Notes in Discrete Mathematics 50 (2015) 391–396
395
only of the complete graphs. And the characterization of Γ≤2 turns out to be simpler than the characterization of G2 .
3
Results
In this section, we give the characterization of the graphs in Γ≤3 with clique number at most 3. A graph G is forbidden for Γ≤k when γ(G) ≥ k + 1. The first step is to compute the set of minimal forbidden graphs for Γ≤3 . We found 49 minimal forbidden graphs with at most 8 vertices. The next step is to characterize the graphs having none of these 49 graphs as induced subgraph. The complete proof can be found in [3]. Theorem 3.1 If a graph G ∈ Γ≤3 has clique number at most 3, then G is an induced subgraph of a graph in the family of graphs C (described in Fig. 1). n5 n2
n1 n7
n6
n4
n1
n3
(i) graph G1
(ii) family of graphs A
(iii) family of graphs B
Fig. 1. The family of graphs C. A black vertex represents a clique of cardinality ni , a white vertex represents a stable set of cardinality nv and a gray vertex represents a single vertex.
The converse is stronger. Theorem 3.2 Each induced subgraph of a graph in C belongs to Γ≤3 . However, a complete characterization of Γ≤3 and G3 still remains. It is also interesting to note that Γ≤2 ⊂ A, that is, the family A already contains the complete tripartite graphs and the Tn2 ∨ (Kn1 + Kn3 ) graphs as induced subgraphs.
References [1] C.A. Alfaro and C.E. Valencia, On the sandpile group of the cone of a graph, Linear Algebra and Its Applications 436 (2012) 1154–1176. [2] C.A. Alfaro and C.E. Valencia, Graphs with two trivial critical ideals, Discrete Applied Mathematics 167 (2014) 33–44.
396
C.A. Alfaro, C.E. Valencia / Electronic Notes in Discrete Mathematics 50 (2015) 391–396
[3] C.A. Alfaro and C.E. Valencia, Small clique number graphs with three trivial critical ideals, preprint arXiv:1311.5927 [math.CO]. [4] C.A. Alfaro, H.H. Corrales and C.E. Valencia, Critical ideals of graphs with twin vertices, in preparation. [5] R. Bacher, P. de la Harpe and T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. France 125 (1997) 167-198. [6] N. Biggs, Chip-firing and the critical group of a graph, J. Alg. Combin. 9 (1999) 25-46. [7] W.H. Chan, Y. Hou, W.C. Shiu, Graphs whose critical groups have larger rank, Acta Math. Sinica 27 (2011) 1663–1670. [8] S. Chen and S.K. Ye, Critical groups for homeomorphic classes of graphs, Discrete Math. 309 (2008) 255–258. [9] J. Clancy, N. Kaplan, T. Leake, S. Payne and M.M. Wood, On a Cohen-Lenstra heuristic for jacobians of random graphs, arXiv:1402.5129 [math.CO]. [10] H.H. Corrales and C.E. Valencia, On the critical ideals, Linear Algebra and its Applications 439 (2013) 3870–3892. [11] R. Grone, R. Merris and W. Watkins, Laplacian unimodular equivalence of graphs. In: R. Brualdi, S. Friedland and V. Klee (Eds.) Combinatorial and Graph-Theoretical Problems in Linear Algebra, Springer-Verlag (1993) 175-180. [12] N. Jacobson, Basic Algebra I, Second Edition, W. H. Freeman and Company, New York, 1985. [13] D.J. Lorenzini, Smith normal form and Laplacians, J. Combin. Theory B 98 (2008) 1271-1300. [14] C. Merino, The chip-firing game, Discrete Math. 302 (2005) 188–210. [15] R. Merris, Unimodular Equivalence of Graphs, Linear Algebra Appl. 173 (1992),181-189 [16] Y. Pan and J. Wang, A note on the third invariant factor of the Laplacian matrix of a graph, preprint arXiv:0912.3608 [math.CO]. [17] D.G. Wagner, The critical group arXiv:math/0010241v1 [math.CO]
of
a
directed
graph,
preprint
[18] M.M. Wood, The distribution of the sandpile groups of random graphs, arXiv:1402.5149 [math.PR].
Electronic Notes in Discrete Mathematics 50 (2015) 391–396 www.elsevier.com/locate/endm
Graphs with few trivial critical ideals Carlos A. Alfaro 1,2 Banco de M´exico Calzada Legaria 691, m´ odulo IV Col. Irrigaci´ on 11500 Mexico City, D.F.
Carlos E. Valencia 3 Departamento de Matem´ aticas Centro de Investigaci´ on y de Estudios Avanzados del IPN Apartado Postal 14–740 07000 Mexico City, D.F.
Abstract The critical ideals of a graph are determinantal ideals of the generalized Laplacian matrix associated to a graph. Let Γ≤i denote the set of simple connected graphs with at most i trivial critical ideals. The main goal is to obtain a characterization of the graphs in Γ≤3 with clique number equal to 2, and the graphs in Γ≤3 with clique number equal to 3. This shows that there exists a strong connection between the structural properties of the graph (like the clique number and the stability number) with its critical ideals. Keywords: Critical ideal, generalized Laplacian matrix, forbidden induced subgraph.
http://dx.doi.org/10.1016/j.endm.2015.07.065 1571-0653/© 2015 Elsevier B.V. All rights reserved.
392
1
C.A. Alfaro, C.E. Valencia / Electronic Notes in Discrete Mathematics 50 (2015) 391–396
Introduction
Given a graph without loops G = (V, E), the Laplacian matrix L(G) of G is the matrix with rows and columns indexed by the vertices of G given by degG (u) if u = v, L(G)uv = −muv otherwise, where degG (u) denote the degree of u, and muv denote the number of edges from u to v. By considering the Laplacian matrix of a connected graph G as a linear operator on Zn , the critical group K(G) of G is the torsion part of the cokernel of L(G). The critical group has been studied intensively over the last 30 years on several contexts: the group of components [13], the Picard group [5,6], the Jacobian group [5,6], the sandpile group [1], chip-firing game [6,14], or Laplacian unimodular equivalence [11,15]. It is known (see [12, Theorem 3.9]) that the critical group of a connected graph G with n vertices can be described as follows: K(G) ∼ = Zf1 ⊕ Zf2 ⊕ · · · ⊕ Zfn−1 , where f1 , f2 , ..., fn−1 are positive integers with fi | fj for all i ≤ j. These integers are called invariant factors of the Laplacian matrix of G. If Δi (G) is the greatest common divisor of the i-minors of the Laplacian matrix L(G) of G, then the i-th invariant factor fi is equal to Δi (G)/Δi−1 (G), where Δ0 (G) = 1. Definition 1.1 Given an integer k, let fk (G) be the number of invariant factors of the Laplacian matrix of G equal to k. The computation of the invariant factors of the Laplacian matrix is an important technique used in the understanding of the critical group. For instance, several researchers addressed the question of how often the critical group is cyclic, that is, if f1 (G) denote the number of invariant factors equal to 1, then the question is how often f1 (G) is equal to n − 2 or n − 1? In [13] and [17] D. Lorenzini and D. G. Wagner, based on numerical data, suggest we could expect to find a substantial proportion of graphs having a cyclic critical group. Based on this, D. G. Wagner conjectured [17] that almost 1
Carlos A. Alfaro was supported by CONACyT grant 166059 and Carlos E. Valencia was supported by SNI and CONACyT grant 166059. 2 Email: [email protected] 3 Email: [email protected]
C.A. Alfaro, C.E. Valencia / Electronic Notes in Discrete Mathematics 50 (2015) 391–396
393
every connected simple graph has a cyclic critical group. A recent study [18] concluded that the probability that the critical group of a random graph is cyclic is asymptotically at most ζ(3)−1 ζ(5)−1 ζ(7)−1 ζ(9)−1 ζ(11)−1 · · · ≈ 0.7935212, where ζ is the Riemann zeta function, differing from Wagner’s conjecture. Besides, it is interesting [8] that for any given connected simple graph, there is an homeomorphic graph with cyclic critical group. The reader interested on this topic may consult [9,13,18] for more questions and results. A deeper study in the same direction is to characterize the graphs with a fixed number of invariant factors equal to 1. For instance, it follows from Kirchoff’s matrix-tree theorem that the order of K(G) is equal to the number κ(G) of spanning trees of G. Therefore, the only graphs with n vertices and n − 1 invariant factors equal to one are the trees. The counterpart of the characterization of the graphs with cyclic critical group is the characterization of the graphs with few invariant factors equal to one. In this sense, we define the following family of graphs. Definition 1.2 Let Gi = {G : G is a connected graph with f1 (G) = i}. The characterization of the family Gi of simple connected graphs has been of great interest. It is easy to see [15] that G1 consists only of the complete graphs. Several researchers (see [2,3,7,14,16]) have posed interest on the characterization of G2 and G3 . In this sense, several developments have been done. In [16] were characterized the graphs in G2 whose third invariant factor is equal to n, n − 1, n − 2, or n − 3. Later in [7], the characterizations of the graphs in G2 with a cut vertex, and the graphs in G2 with number of independent cycles equal to n − 2 were given. Recently, a complete characterization of G2 was obtained in [2]. This result was obtained by means of the critical ideals. The main result we present is the characterization of the graphs with clique number less or equal to 3 and at most 3 trivial critical ideals. The converse is more interesting, we find a family of graphs with at most 3 trivial critical ideals which the authors believe is the biggest family of the complete characterization.
2
Critical ideals
Critical ideals were defined in [10] as a generalization of the critical group and have been studied in [2,3,4,10]. Given a graph G = (V, E) and a set of indeterminates XG = {xu : u ∈ V (G)}, the generalized Laplacian matrix
394
C.A. Alfaro, C.E. Valencia / Electronic Notes in Discrete Mathematics 50 (2015) 391–396
L(G, XG ) of G is the matrix with rows and columns indexed by the vertices of G given by if u = v, xu L(G, XG )uv = −muv otherwise, where muv is the number of edges connecting u and v. Definition 2.1 For all 1 ≤ i ≤ |V (G)|, the i-th critical ideal of G is the determinantal ideal given by Ii (G, XG ) = {det(m) : m is an i × i submatrix of L(G, XG )} ⊆ Z[XG ]. We say that a critical ideal is trivial when it is equal to 1. The counterpart, in critical ideals, to the number f1 (G) of invariant factors equal to 1 is the algebraic co-rank defined as follows. Definition 2.2 The algebraic co-rank γ(G) of a graph G is the number of trivial critical ideals of G. Most of the basic properties of the critical ideals were obtained in [10]. For instance, it was proven that if H is an induced subgraph of G, then Ii (H, XH ) ⊆ Ii (G, XG ) for all i ≤ |V (H)|. Thus γ(H) ≤ γ(G). The algebraic co-rank allows to define the following graph family: Definition 2.3 Γ≤i = {G : G is a simple connected graph with γ(G) ≤ i}, We have that Γ≤i is closed under induced subgraphs. The following result serves as a bridge between the critical groups and critical ideals. Theorem 2.4 [10] If deg(G) = (degG (v1 ), ..., degG (vn )) is the degree vector of G, and f1 | · · · | fn−1 are the invariant factors of K(G), then i fj = Δi (G) for all 1 ≤ i ≤ n − 1. Ii (G, deg(G)) = j=1
Thus if the critical ideal Ii (G, XG ) is trivial, then Δi (G) and fi are equal to 1. Equivalently, if Δi (G) and fi are not equal to 1, then the critical ideal Ii (G, XG ) is not trivial. Hence, Gi ⊆ Γ≤i for all i ≥ 0. Therefore, after an analysis of the i-th invariant factor of the Laplacian matrix of the graphs in Γ≤i , the characterization of Gi can be obtained. In [2] these ideas were used to obtain a characterization of Γ≤1 and Γ≤2 . It was found that Γ≤1 consists
C.A. Alfaro, C.E. Valencia / Electronic Notes in Discrete Mathematics 50 (2015) 391–396
395
only of the complete graphs. And the characterization of Γ≤2 turns out to be simpler than the characterization of G2 .
3
Results
In this section, we give the characterization of the graphs in Γ≤3 with clique number at most 3. A graph G is forbidden for Γ≤k when γ(G) ≥ k + 1. The first step is to compute the set of minimal forbidden graphs for Γ≤3 . We found 49 minimal forbidden graphs with at most 8 vertices. The next step is to characterize the graphs having none of these 49 graphs as induced subgraph. The complete proof can be found in [3]. Theorem 3.1 If a graph G ∈ Γ≤3 has clique number at most 3, then G is an induced subgraph of a graph in the family of graphs C (described in Fig. 1). n5 n2
n1 n7
n6
n4
n1
n3
(i) graph G1
(ii) family of graphs A
(iii) family of graphs B
Fig. 1. The family of graphs C. A black vertex represents a clique of cardinality ni , a white vertex represents a stable set of cardinality nv and a gray vertex represents a single vertex.
The converse is stronger. Theorem 3.2 Each induced subgraph of a graph in C belongs to Γ≤3 . However, a complete characterization of Γ≤3 and G3 still remains. It is also interesting to note that Γ≤2 ⊂ A, that is, the family A already contains the complete tripartite graphs and the Tn2 ∨ (Kn1 + Kn3 ) graphs as induced subgraphs.
References [1] C.A. Alfaro and C.E. Valencia, On the sandpile group of the cone of a graph, Linear Algebra and Its Applications 436 (2012) 1154–1176. [2] C.A. Alfaro and C.E. Valencia, Graphs with two trivial critical ideals, Discrete Applied Mathematics 167 (2014) 33–44.
396
C.A. Alfaro, C.E. Valencia / Electronic Notes in Discrete Mathematics 50 (2015) 391–396
[3] C.A. Alfaro and C.E. Valencia, Small clique number graphs with three trivial critical ideals, preprint arXiv:1311.5927 [math.CO]. [4] C.A. Alfaro, H.H. Corrales and C.E. Valencia, Critical ideals of graphs with twin vertices, in preparation. [5] R. Bacher, P. de la Harpe and T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. France 125 (1997) 167-198. [6] N. Biggs, Chip-firing and the critical group of a graph, J. Alg. Combin. 9 (1999) 25-46. [7] W.H. Chan, Y. Hou, W.C. Shiu, Graphs whose critical groups have larger rank, Acta Math. Sinica 27 (2011) 1663–1670. [8] S. Chen and S.K. Ye, Critical groups for homeomorphic classes of graphs, Discrete Math. 309 (2008) 255–258. [9] J. Clancy, N. Kaplan, T. Leake, S. Payne and M.M. Wood, On a Cohen-Lenstra heuristic for jacobians of random graphs, arXiv:1402.5129 [math.CO]. [10] H.H. Corrales and C.E. Valencia, On the critical ideals, Linear Algebra and its Applications 439 (2013) 3870–3892. [11] R. Grone, R. Merris and W. Watkins, Laplacian unimodular equivalence of graphs. In: R. Brualdi, S. Friedland and V. Klee (Eds.) Combinatorial and Graph-Theoretical Problems in Linear Algebra, Springer-Verlag (1993) 175-180. [12] N. Jacobson, Basic Algebra I, Second Edition, W. H. Freeman and Company, New York, 1985. [13] D.J. Lorenzini, Smith normal form and Laplacians, J. Combin. Theory B 98 (2008) 1271-1300. [14] C. Merino, The chip-firing game, Discrete Math. 302 (2005) 188–210. [15] R. Merris, Unimodular Equivalence of Graphs, Linear Algebra Appl. 173 (1992),181-189 [16] Y. Pan and J. Wang, A note on the third invariant factor of the Laplacian matrix of a graph, preprint arXiv:0912.3608 [math.CO]. [17] D.G. Wagner, The critical group arXiv:math/0010241v1 [math.CO]
of
a
directed
graph,
preprint
[18] M.M. Wood, The distribution of the sandpile groups of random graphs, arXiv:1402.5149 [math.PR].