On graphs of minimum skew rank 4

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1 On graphs of minimum skew rank 4 Sudipta Mallik a and Bryan L. Shader b a Department of Mathematics & Statistics, Northern Arizona University, 85 S. Osborne Dr. PO Box: 5717, Flagstaff, AZ 8611, USA b Department of Mathematics, University of Wyoming, 1 E University Avenue, Laramie,WY 8271, USA Abstract The zero-nonzero pattern of a skew-symmetric matrix defines a graph. The minimum rank of all real skew-symmetric matrices with a given graph is studied. The probabilistic method is used to show that for sufficiently large n, there is a regular graph G of order n whose complement has girth at least 5 and for which every skew-symmetric matrix with graph G has rank at least 6. This settles an open problem. It is also proved that almost all graphs on n vertices do not allow a skew-symmetric matrix whose rank is less than n 25. AMS classification: 5C5; 15A3. Keywords: Skew-symmetric matrix, skew rank, regular graph, average minimum skew rank, probabilistic method. 1 Introduction We begin by establishing some notation that follows [4]. Throughout all matrices are real. An n n skew-symmetric matrix A determines a graph G(A that has vertex set {1, 2,..., n} and edge set {{i, j} : a i,j, 1 i < j n}. For a graph G, S (G denotes the set of all skew-symmetric Corresponding author. sudipta.mallik@nau.edu 1

2 matrices whose graph is G. The minimum skew rank of the graph G is the minimum rank of a matrix in S (G and is denoted by mr (G. The edge joining vertices u and v is simply denoted by uv. For a vertex v of G, N(v denotes the neighbourhood of v, i.e., the set of all vertices u such that uv is an edge. Two non-adjacent vertices u and v, u v, of a graph G are called duplicate vertices if N(u = N(v. The complete graph on n vertices is denoted by K n. The graph obtained from K 4 by deleting an edge is called a diamond. The complement graph of a diamond is a co-diamond. The union of k vertex disjoint copies of the graph H is denoted by kh. The graph 2K 3 + e is obtained from 2K 3 by inserting one additional edge (see Figure 1. Figure 1: co-diamond and 2K 3 + e The problem of characterizing graphs with minimum skew rank 4 is introduced in [3]. Deleting isolated vertices or duplicate vertices from a graph does not change the minimum skew rank of the graph. The following necessary conditions for a graph G of order n 6 with neither duplicate vertices nor isolated vertices to have minimum skew rank at most 4 are given in [4]. If H is not an induced subgraph of a graph G, then G is H-free. Theorem 1.1. [4, Thm 2.6] Let G be a graph of order n 6 with neither duplicate vertices nor isolated vertices. If mr (G 4, then G is co-diamondfree and (2K 3 + e-free. It has been shown in [4] that these conditions are sufficient for several families of graphs, e.g., the complement of trees. In Section 2, we investigate the sufficiency of these conditions and show sufficiency for graphs of order n 8, and all but one graph of order 9. In Section 3, we prove that for n sufficiently large these conditions are far from sufficient. In Section 4, we give a lower bound on the average minimum skew rank of graphs of order n. A matching of a graph G is a set of edges of G where no two edges share a common vertex. A perfect matching of a graph G of order 2k is a matching consisting of k edges. Graphs of order n whose minimum skew rank is n are characterized as follows. 2

3 Theorem 1.2. [3, Theorem 2.6] Let G be a graph of order n. Then mr (G = n if and only if G has a unique perfect matching. We note that Theorem 1.2 implies that a graph of odd order n has minimum skew rank at most n 1. A cycle on n vertices is denoted by C n. The girth of a graph G is the length of a shortest cycle in G if G contains a cycle, and is infinity otherwise. For example, a graph G with girth at least 5 is C 3 -free and C 4 -free. 2 Necessary conditions for graphs to have minimum skew rank at most 4 It is natural to ask if the converse of Theorem 1.1 is true. In this section we study this question for graphs of small order. It is often more convenient to study the equivalent problem: If mr (G 6 and G has neither duplicate vertices nor isolated vertices, then does G have a diamond or 2K 3 + e as an induced subgraph? First we show that the converse of Theorem 1.1 is true for disconnected graphs by using the following lemma. Lemma 2.1. [3, Obs 1.6, Thm 2.1] (a mr (K n = 2 for all n 2. (b If the connected components of G are G 1,..., G t, then mr (G = t mr (G i. Theorem 2.2. Let G be a disconnected graph of order n 6 with neither duplicate vertices nor isolated vertices. If G is co-diamond-free and (2K 3 +e- free, then G is a disjoint union of two complete graphs and mr (G = 4. Proof. Suppose that G is co-diamond-free, (2K 3 + e-free, and disconnected graph of order n 6 with neither duplicate vertices nor isolated vertices. Since G has no isolated vertices, each connected component of G has at 3

4 least one edge. If G has at least three connected components, then G has an induced co-diamond. Thus G has exactly two connected components, each containing at least one edge. If any connected component contains two non-adjacent vertices, then these two vertices along with an edge of the other connected component will form a co-diamond in G. Thus each of two connected components of G is a complete graph. By Lemma 2.1, we get mr (G = 4. Now we verify the converse of Theorem 1.1 for connected graphs of small order. Let us start with connected graphs on 6 vertices. Suppose that connected graph G has 6 vertices with mr (G = 6 and with no duplicate vertices. By Theorem 1.2, G has a unique perfect matching. If G has a pendant edge uv with pendant vertex v, then the graph obtained from G by deleting u and v is a graph on 4 vertices with a perfect matching. It can be verified that G is a graph of the form of one of the graphs of Figure 4 other than 2K 3 + e. Now assume that G has minimum degree at least 2. Then G has at least three edges different from three edges of the perfect matching. Since G has a unique perfect matching and minimum degree at least 2, after appending two more edges to two vertices of an edge, say 12 of the perfect matching, G would have a subgraph of the form of one of the graphs of Figure Figure 2: Possible subgraphs of G on 6 vertices with 5 edges Since G has a unique perfect matching and the degree of vertex 4 is at least 2, after appending an edge to vertex 4, G would have a subgraph of the form of one of the graphs of Figure Figure 3: Possible subgraphs of G on 6 vertices with 6 edges 4

5 Since G has a unique perfect matching and the degree of vertex 5 is at least 2, after appending an edge to vertex 5, G is 2K 3 + e. Now it can be checked that except 2K 3 + e each of the graphs in Figure 4 has an induced co-diamond. Thus the converse of Theorem 1.1 holds for graphs on 6 vertices. any but at least 1 any but at least 1 Figure 4: Connected graphs on 6 vertices with a unique perfect matching Next we study the converse of Theorem 1.1 for connected graphs on more than 6 vertices using a technique from [4]. The matrix N is a basis matrix of the left null space of a matrix A if the rows of N form a basis of the left null space of A. A collection {N i : i I} of vectors is a minimal dependent set of vectors if it is a linearly dependent set of vectors and for each j I, {N i : i j, i I} is a linearly independent set of vectors. We denote the support of a row or column vector v R n by Z(v and define by Z(v = {i : 1 i n, v i }. Theorem 2.3. [4, Thm 4.8] Let A be a real skew-symmetric matrix of order n 2. Let N be a basis matrix of the left null space of A. If Z is a union of indices of minimal dependent columns of N, then there exists a vector w with support Z such that ([ ] A w rank w T = rank(a. Corollary 2.4. [4, Cor 4.9] Let G be a graph on n vertices. Let A S (G with rank(a = mr (G and N be a basis matrix of the left null space of A. Let G + v be a graph obtained by adjoining a new vertex v and some edges from v to G. If N(v is a union of indices of minimal dependent columns of N, then mr (G + v = mr (G. Enabled by Sage [8] and Brendan McKay s lists [6] of non-isomorphic connected graphs of small order, we are able to use Corollary 2.4 to show that for n 8, the necessary conditions in Theorem 1.1 are also sufficient. More precisely, for n 5, all graphs of order n have minimum skew rank at most 4. As argued earlier in this section, each graph of order 6 that has no 5

6 duplicate or isolated vertices, and no induced co-diamond or 2K 3 + e, has minimum skew rank at most 4. Next consider graphs of order 7. Starting with McKay s list, G(6, of connected graphs on 6 vertices, we eliminate all those that have a duplicate vertex, a co-diamond or a 2K 3 + e, to obtain a list Ĝ(6, which we know to have all connected graphs on 6 vertices minimum skew rank at most 4. For each H in Ĝ(6 we find (by hand a matrix A H in S (H of rank at most 4. These graphs and matrices are available at [8]. Starting with McKay s list, G(7, of connected graphs on 7 vertices, we eliminate all those that have a duplicate vertex, a co-diamond or a 2K 3 + e, to obtain a list Ĝ(7, which we know contains all counterexamples of order 7, if any exist. For each H in Ĝ(6, we use Corollary 2.4 to construct all graphs of order 7 which contain H as an induced subgraph and have a rank 4 realization whose leading 6 by 6 matrix is A H. Our Sage-supported computations show that all graphs in Ĝ(7 arise in this way, and hence the necessary conditions are also sufficient when n = 7. As we use Theorem 2.3 to build a graph G in Ĝ(7, we also find an A G in S (G with rank at most 4. These graphs and matrices are available at [8]. We move to graphs of order 8, and proceed in a similar manner. Our Sage-supported computations show that all graphs in Ĝ(8 arise this way, and hence the necessary conditions are also sufficient when n = 8. These graphs and matrices are available at [8]. In summary, we have shown the following. Theorem 2.5. Let G be a graph of order at most 8 with neither duplicate vertices nor isolated vertices. If G is co-diamond-free and (2K 3 + e-free, then mr (G 4. For graphs of order 9, we follow the similar procedure as before and we get two potential counterexamples which are shown in Figure 5, and are HEhbtjK and HEjbvj[ in graph6 format (see data/formats.html for details on this format. We note that the first graph is isomorphic to the Cartesian product K 3 K 3. 6

7 Figure 5: HEjbvj[ and HEhbtjK It can be verified that the skew-symmetric matrix has graph HEjbvj[ and rank 4. Hence mr (HEjbvj[ = 4. We now show that K 3 K 3 is indeed a counterexample. The adjacency matrix of K 3 K 3 is Note that this adjacency matrix is (J 3 I 3 (J 3 I 3, the Kronecker product of J 3 I 3 with itself where J 3 and I 3 are the all ones matrix and the identity matrix of order 3. Since K 3 K 3 has P 4 as an induced subgraph (on vertices 7.

8 2,3,4 and 5 in Figure 5, K 3 K 3 has minimum skew rank at least 4. Suppose that there exists a B S (K 3 K 3 of rank 4. Then columns 3i+1, 3i+2 and 3i + 3 are minimally linearly dependent for i =, 1, 2. Hence the nullspace of B contains vectors of the form a b c, d e f, and for some a, b, c, d, e, f, g, h, i each of which is nonzero. By replacing B by DBD where D is the diagonal matrix with main diagonal (a, b, c, d, e, f, g, h, i matrix, we may assume without loss of generality that a = b = c = d = e = f = g = h = i = 1. Thus, B has the form: x x y y x x y y x x y y x x z z x x z z x x z z y y z z y y z z y y z z for some nonzero x, y, z. Calculation shows that det B[{1, 2, 4, 5, 7, 8}] = 4x 2 y 2 z 2, which contradicts the assumption that B has rank 4. Hence mr (K 3 K 3 6. Thus, we have proven the following. Theorem 2.6. Let G be a graph of order at most 9 with neither duplicate vertices nor isolated vertices. If G is co-diamond-free and (2K 3 +e-free, and G is not isomorphic to K 3 K 3 then mr (G 4. g h i 8

9 3 Existence of co-diamond-free and (2K 3 + e- free graphs with minimum skew rank greater than 4 In this section we show that the converse of Theorem 1.1 is far from true for n sufficiently large. The technique used is a common technique in combinatorics, known as the probabilistic method (see [1]. This method is a non-constructive method that proves the existence (or ubiquity of mathematical object by showing that if one selects the object from a specified class the probability the the object has the desired property is greater than. We will use the following lemma to give an upper bound on the number of labelled graphs of order v with minimum skew rank at most 2k. Recall from Section 2 that the support of a row or column vector v R n is Z(v = {i : 1 i n, v i }. If f = (f 1,..., f m is a sequence of polynomials in n variables over the field F, then the zero-pattern set of f is {Z(f 1 (x,..., f m (x : x F n } and the cardinality of this set is denoted by Z F (f. Throughout the remainder of this article e is the base of the natural logarithm. Lemma 3.1. [7, Cor. 1.5] Let f = (f 1,..., f m be a sequence of polynomials of degree at most d in n variables over the field F, where d 1. Then ( n emd Z F (f <. n We now explain how the graphs of order n with minimum skew rank at most 2k corresponds to the zero-pattern set of a family of polynomials. First note that if A is a real skew-symmetric matrix of order v with rank at most 2k, then there exist v k matrices X and Ŷ such that A = XŶ T Ŷ X T [3, Lemma 1.3]. Now let x i,j and y i,j be distinct variables for 1 i v and 1 j k and let X = [x i,j ] and Y = [y i,j ]. Set p i,j = (XY T Y X T i,j. Consider the m-tuple f = (p i,j : 1 i < j v of polynomials in n variables over R where m = v(v 1/2 and n = 2vk. Now each zero pattern of f obtained by p i,j = a i,j, 1 i < j v corresponds to a skew-symmetric matrix A = [a i,j ] of order v and rank at most 2k whose graph G has mr (G 2k. Also each graph G of order v with mr (G 2k has a skew-symmetric matrix of rank at most 2k whose zero pattern corresponds to a unique zero pattern of f. Thus Z R (f is the number of labelled graphs of order v with minimum skew rank at most 2k. Lemma 3.1 now implies the following. 9

10 Theorem 3.2. The number of labelled graphs of order v with minimum skew ( 2vk e(v 1 rank at most 2k is less than. 2k Corollary 3.3. The number ( of labelled graphs of order n with minimum skew 4n e(n 1 rank at most 4 is less than. 4 Now we identify a family of co-diamond-free and (2K 3 + e-free graphs without duplicate vertices. Proposition 3.4. Let H be a (k + 1-regular graph of order n with girth at least 5 where 2 k n 3. Let G be the complement of H. Then G is a co-diamond-free and (2K 3 + e-free graph with neither duplicate vertices nor isolated vertices. Proof. Since H has girth at least 5, H is C 4 -free and C 3 -free. Consequently G is (2K 3 + e-free and co-diamond-free. Since n k 2 1 and G is (n k 2-regular graph, G has no isolated vertices. We will show that G has no duplicate vertices. Suppose to the contrary that u and v are duplicate vertices of G. Since H is C 3 -free, each vertex w of G other than u and v is adjacent to both u and v in G. But then the degree of u in G is n 2 which contradicts the fact that G is (n k 2-regular graph with 2 k n 3. The following lemmas lead us to a lower bound of the number of certain regular labelled graphs of order n with girth greater than 4. Throughout our calculations we denote log 2 by lg. Lemma 3.5. [5, Corollary 2] For (d 1 2g 1 = o(n, the number of d-regular labelled graphs of order n with girth greater than g is ( g (d 1 i (nd! exp + o(1 2i as n. It is well known that (nd/2!2 nd/2 (d! n t ln(t t + 1 ln(t! t ln(t 1

11 for all t >. This implies that for positive integers k we have lg(k! k lg k and (3.1 lg((k 9 + k 8! > (k 9 + k 8 lg(k 9 + k 8 (k9 + k 8 (3.2 ln 2 Corollary 3.6. Let k be a positive integer and n = k 8. For n sufficiently large, the number of (1 + k-regular labelled graphs of order n with girth greater than 4 is at least n n9/8 3. Proof. Set g = 4 and d = 1 + k. Then (d 1 2g 1 = n 7 8 = o(n. Let N be the number of (1 + k-regular graphs of order n with girth greater than 4. By Lemma 3.5, there exist constants c k such that ( 4 (k 9 + k 8 k i! exp 2i + c k N = ((k 9 + k 8 /2!2 (k9 +k 8 /2 ((k + 1! k8 and lim k c k =. Taking lg of both sides we get lg N = lg((k 9 + k 8! ( 4 k i lg e + c k lg e 2i lg(((k 9 + k 8 /2! (k 9 + k 8 /2 k 8 lg((k + 1! Using (3.1 and (3.2 in the preceding equation, we get ( 4 lg N > (k 9 + k 8 lg(k 9 + k 8 (k9 + k 8 k i lg e + c k lg e ln 2 2i ( k9 + k 8 k 9 + k 8 lg k9 + k 8 k 8 (k + 1 lg(k ( 4 = (k 9 + k 8 lg(k 9 + k 8 (2 + ln 2(k9 + k 8 k i lg e + c k lg e 2 ln 2 2i ( k9 + k 8 k 9 + k 8 lg (k 9 + k 8 lg(k + 1. (

12 ( Since k9 +k 8 k lg 9 +k 8 > k9 +k 8 lg (k 9 + k 8, from (3.3 we get lg N > k9 + k 8 Since k9 +k 8 2 lg(k 9 + k 8 (2 + ln 2(k9 + k 8 2 ln 2 ( 4 k i lg e + c k lg e 2i (k 9 + k 8 lg(k + 1. (3.4 > k9 2 2, from (3.4 we get lg N > k9 2 lg(k9 + k 8 (2 + ln 2k9 ln 2 ( 4 k i lg e + c k lg e 2i (k 9 + k 8 lg(k + 1. (3.5 ( 4 k i Since lg e = 3k8 + 4k 3 + 6k k lg e > 2k 8, from (3.5 we 2i 24 get lg N > k9 2 lg(k9 + k 8 (2 + ln 2k9 2k 8 + c k lg e (k 9 + k 8 lg(k + 1 ln 2 = k9 ( lg(k 8 + lg(k + 1 (2 + ln 2k9 2k 8 + c k lg e (k 9 + k 8 lg(k ( ln 2 = k9 lg(k 8 2(2 + ln 2 lg(k + 1 2k 8 + c k lg e k 8 lg(k + 1. (3.6 2 ln 2 Thus So for r > 2, say r = 3, lg N k 9 lg(k 8 /r = r + terms that go to, as k. 2 Then lg N > k 9 lg(k 8 /3 for k sufficiently large. lg N > n 9/8 lg(n/3 for n sufficiently large. Hence N is at least 2 n9/8 lg n 3 = (2 lg n n9/8 3 = n n9/8 3 for sufficiently large n. Now we show that for n sufficiently large, the converse of Theorem 1.1 fails to hold. 12

13 Theorem 3.7. For n sufficiently large there exists a co-diamond-free and (2K 3 + e-free graph of order n with neither duplicate vertices nor isolated vertices and minimum skew rank at least 6. Proof. By Corollary 3.3, the number of labelled graphs of order n with minimum skew rank at most 4 is less than n 4n. Now by Corollary 3.6, the number of (1 + k-regular labelled graphs of order n = k 8 with girth greater than 4 is at least n n9/8 3 for sufficiently large n. Since n 4n < n n9/8 3 as n, for sufficiently large n there exists a (k + 1-regular graph H of order n = k 8 with girth at least 5 whose complement G has minimum skew rank at least 6. By Proposition 3.4, G is co-diamond-free and (2K 3 + e-free graph with neither duplicate vertices nor isolated vertices. 4 Average minimum skew rank The average minimum rank of graphs of order v is introduced and studied in [2]. The average minimum skew rank of graphs of order v is denoted by amr (v and defined by amr (v = G mr (G, 2 (v 2 where the sum runs over all labelled graphs of order v. Thus, amr (v is the expected minimum skew rank of all labelled graphs on v vertices. In Theorem 3.2 we obtained an upper bound of the number of labelled graphs of order v with minimum skew rank at most 2k. This leads us to the following lower bound of the average minimum skew rank of graphs of order v. Theorem 4.1. Asymptotically almost all graphs on v vertices have minimum skew rank at least.4v. Proof. If N(v, 2k is the number of labelled graphs of order v with minimum skew rank at most 2k, then by Theorem 3.2, Thus N(v, 2k 2 (v 2 < 1 2 (v 2 ( 2vk e(v 1 N(v, 2k <. 2k ( 2vk ( e(v 1 e(v 1 = 2k 13 2k2 (v 1/(4k 2vk (4.1

14 Set k = v/5. Then from (4.1 we get N(v, v/25 2 (v 2 ( e(v 1 < 2k2 (v 1/(4k 2vk ( 25e(v 1 = v2 25(v 1/(2v v 2 25 = ( 25e(1 1/v 2 25(1 1/v/2 v 2 25 (4.2 25e(1 1/v Note that < 25e <.9 for v 2. Thus from (4.2 we get 2 25(1 1/v/2 225/4 N(v, v/25 as v. 2 (v 2 Corollary 4.2. For v sufficiently large, amr (v.4v. References [1] Noga Alon and Joel H. Spencer, The probabilistic method (2ed, New York: Wiley-Interscience, (2. [2] H. Tracy Hall, Leslie Hogben, Ryan Martin and Bryan Shader, Expected values of parameters associated with the minimum rank of a graph, Linear Algebra Appl. 433 ( [3] IMA-ISU research group on minimum rank (Mary Allison, Elizabeth Bodine, Luz Maria DeAlba, Joyati Debnath, Laura DeLoss, Colin Garnett, Jason Grout, Leslie Hogben, Bokhee Im, Hana Kim, Reshmi Nair, Olga Pryporova, Kendrick Savage, Bryan Shader, Amy Wangsness Wehe, Minimum rank of skew-symmetric matrices described by a graph, Linear Algebra Appl. 432 ( [4] Sudipta Mallik and Bryan L. Shader, Classes of graphs with minimum skew rank 4, Linear Algebra Appl. 439 ( [5] Brendan D. McKay, Nicholas C. Wormald and Beata Wysocka, Short cycles in random regular graphs, The Electronic Journal of Combinatorics 11 (24, # R66 [6] Brendan D. McKay, Collection of graphs as strings, [7] L. Rónyai, L. Babai, M.K. Ganapathy, On the number of zero-patterns of a sequence of polynomials, J. Amer. Math. Soc. 14 (3 ( (electronic. 14

15 [8] Minimum skew-rank of small order graphs website: edu/bshader/skew 15

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