Growth rates of geometric grid classes of permutations

Transcription

1 Growth rates of geometric grid classes of permtations David Bevan Department of Mathematics and Statistics The Open University Milton Keynes, England Sbmitted: Nov 17, 2014; Accepted: Nov 22, 2014; Pblished: Dec 4, 2014 Mathematics Sbject Classifications: 05A05, 05A16, 05C31 Abstract Geometric grid classes of permtations have proven to be key in investigations of classical permtation pattern classes. By considering the representation of gridded permtations as words in a trace monoid, we prove that every geometric grid class has a growth rate which is given by the sqare of the largest root of the matching polynomial of a related graph. As a conseqence, we characterise the set of growth rates of geometric grid classes in terms of the spectral radii of trees, explore the inflence of cycle parity on the growth rate, compare the growth rates of geometric grid classes against those of the corresponding monotone grid classes, and present new reslts concerning the effect of edge sbdivision on the largest root of the matching polynomial. 1 Introdction Following the proof by Marcs & Tardos [21] of the Stanley Wilf conjectre, there has been particlar interest in the growth rates of permtation classes. Kaiser & Klazar [18] determined the possible growth rates less than 2, and then Vatter [27] characterised all the (contably many) permtation classes with growth rates below κ and established that there are ncontably many permtation classes with growth rate κ. Critical to these reslts has been the consideration of grid classes of permtations, and particlarly of geometric grid classes. Geometric grid classes have also been sed to achieve the enmeration of some specific permtation classes [1, 3, 24]. Following initial work on particlar geometric grid classes by Waton [30], Vatter & Waton [29], and Elizalde [8], their general strctral properties have been investigated in articles by Albert, Atkinson, Bovel, Rškc & Vatter [2] and Albert, Rškc & Vatter [4]. We bild on their work the electronic jornal of combinatorics 21(4) (2014), #P4.51 1

2 r 2 c 1 c 3 r 1 c 2 Figre 1: At left: The standard figre for Geom ( ), showing two plots of the permtation with distinct griddings. At right: Its row-colmn graph; positive edges are shown as solid lines, negative edges are dashed. to establish the growth rate of any given geometric grid class. Before we can state or reslt, we need a nmber of definitions. A geometric grid class is specified by a 0/±1 matrix which represents the shape of plots of permtations in the class. To match the Cartesian coordinate system, we index these matrices from the lower left, by colmn and then by row. If M is sch a matrix, then we say that the standard figre of M, denoted Λ M, is the sbset of R 2 consisting of the nion of obliqe open line segments L i,j with slope M i,j for each i, j for which M i,j is nonzero, where L i,j extends from (i 1, j 1) to (i, j) if M i,j = 1, and from (i 1, j) to (i, j 1) if M i,j = 1. The geometric grid class Geom(M) is then defined to be the set of permtations σ 1 σ 2... σ n that can be plotted as a sbset of the standard figre, i.e. for which there exists a seqence of points (x 1, y 1 ),..., (x n, y n ) Λ M sch that x 1 < x 2 <... < x n and the seqence y 1,..., y n is order-isomorphic to σ 1,..., σ n. See Figre 1 for an example. If g n is the nmber of permtations of length n in Geom(M), then the growth rate of the class is given by gr(geom(m)) = lim. We will demonstrate that this limit exists gn 1/n n for geometric grid classes 1 and determine its vale for any given 0/±1 matrix M. Mch of the strctre of a geometric grid class is reflected in a graph that we associate with the nderlying matrix. If M is a 0/±1 matrix of dimensions t, the row-colmn graph G(M) of M is the bipartite graph with vertices r 1,..., r t, c 1,..., c and an edge between r i and c j if and only if M i,j 0. We label each edge r i c j with the vale of M i,j. Edges labelled +1 are called positive; edges labelled 1 are called negative. See Figre 1 for an example. We need one final definition related to geometric grid classes. If M is a 0/±1 matrix of dimensions t, we define the doble refinement M 2 of M to be the 0/±1 matrix of dimensions 2t 2 obtained from M by replacing each 0 with ( ), each 1 with ( ), and each 1 with ( ). See Figre 2 for an example. Note that the standard figre of M 2 is essentially a scaled copy of the standard figre of M, so we have: Observation 1. Geom(M 2 ) = Geom(M) for any 0/±1 matrix M. We will demonstrate a connection between the growth rate of Geom(M) and the matching polynomial of the graph G(M 2 ), the row-colmn graph of the doble refinement of M. 1 It is widely believed that all permtation classes have growth rates. The proof of the Stanley Wilf conjectre by Marcs & Tardos [21] establishes only that each has an pper growth rate (lim sp g 1/n n ). the electronic jornal of combinatorics 21(4) (2014), #P4.51 2

3 c 2 r 1 c 1 r 2 c 3 r 1 c 2 c 3 r 2 c 1 Figre 2: At left: The standard figre of ( , 1 1 1) with a consistent orientation marked. At right: Its row-colmn graph. A k-matching of a graph is a set of k edges, no pair of which have a vertex in common. For example, the negative (dashed) edges in the graph in Figre 2 constitte a 4-matching. If, for each k, m k (G) denotes the nmber of distinct k-matchings of a graph G with n vertices, then the matching polynomial µ G (z) of G is defined to be µ G (z) = k 0( 1) k m k (G)z n 2k. (1) Note that this is a polynomial since m k (G) = 0 for k > n/2. Observe also that the exponents of the variable z enmerate defects in k-matchings: the nmber of vertices which are not endvertices of an edge in sch a matching. If n is even, µ G (z) is an even fnction; if n is odd, µ G (z) is an odd fnction. With the relevant definitions complete, we can now state or theorem: Theorem 2. The growth rate of geometric grid class Geom(M) exists and is eqal to the sqare of the largest root of the matching polynomial µ G(M 2 )(z), where G(M 2 ) is the row-colmn graph of the doble refinement of M. In the next section, we prove this theorem by tilizing the link between geometric grid classes and trace monoids, and their connection to rook nmbers and the matching polynomial. Then, in Section 3 we investigate a nmber of implications of this reslt by tilizing properties of the matching polynomial, especially the fact that the moments of µ G (z) enmerate certain closed walks on G. Firstly, we characterise the growth rates of geometric grid classes in terms of the spectral radii of trees. Then, we explore the inflence of cycle parity on growth rates and relate the growth rates of geometric grid classes to those of monotone grid classes. Finally, we consider the effect of sbdividing edges in the row-colmn graph, proving some new reslts regarding how edge sbdivision affects the largest root of the matching polynomial. 2 Proof of Theorem 2 In order to prove or reslt, we make se of the connection between geometric grid classes and trace monoids. This relationship was first sed by Vatter & Waton [28] to establish certain strctral properties of grid classes, and was developed frther in [2] from where we se a nmber of reslts. To begin with, we need to consider griddings of permtations. the electronic jornal of combinatorics 21(4) (2014), #P4.51 3

4 If M has dimensions t, then an M-gridding of a permtation σ 1... σ n of length n in Geom(M) consists of two seqences c 1,..., c t and r 1,..., r sch that there is some plot (x 1, y 1 ),..., (x n, y n ) of σ for which c i is the nmber of points (x k, y k ) in colmn i (with i 1 < x k < i), and r j is the nmber of points in row j (with j 1 < y k < j). 2 Note that a permtation may have mltiple distinct griddings in a given geometric grid class; see Figre 1 for an example. We call a permtation together with one of its M-griddings an M- gridded permtation. We se Geom # (M) to denote the set of all M-gridded permtations. From an enmerative perspective, it can be mch easier working with M-gridded permtations than directly with the permtations themselves. The following observation means that we can, in fact, restrict or considerations to M-gridded permtations: Lemma 3 (see Vatter [27] Proposition 2.1). If it exists, the growth rate of Geom(M) is eqal to the growth rate of the corresponding class of M-gridded permtations Geom # (M). Proof. Sppose that M has dimensions t. Each permtation in Geom(M) has at least one gridding in Geom # (M), bt no permtation of length n in Geom(M) can have more than ( )( n+t 1 n+ 1 ) t 1 1 griddings in Geom # (M) becase that is the nmber of ways of choosing the nmber of points in each colmn and row. Ths the nmber of M- gridded permtations of length n is no more than a polynomial mltiple of the nmber of n-permtations in Geom(M); the reslt follows immediately from the definition of the growth rate. To determine the growth rate of Geom # (M), we relate M-gridded permtations to words in a trace monoid. To achieve this, one additional concept is reqired, that of a consistent orientation of a standard figre. If Λ M = {L i,j : M i,j 0} is the standard figre of a 0/±1 matrix M, then a consistent orientation of Λ M consists of an orientation of each obliqe line L i,j sch that in each colmn either all the lines are oriented leftwards or all are oriented rightwards, and in each row either all the lines are oriented downwards or all are oriented pwards. 3 See Figres 2 and 3 for examples. It is not always possible to consistently orient a standard figre. The ability to do so depends on the cycles in the row-colmn graph. We say that the parity of a cycle in G(M) is the prodct of the labels of its edges, a positive cycle is one which has parity +1, and a negative cycle is one with parity 1. The following reslt relates cycle parity to consistent orientations: Lemma 4 (see Vatter & Waton [28] Proposition 2.1). The standard figre Λ M has a consistent orientation if and only if its row-colmn graph G(M) contains no negative cycles. For example, G ( ) contains a negative cycle so its standard figre has no consistent orientation (see Figre 1), whereas G ( ) has no negative cycles so its standard figre has a consistent orientation (see Figre 3). 2 This definition of an M-gridding is eqivalent to the traditional one given in terms of the positions of the cell dividers relative to the points (k, σ k ). 3 For ease of exposition, we se the concept of a consistent orientation rather than the approach sed previosly involving partial mltiplication matrices; reslts from [2] follow mtatis mtandis. the electronic jornal of combinatorics 21(4) (2014), #P4.51 4

5 On the other hand, we can always consistently orient the standard figre of the doble refinement of a matrix by orienting each obliqe line towards the centre of its 2 2 block (as in Figre 2). So we have the following: Lemma 5 (see [2] Proposition 4.1). If M is any 0/±1 matrix, then Λ M 2 has a consistent orientation. Ths, by Lemma 4, the row-colmn graph of the doble refinement of a matrix never contains a negative cycle. Figre 2 shows a consistent orientation of the standard figre of the doble refinement of a matrix whose standard figre (shown in Figre 1) doesn t itself have a consistent orientation Figre 3: The plots of permtation in Geom ( ) associated with the words a 32 a 32 a 11 a 12 a 21 a 31 a 32 and a 11 a 32 a 21 a 32 a 31 a 12 a 32. Both plots correspond to the same gridding. We are now in a position to describe the association between words and M-gridded permtations. If M is a 0/±1 matrix, then we let Σ M = {a ij : M i,j 0} be an alphabet of symbols, one for each nonzero cell in M. If we have a consistent orientation for Λ M, then we can associate to each finite word w 1... w n over Σ M a specific plot of a permtation in Geom(M) as follows: If w k = a ij, inclde the point at distance k 2/(n + 1) along line segment L i,j according to its orientation. See Figre 3 for two examples. Clearly, this indces a mapping from the set of all finite words over Σ M to Geom # (M). In fact, it can readily be shown that this map is srjective, every M-gridded permtation corresponding to some word over Σ M ([2] Proposition 5.3). As can be seen in Figre 3, distinct words may be mapped to the same gridded permtation. This occrs becase the order in which two consective points are inclded is immaterial if they occr in cells that are neither in the same colmn nor in the same row. From the perspective of the words, adjacent symbols corresponding to sch cells may be interchanged withot changing the gridded permtation. This corresponds to a strctre known as a trace monoid. If we have a consistent orientation for standard figre Λ M, then we define the trace monoid of M, which we denote by M(M), to be the set of eqivalence classes of words over Σ M in which a ij and a kl commte (i.e. a ij a kl = a kl a ij ) whenever i k and j l. It is then relatively straightforward to show eqivalence between gridded permtations and elements of the trace monoid: the electronic jornal of combinatorics 21(4) (2014), #P4.51 5

6 Lemma 6 (see [2] Proposition 7.1). If the standard figre Λ M has a consistent orientation, then gridded n-permtations in Geom # (M) are in bijection with eqivalence classes of words of length n in M(M). ence, by combining Lemmas 3, 5 and 6 with Observation 1, we know that the growth rate of Geom(M) is eqal to the growth rate of M(M 2 ) if it exists. All that remains is to determine the growth rate of the trace monoid of a matrix. Trace monoids were first stdied by Cartier & Foata [6]. Using extended Möbis inversion, they determined the general form of the generating fnction, as follows: Lemma 7 ([6]; see also Flajolet & Sedgewick [10] Note V.10). The ordinary generating fnction for M(M) is given by f M (z) = 1 k 0 ( 1)k r k (M)z k where r k (M) is the nmber of k-sbsets of Σ M whose elements commte pairwise. Since symbols in M(M) commte if and only if they correspond to cells that are neither in the same colmn nor in the same row, it is easy to see that r k (M) is the nmber of distinct ways of placing k chess rooks on the nonzero entries of M in sch a way that no two rooks attack each other by being in the same colmn or row. The nmbers r k (M) are known as the rook nmbers for M (see Riordan [25]). Moreover, a matching in the row-colmn graph G(M) also corresponds to a set of cells no pair of which share a colmn or row. So the rook nmbers for M are the same as the nmbers of matchings in G(M): Observation 8. For all k 0, r k (M) = m k (G(M)). Now, by elementary analytic combinatorics, we know that the growth rate of M(M) is given by the reciprocal of the root of the denominator of f M (z) that has least magnitde (see [10] Theorem IV.7). The fact that this polynomial has a niqe root of smallest modls was proved by Goldwrm & Santini in [14]. It is real and positive by Pringsheim s Theorem. Bt the reciprocal of the smallest root of a polynomial is the same as the largest root of the reciprocal polynomial (obtained by reversing the order of the coefficients). ence, if M has dimensions t and n = t +, then the growth rate of M(M) is the largest (positive real) root of the polynomial g M (z) = 1 z n/2 f M ( 1 z ) = n/2 ( 1) k r k (M)z n/2 k. (2) k=0 ere, g M (z) is the reciprocal polynomial of (f M (z)) 1 mltiplied by some nonnegative power of z, since r k (M) = 0 for all k > n/2. Note also that n is the nmber of vertices in G(M). the electronic jornal of combinatorics 21(4) (2014), #P4.51 6

7 If we now compare the definition of g M (z) in (2) with that of the matching polynomial µ G (z) in (1) and se Observation 8, then we see that: g M (z 2 ) = { µg(m) (z), if n is even; z 1 µ G(M) (z), if n is odd. ence, the largest root of g M (z) is the sqare of the largest root of µ G(M) (z). We now have all we need to prove Theorem 2: The growth rate of Geom(M) is eqal to the growth rate of M(M 2 ) which eqals the sqare of the largest root of µ G(M 2 )(z). In the above argment, we only employ the doble refinement M 2 to ensre that a consistent orientation is possible. By Lemma 4, we know that if G(M) is free of negative cycles then Λ M can be consistently oriented. Ths, we have the following special case of Theorem 2: Corollary 9. If G(M) contains no negative cycles, then the growth rate of Geom(M) is eqal to the sqare of the largest root of µ G(M) (z). 3 Conseqences In this final section, we investigate some of the implications of Theorem 2. By considering properties of the matching polynomial, we characterise the growth rates of geometric grid classes in terms of the spectral radii of trees, prove a monotonicity reslt, and explore the inflence of cycle parity on growth rates. We then compare the growth rates of geometric grid classes with those of monotone grid classes. Finally, we consider the effect of sbdividing edges in the row-colmn graph. Let s begin by introdcing some notation. We se G+ to denote the graph composed of two disjoint sbgraphs G and. The graph reslting from deleting the vertex v (and all edges incident to v) from a graph G is denoted G v. Generalising this, if is a sbgraph of G, then G is the graph obtained by deleting the vertices of from G. In contrast, we se G \e to denote the graph reslting from deleting the edge e from G. The nmber of connected components of G is represented by comp(g). The characteristic polynomial of a graph G is denoted Φ G (z). We se ρ(g) to denote the spectral radis of G, the largest root of Φ G (z). Finally, we se λ(g) for the largest root of the matching polynomial µ G (z). The matching polynomial was independently discovered a nmber of times, beginning with eilmann & Lieb [16] when investigating monomer-dimer systems in statistical physics. It was first stdied from a combinatorial perspective by Farrell [9] and Gtman [15]. The theory was then frther developed by Godsil & Gtman [13] and Godsil [11]. An introdction can be fond in the books by Godsil [12] and Lovász & Plmmer [20]. The facts concerning the matching polynomial that we se are covered by three lemmas. As a conseqence of the first, we only need to consider connected graphs: the electronic jornal of combinatorics 21(4) (2014), #P4.51 7

8 Lemma 10 (Farrell [9], Gtman [15]). The matching polynomial of a graph is the prodct of the matching polynomials of its connected components. Ths, in particlar: Corollary 11. For any graphs G and, we have λ(g + ) = max(λ(g), λ()). The second lemma relates the matching polynomial to the characteristic polynomial. Lemma 12 (Godsil & Gtman [13]). If C G consists of all nontrivial sbgraphs of G which are nions of vertex-disjoint cycles (i.e., all sbgraphs of G which are reglar of degree 2), then µ G (z) = Φ G (z) + C C G 2 comp(c) Φ G C (z), where Φ G C (z) = 1 if C = G. As an immediate conseqence, we have the following: Corollary 13 (Sachs [26], Mowshowitz [23], Lovász & Pelikán [19]). The matching polynomial of a graph is identical to its characteristic polynomial if and only if the graph is acyclic. In particlar, their largest roots are identical: Corollary 14. If G is a forest, then λ(g) = ρ(g). Ths, sing Corollaries 9 and 11, we have the following alternative characterisation for the growth rates of acyclic geometric grid classes: Corollary 15. If G(M) is a forest, then gr(geom(m)) = ρ(g(m)) 2. The last, and most important, of the three lemmas allows s to determine the largest root of the matching polynomial of a graph from the spectral radis of a related tree. It is a conseqence of the fact, determined by Godsil in [11], that the moments (sms of the powers of the roots) of µ G (z) enmerate certain closed walks on G, which he calls tree-like. This is analogos to the fact that the moments of Φ G (z) cont all closed walks on G. On a tree, all closed walks are tree-like. Lemma 16 (Godsil [11]; see also [12] and [20]). Let G be a graph and let and v be adjacent vertices in a cycle of G. Let be the component of G that contains v. Now let K be the graph constrcted by taking a copy of G \v and a copy of and joining the occrrence of in the copy of G \v to the occrrence of v in the copy of (see Figre 4). Then λ(g) = λ(k). The process that is described in Lemma 16 we will call expanding G at along v. Each sch expansion of a graph G prodces a graph with fewer cycles than G. Repeated application of this process will ths eventally reslt in a forest F sch that λ(f ) = λ(g). We shall say that F reslts from flly expanding G. ence, by Corollaries 11 and 14, the the electronic jornal of combinatorics 21(4) (2014), #P4.51 8

9 G: v K: v Figre 4: Expanding G at along v; is the component of G that contains v largest root of the matching polynomial of a graph eqals the spectral radis of some tree: for any graph G, there is a tree T sch that λ(g) = ρ(t ). It is readily observed that every tree is the row-colmn graph of some geometric grid class. Ths we have the following characterisation of geometric grid class growth rates. Corollary 17. The set of growth rates of geometric grid classes consists of the sqares of the spectral radii of trees. The spectral radii of connected graphs satisfy the following strict monotonicity condition: Lemma 18 ([7] Proposition ). If G is connected and is a proper sbgraph of G, then we have ρ() < ρ(g). Lemma 16 enables s to prove the analogos fact for the largest roots of matching polynomials, from which we can dedce a monotonicity reslt for geometric grid classes: Corollary 19. If G is connected and is a proper sbgraph of G, then λ() < λ(g). Proof. Sppose we flly expand (at vertices 1,..., k, say), then the reslt is a forest F sch that λ() = ρ(f ). Now sppose that we repeatedly expand G analogosly at 1,..., k, and then contine to flly expand the reslting graph. The otcome is a tree T (since G is connected) sch that F is a proper sbgraph of T and λ(g) = ρ(t ). The reslt follows from Lemma 18. Adding a non-zero cell to a 0/±1 matrix M adds an edge to G(M). Ths, geometric grid classes satisfy the following monotonicity condition: Corollary 20. If G(M) is connected and M reslts from adding a non-zero cell to M in sch a way that G(M ) is also connected, then gr(geom(m )) > gr(geom(m)). 3.1 Cycle parity The growth rate of a geometric grid class depends on the parity of its cycles. Consider the case of G(M) being a cycle graph C n. If G(M) is a negative cycle, then G(M 2 ) = C 2n. the electronic jornal of combinatorics 21(4) (2014), #P4.51 9

10 Now, by Lemma 16, we have λ(c n ) = ρ(p 2n 1 ), where P n is the path graph on n vertices. The spectral radis of a graph on n vertices is 2 cos π n+1. So, gr(geom(m)) = { 4 cos 2 π 2n if G(M) is a positive cycle; (3) 4 cos 2 π, 4n if G(M) is a negative cycle. Ths the geometric grid class whose row-colmn graph is a negative cycle has a greater growth rate than the class whose row-colmn graph is a positive cycle. As another example, gr ( Geom ( )) = , (4) whereas gr ( Geom ( )) = 4. (5) The former, containing a negative cycle, has a greater growth rate than the latter, whose cycle is positive. This is typical; we will prove the following reslt: Corollary 21. If G(M) is connected and contains no negative cycles, and M 1 reslts from changing the sign of a single entry of M that is in a cycle (ths making one or more cycles in G(M 1 ) negative), then gr(geom(m 1 )) > gr(geom(m)). In order to do this, we need to consider the strctre of G(M 2 ). The graph G(M 2 ) can be constrcted from G(M) as follows: If G(M) has vertex set {v 1,..., v n }, then we let G(M 2 ) have vertices v 1,..., v n and v 1,..., v n. If v i v j is a positive edge in G(M), then in G(M 2 ) we add an edge between v i and v j and also between v i and v j. On the other hand, if v i v j is a negative edge in G(M), then in G(M 2 ) we join v i to v j and v i to v j. The correctness of this constrction follows directly from the definitions of doble refinement and of the row-colmn graph of a matrix. For an illstration, compare the graph in Figre 2 against that in Figre 1. Note that if v 1,..., v k is a positive k-cycle in G(M), then G(M 2 ) contains two vertexdisjoint positive k-cycles, the nion of whose vertices is {v 1,..., v k, v 1,..., v k }. In contrast, if v 1,..., v l is a negative l-cycle in G(M), then G(M 2 ) contains a (positive) 2l-cycle on {v 1,..., v l, v 1,..., v l } in which v i is opposite v i (i.e. v i is at distance l from v i arond the cycle) for each i, 1 i l. We make the following additional observations: Observation 22. If G(M) has no odd cycles, then G(M 2 ) = G(M) + G(M). Observation 23. If G(M) is connected and has an odd cycle, then G(M 2 ) is connected. We now have all we reqire to prove or cycle parity reslt. Proof of Corollary 21. Let G = G(M) and G 1 = G(M 1 2 ), and let v be the edge in G corresponding to the entry in M that is negated to create M 1. Since G contains no negative cycles, by Observation 22, G(M 2 ) = G + G. Ths, since G is connected, it has the form at the left of Figre 5, in which is the component of G containing v. Moreover, we have gr(geom(m)) = λ(g). (This also follows from Corollary 9.) Now, if the electronic jornal of combinatorics 21(4) (2014), #P

11 v v v v v G(M 2 ) = G + G K G 1 = G(M 1 2 ) Figre 5: Graphs sed in the proof of Corollary 21 we expand G at along v, by Lemma 16, λ(g) = λ(k), where K is the graph in the centre of Figre 5. On the other hand, G 1 is obtained from G(M 2 ) by removing the edges v and v, and adding v and v, as shown at the right of Figre 5. It is readily observed that K is a proper sbgraph of G 1 (see the shaded box in Figre 5), and hence, by Corollary 19, λ(k) < λ(g 1 ). Since gr(geom(m 1 )) = λ(g 1 ), the reslt follows. Ths, making the first negative cycle increases the growth rate. We sspect, in fact, that the following stronger statement is also tre: Conjectre 24. If G(M) is connected and M 1 reslts from negating a single entry of M that is in one or more positive cycles bt in no negative cycle, then we have gr(geom(m 1 )) > gr(geom(m)). To prove this more general reslt seems to reqire some new ideas. If G(M) already contains a negative cycle, then G(M 2 ) is connected, and, when this is the case, there appears to be no obvios way to generate a sbgraph of G(M 1 2 ) by expanding G(M 2 ). 3.2 Monotone grid classes In a recent paper [5], we established the growth rates of monotone grid classes. If M is a 0/±1 matrix, then the monotone grid class Grid(M) consists of those permtations that can be plotted as a sbset of some figre consisting of the nion of any monotonic crves Γ i,j with the same endpoints as the L i,j in Λ M. This permits greater flexibility in the positioning of points in the cells, so Geom(M) is a sbset of Grid(M) and we have gr(geom(m)) gr(grid(m)). In fact, the geometric grid class Geom(M) and the monotone grid class Grid(M) are identical if and only if G(M) is acyclic (Theorem 3.2 in [2]). ence, if G(M) is a forest, gr(geom(m)) = gr(grid(m)). We determined in [5] that the growth rate of monotone grid class Grid(M) is eqal to the sqare of the spectral radis of G(M). For acyclic G(M), this is consistent with the growth rate of the geometric grid class as given by Corollary 15. Typically, the growth rate of a monotone grid class will be greater than that of the corresponding geometric grid class. For example, if G(M) is a cycle then gr(grid(m)) = 4, the electronic jornal of combinatorics 21(4) (2014), #P

12 whereas from (3) we have gr(geom(m)) < 4. And we have gr ( Grid ( )) ( ( = gr Grid )) = 1 (5 + 17) , 2 which shold be compared with (4) and (5). The fact that the growth rate of the monotone grid class is strictly greater is a conseqence of the fact that, if G is connected and not acyclic, then λ(g) and ρ(g) are distinct: Lemma 25 (Godsil & Gtman [13]). If G is connected and contains a cycle, then we have λ(g) < ρ(g). Proof. By Lemma 18, if C is a nonempty sbgraph of G, then ρ(g C) < ρ(g). So we have Φ G C (z) > 0 for all z ρ(g). Moreover, Φ G (z) 0 for z ρ(g). So, since G contains a cycle, from Lemma 12 we can dedce that µ G (z) > 0 if z ρ(g), and ths λ(g) < ρ(g). Note that, analogosly to Observation 1, Grid(M 2 ) = Grid(M). ence it mst be the case that ρ(g(m 2 )) = ρ(g(m)), the growth rate of a monotone grid class ths being independent of the parity of its cycles. As a conseqence, from Lemma 25 we can dedce that in the non-acyclic case there is a strict ineqality between the growth rate of a geometric grid class and the growth rate of the corresponding monotone grid class: Corollary 26. If G(M) is connected, then gr(geom(m)) < gr(grid(m)) if and only if G(M) contains a cycle. 3.3 Sbdivision of edges One particlarly srprising reslt in [5] concerning the growth rates of monotone grid classes is the fact that classes whose row-colmn graphs have longer internal paths or cycles exhibit lower growth rates. An edge e of a graph G is said to lie on an endpath of G if G \e is disconnected and one of its components is a (possibly trivial) path. An edge that does not lie on an endpath is said to be internal. The following reslt of offman & Smith states that the sbdivision of an edge increases or decreases the spectral radis of the graph depending on whether the edge lies on an endpath or is internal: Lemma 27 (offman & Smith [17]). Let G be a connected graph and G be obtained from G by sbdividing an edge e. If e lies on an endpath, then ρ(g ) > ρ(g). Otherwise (if e is an internal edge), ρ(g ) ρ(g), with eqality if and only if G is a cycle or has the following form (which we call an graph : Ths for monotone grid classes, if G(M) is connected, and G(M ) is obtained from G(M) by the sbdivision of one or more internal edges, then gr(grid(m )) gr(grid(m)). the electronic jornal of combinatorics 21(4) (2014), #P

13 As we will see, the sitation is not as simple for geometric grid classes. The effect of edge sbdivision on the largest root of the matching polynomial does not seem to have been addressed previosly. In fact, the sbdivision of an edge that is in a cycle may case λ(g) to increase or decrease, or may leave it nchanged. See Figres 8 10 for illstrations of the three cases. We investigate this frther below. owever, if the edge being sbdivided is not on a cycle in G, then the behavior of λ(g) mirrors that of ρ(g), as we now demonstrate: Lemma 28. Let G be a connected graph and G be obtained from G by sbdividing an edge e. If e lies on an endpath, then λ(g ) > λ(g). owever, if e is an internal edge and not on a cycle, then λ(g ) λ(g), with eqality if and only if G is an graph. G: e v 1 2 K: v 1 1 e 1 e Figre 6: Graphs sed in the proof of Lemma 28 Proof. If e lies on an endpath, then G is a proper sbgraph of G and so the reslt follows from Corollary 19. On the other hand, if e is internal and G is acyclic, the conclsion is a conseqence of Corollary 14 and Lemma 27. Ths, we need only consider the sitation in which e is internal and G contains a cycle. We proceed by indction on the nmber of cycles in G, acyclic graphs constitting the base case. Let v be an edge in a cycle of G sch that is not an endvertex of e. Now, let K be the reslt of expanding G at along v, and let K, analogosly, be the reslt of expanding G at along v. We consider the effect of the expansion of G pon e and the effect of the expansion of G pon the two edges reslting from the sbdivision of e. If e is in the component of G containing v, then e is dplicated in K, both copies of e remaining internal (see Figre 6). Moreover, K reslts from sbdividing both copies of e in K. Conversely, if e is in a component of G not containing v, then e is not dplicated in K (and remains internal). In this case, K reslts from sbdividing e in K. In either case, K is the reslt of sbdividing internal edges of K (a graph with fewer cycles than G), and so the reslt follows from the indction hypothesis. Now, the sbdivision of an edge of a row-colmn graph that is not on a cycle has no effect on the parity of the cycles. ence, we have the following conclsion for the growth rates of geometric grid classes: Corollary 29. If G(M) is connected, and G(M ) is obtained from G(M) by the sbdivision of one or more internal edges not on a cycle, then gr(geom(m )) gr(geom(m)), with eqality if and only if G(M) is an graph. the electronic jornal of combinatorics 21(4) (2014), #P

14 G: 1 x 1 e K: 1 2 x2 1 2 x2 2 x x2 2 1 x 1 Figre 7: Graphs sed in Lemma 30 Let s now investigate the effect of sbdividing an edge e that lies on a cycle. We restrict or attention to graphs in which there is a vertex sch that the two endvertices of e are in distinct components of (G \e). See the graph at the left of Figre 7 for an illstration. We leave the consideration of mltiply-connected graphs that fail to satisfy this condition for ftre stdy. Lemma 30. Let G be a connected graph and e = x 1 x 2 an edge on a cycle C of G. Let be a vertex on C, and let 1 and 2 be the distinct components of (G \e) that contain x 1 and x 2 respectively. Finally, let G be the graph obtained from G by sbdividing e. (a) If, for i {1, 2}, i is a (possibly trivial) path of which x i is an endvertex, then λ(g ) > λ(g). (b) If, for i {1, 2}, i is not a path or is a path of which x i is not an endvertex, then λ(g ) < λ(g). Proof. Let K be the reslt of repeatedly expanding G at along every edge joining to 1. K has the form shown at the right of Figre 7. Also let K be the reslt of repeatedly expanding G (G with edge e sbdivided) in an analogos way at. Clearly K is the same as the graph that reslts from sbdividing the copies of e in K. Now, for part (a), since 1 is a path with an end at x 1, and also 2 is a path with an end at x 2, we see that K is the reslt of sbdividing edges of K that are on endpaths. ence, by the first part of Lemma 28, we have λ(g ) > λ(g) as reqired. For part (b), since 1 is not a path with an end at x 1, and nor is 2 a path with an end at x 2, we see that K is the reslt of sbdividing internal edges of K. Since K is not an graph, by Lemma 28, we have λ(g ) < λ(g) as reqired. If the conditions for parts (a) and (b) of this lemma both fail to be satisfied (i.e. 1 is a sitable path and 2 isn t, or vice versa), then the proof fails. This is de to the fact that expansion leads to at least one copy of e in K being internal and to another copy of e in K being on an endpath. Sbdivision of the former decreases λ(g) whereas sbdivision of the latter cases it to increase. Sometimes, as in Figre 9, these effects balance exactly; the electronic jornal of combinatorics 21(4) (2014), #P

15 Figre 8: Standard figres and row-colmn graphs of geometric grid classes whose growth rates increase from left to right Figre 9: Standard figres and row-colmn graphs of geometric grid classes whose growth rates are all the same (eqal to 5) Figre 10: Standard figres and row-colmn graphs of geometric grid classes whose growth rates decrease from left to right on other occasions one or the other dominates. We leave a detailed analysis of sch cases for later stdy. To conclde, we state the conseqent reslt for the growth rates of geometric grid classes. To simplify its statement and avoid having to concern orselves directly with cycle parities, we define G (M) to be G(M) when G(M) has no odd cycles and G (M) to be G(M 2 ) otherwise. Corollary 31. Sppose G (M) is connected. (a) If G (M ) is obtained from G (M) by sbdividing one or more edges that satisfy the conditions of part (a) of Lemma 30, then gr(geom(m )) > gr(geom(m)). (b) If G (M ) is obtained from G (M) by sbdividing one or more edges that satisfy the conditions of part (b) of Lemma 30, then gr(geom(m )) < gr(geom(m)). Figre 8 provides an illstration of part (a) and Figre 10 an illstration of part (b). Acknowledgements The athor wold like to thank Penny Bevan and Robert Brignall for reading earlier drafts of this paper. Their feedback led to significant improvements in its presentation. Thanks are also de to an anonymos referee whose comments reslted in a mch expanded final section, and to Brendan McKay, whose helpfl response to a qestion on MathOverflow [22] alerted the athor to the significance of Godsil s Lemma 16. S.D.G. the electronic jornal of combinatorics 21(4) (2014), #P

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