CHAPTER 3. DETERMINATION OF POSITIVE SEMIDEFINITE MAX-
3.2 Positive semidefinite maximum nullity and zero forcing number of dual planar
Members of the ISU Math Research Experiences for Undergraduates (REU) 2009 [3] intro-duced the ciclo and estrella graph families to show that the maximum nullity as well as the zero forcing number of a 3-connected planar graph and its dual may differ. In this section, we include the definitions of the ciclo and estrella graph families, and we determine the positive semidefinite maximum nullity and zero forcing number for many of these families. The results are summarized in Table 3.1. In addition, we show that the maximum positive semidefinite nullity as well as the positive semidefinite zero forcing number of a 3-connected planar graph and its dual may differ.
Definition 3.2.1. Let G be a graph and specify an edge e of G. A t-ciclo of G with e, denoted Ct(G, e), is constructed from a t-cycle Ct and t copies of G as follows: identify each edge of Ct with the edge e in one copy of G. The notation Ct(G) is used in place of Ct(G, e) if a symbol for the graph identifies a specific edge or the graph is edge transitive. A vertex belonging to Ctis called a cycle vertex.
The ciclo C4(K4) is shown in Figure 3.21. The ciclo formed from a complete graph has a special name, as stated in the next definition.
Figure 3.21 The complete ciclo C4(K4) and the complete estrella S4(K4)
Definition 3.2.2. The complete ciclo, denoted Ct(Kr), is the ciclo of the complete graph Kr, with t, r ≥ 3. A vertex not belonging to Ct is called a noncycle vertex.
Next, we include the definition of the estrella graph family.
Definition 3.2.3. Let G be a graph and specify an edge e of G and a vertex v of G that is not an endpoint of e. A t-estrella of G with e and v, denoted St(G, e, v), is the union of a t-ciclo Ct(G) and the complete bipartite graph K1,t with each degree one vertex of K1,t identified with one copy of v. The notation St(G) is used in place of St(G, e, v) if a symbol for the graph identifies a specific edge and vertex or the graph is vertex and edge transitive. The degree t vertex of K1,t is called the star vertex, and each neighbor of the star vertex is called a starneighbor vertex. A cycle vertex in the ciclo is called a cycle vertex in the estrella.
The estrella S4(K4) is shown in Figure 3.21. As with the ciclo, the estrella formed from a complete graph has a special name.
Definition 3.2.4. The complete estrella, denoted St(Kr), is the estrella of the complete graph Kr, with t, r ≥ 3. A vertex in St(Kr) that is not the star vertex, a starneighbor vertex, or a cycle vertex is called a standard vertex.
Next, we consider ciclos and estrellas formed from house graphs.
Definition 3.2.5. A house H0 (also called an empty house) is the union of a 3-cycle and a 4-cycle with one edge in common. The symbol H0 designates a specific edge e and vertex v as shown in Figure 3.22. A house ciclo is Ct(H0) = Ct(H0, e). A house estrella is St(H0) = St(H0, e, v).
e v
Figure 3.22 The house H0
The house ciclo C4(H0) and the house estrella S4(H0) are shown in Figure3.23.
Figure 3.23 The house ciclo C4(H0) and the house estrella S4(H0)
Definition 3.2.6. A half-house H1 is a house with one diagonal in the 4-cycle. The symbol H1 designates a specific edge e and vertex v as shown on the left in Figure3.24. A half-house ciclo is a ciclo of half-houses Ct(H1) = Ct(H1, e).
e v
Figure 3.24 The half-house H1 and the half-house ciclo C4(H1)
The half-house ciclo C4(H1) is shown in Figure 3.24.
Definition 3.2.7. A full house H2 is a house with both diagonals in the 4-cycle. The symbol H2 designates a specific edge e and vertex v as shown on the left in Figure 3.25. A full house ciclo is a ciclo of full houses Ct(H2) = Ct(H2, e).
The full house ciclo C4(H2) is shown in Figure 3.25. Finally, we consider the ciclo formed from a cycle graph.
Definition 3.2.8. A cycle ciclo is a ciclo of cycles Ct(Cr), r ≥ 4.
e v
Figure 3.25 The full house H2 and the full house ciclo C4(H2)
Figure 3.26 The cycle ciclo C3(C6)
The cycle ciclo C3(C6) is shown in Figure3.26.
Observe that the maximum nullity, maximum positive semidefinite nullity, zero forcing number, and positive semidefinite zero forcing number of the complete ciclo Ct(Kr) are all equal by [3]. For the complete estrella with r ≥ 4, the maximum nullity and maximum positive semidefinite nullity differ, and similarly for the zero forcing number.
Proposition 3.2.9. For t ≥ 3 and r ≥ 4, M+(St(Kr)) = Z+(St(Kr)) = (r − 3)t + 1 and mr+(St(Kr)) = 2t.
Proof. Note that |St(Kr)| = (r − 1)t + 1. Since St(Kr) can be covered by t copies of Kr (each of minimum positive semidefinite rank 1) and one K1,t (of minimum positive semidefinite rank t), mrR+(St(Kr)) ≤ 2t and so (r − 3)t + 1 ≤ MR+(St(Kr)).
Define a set B consisting of all cycle vertices, the star vertex, and all but one standard vertex in each Kr; note that |B| = (r − 3)t + 1. We claim that B is a positive semidefinite zero
forcing set. Since B contains all cycle vertices and the star vertex, there are t components of St(Kr) − B. Let W1, ..., Wt be the sets of vertices of the t components of St(Kr) − B. Each starneighbor vertex is the only white neighbor of the star vertex in St(Kr)[Wi∪B], i ∈ {1, ..., t}, so the star vertex forces the starneighbor vertex. Now, each starneighbor vertex can force the one white standard vertex in St(Kr)[Wi ∪ B], i ∈ {1, ..., t}. So the entire graph is black, establishing the claim. Thus, (r − 3)t + 1 ≤ MR+(St(Kr)) ≤ Z+(St(Kr)) ≤ (r − 3)t + 1. Hence, all inequalities are equalities.
For the complete estrella with r = 3, the maximum nullity, maximum positive semidefinite nullity, zero forcing number, and positive semidefinite zero forcing number are all equal.
Proposition 3.2.10. For t ≥ 3, M+(St(K3)) = Z+(St(K3)) = t + 1 and mr+(St(K3)) = t.
Proof. Since mrR+(Ct(K3)) = t [3] and Ct(K3) is an induced subgraph of St(K3),
t = mrR+(Ct(K3)) ≤ mrR+(St(K3)). To show that mr(St(K3)) ≤ t, the authors in [3] constructed a matrix of rank t in S(Ct(K3)) and extended it to a matrix in S(St(K3)) without changing the rank of the matrix. The matrix constructed in [3] is in fact positive semidefinite, arising naturally from a vector representation. Number the vertices of Ct(K3) as in Figure 3.27 [3].
The vertices {1, ..., t} are independent.
1
2
t
t-1 t+1 t+2
2t-2 2t
2t-1
Figure 3.27 Numbering for Ct(K3)
Define the t × t matrix B to be
and let 1t denote the t × 1 column vector consisting of 10s. Then the set of column vectors of
I B 1t
is a vector representation of St(K3) in Rt. Hence, mrR+(St(K3)) ≤ t.
For the house ciclo Ct(H0), the half-house ciclo Ct(H1), the full house ciclo Ct(H2), and the cycle ciclo Ct(Cr), the maximum nullity and maximum positive semidefinite nullity are equal. In particular, the maximum positive semidefinite nullity of the house ciclo Ct(H0), the half-house ciclo Ct(H1), and the cycle ciclo Ct(Cr) is t, and the maximum positive semidefinite nullity of the full house ciclo Ct(H2) is 2t. The zero forcing number and the positive semidefinite zero forcing number of the house ciclo Ct(H0), half-house ciclo Ct(H1), and cycle ciclo Ct(Cr) may differ. The zero forcing number for the house ciclo Ct(H0) is known to be t when t ≥ 4 is even, and it has been established by use of software [14] that the zero forcing number is t + 1 for odd t ≤ 9. In contrast, we show below that the positive semidefinite zero forcing number of Ct(H0) is t for all t. Although the zero forcing number for the half-house ciclo Ct(H1) is known to be t when t is even, the positive semidefinite zero forcing number is t for all t, as shown below. The zero forcing number of the cycle ciclo Ct(Cr) is known to be t when t ≥ 4 is even, while the positive semidefinite zero forcing number is t for all t, as shown below. For the full house ciclo Ct(H2), the zero forcing number and the positive semidefinite zero forcing number are equal (this follows from the proof by clique covering and zero forcing given in [3]).
Proposition 3.2.11. For t ≥ 3, M+(Ct(H0)) = Z+(Ct(H0)) = t and mr+(Ct(H0)) = 3t.
Proof. The house ciclo Ct(H0) can be covered by t copies of H0. Since H0 is a polygonal path, mrR+(H0) = 3 by Proposition 3.1.5. So mrR+(Ct(H0)) ≤ 3t by Observation 3.1.3. Hence, t ≤ MR+(Ct(H0)) ≤ Z+(Ct(H0)) ≤ t as the set of t cycle vertices form a positive semidefinite zero forcing set of Ct(H0). Hence, all inequalities are equalities.
Proposition 3.2.12. For t ≥ 3, M+(Ct(H1)) = Z+(Ct(H1)) = t and mr+(Ct(H1)) = 3t.
Proof. The half-house ciclo Ct(H1) can be covered by t copies of H1. Since H1 is a polygonal path, mrR+(H1) = 3 by Proposition 3.1.5. So mrR+(Ct(H1)) ≤ 3t by Observation 3.1.3. Hence, t ≤ MR+(Ct(H1)) ≤ Z+(Ct(H1)) ≤ t as the set of t cycle vertices form a positive semidefinite zero forcing set of Ct(H1). Hence, all inequalities are equalities.
Proposition 3.2.13. For t ≥ 3, r ≥ 4, M+(Ct(Cr)) = Z+(Ct(Cr)) = t and mr+(Ct(Cr)) = (r − 2)t.
Proof. The cycle ciclo Ct(Cr) can be covered by t copies of Cr. We know mrR+(Cr) = r − 2. So mrR+(Ct(Cr)) ≤ (r − 2)t by Observation 3.1.3. Hence, (r − 1)t − (r − 2)t = t ≤ MR+(Ct(Cr)) ≤ Z+(Ct(Cr)) ≤ t as the set of t cycle vertices form a positive semidefinite zero forcing set of Ct(Cr). Hence, all inequalities are equalities.
A connected graph G is k-connected if κ(G) ≥ k. To construct the dual Gdof a 3-connected planar graph G, place a dual vertex in each region of a plane drawing of G and an edge between dual vertices if the regions containing the dual vertices share an original edge (we assume the graph is 3-connected to ensure that the dual is determined by the graph rather than a particular plane embedding).
Theorem 3.2.14. [3] The dual of the complete estrella St(K4) is the house estrella St(H0).
Corollary 3.2.15. [3] Since M(S4(K4)) = Z(S4(K4)) = 7 and M(S4(H0)) = Z(S4(H0)) = 5, the maximum nullity of a 3-connected planar graph and its dual may differ, as well as the zero forcing number of a 3-connected planar graph and its dual.
It is only natural to pose the following questions:
Question 3.2.16. If G is a 3-connected planar graph, is it true that MR+(Gd) = MR+(G)?
Question 3.2.17. If G is a 3-connected planar graph, is it true that Z+(Gd) = Z+(G)?
By Proposition 3.2.9, Z+(S4(K4)) = 5. It has been established by use of software [14]
that Z+(S4(H0)) = 5. So although the zero forcing number of S4(K4) and its dual S4(H0) differ, the positive semidefinite zero forcing number of the two are the same. However, we have found an example where the positive semidefinite zero forcing number and maximum positive semidefinite nullity of a graph and its dual differ.
Example 3.2.18. The graph G shown in Figure3.28is 3-connected, so its dual is independent of how it is drawn in the plane. The dual Gdis also shown in Figure3.28. It has been established by use of software [14] that Z(G) = Z(Gd) = 5 while Z+(G) = 5 and Z+(Gd) = 4. Furthermore, MR+(G) ≤ Z+(G) = 5. So mrR+(G) = |G| − MR+(G) ≥ 12 − 5 = 7. We know mrR+(G) ≤ cc(G) ≤ 7 as the four K4’s along with the remaining three edges form a clique cover of G [19, Observation 3.14]. Hence, mrR+(G) = 7, and so MR+(G) = 5. Now, MR+(Gd) ≤ Z+(Gd) = 4. This example answers negatively Question 3.2.16and Question3.2.17.
Figure 3.28 The graph G and its dual Gd
3.3 Summary table
The results of this chapter are summarized in Table 3.1. Column 1 of the table lists the result number or the reference for results previously established. The positive semidefinite zero forcing number, maximum nullity, and minimum rank are listed in separate columns. Column 6 lists whether or not M(G) = Z(G) = M+(G) = Z+(G) for each graph G. Often, when the response in Column 6 is No, it is still the case that M+(G) = Z+(G). In most cases, the standard and positive semidefinite parameters are equal.
Table3.1Summaryofpositivesemidefinitezeroforcingnumber,minimumrank,andmaximumnullityresults result#GorderZ+(G)M+(G)M(G)=Z(G)=mr+(G) M+(G)=Z+(G)? [24]Pnn11Yn−1 [24]Cnn22Yn−2 [19]Knnn−1n−1Y1 3.1.4Kn1,n2,...,nk,n1+n2+...+nkn2+...+nkn2+...+nkNn1 n1≥n2≥...≥nk>0 [24]T(tree)n11Nn−1 3.1.5polygonalpathG|G|22Y|G|−2 3.1.8Qn(hypercube)2n 2n−1 2n−1 Y2n−1 3.1.12KsPtstssYs(t−1) 3.1.13CsP2,s≥42s44Y2s−4 3.1.14C4Pt4t44Y4t−4 ConjectureCsPtstmin{s,2t}min{s,2t}Yst−min{s,2t} 3.1.15PsP2,s≥22s22Y2s−2 ConjecturePsPt,s≥tstttY(s−1)t 3.1.16K3K3955Y4 ConjectureKsKtstst−s−t+2st−s−t+2Ys+t−2 3.1.17,3.1.18CsK3,s≥43s66Y3s−6 ConjectureCsKt,s≥4st2t2tY(s−2)t
Table3.1(Continued) Summaryofpositivesemidefinitezeroforcingnumber,minimumrank,andmaximumnullityresults result#GorderZ+(G)M+(G)M(G)=Z(G)=mr+(G) M+(G)=Z+(G)? [35]M¨obiusladder2n
3ifn=3, 4ifn=4, 4else.
3ifn=3, 3ifn=4, 4else.N,n=3,41
3ifn=3, 5ifn=4, 2n−4else.Y,n>4 [1]Tn(supertriangle)1 2n(n+1)nnY1 2n(n−1) [1]PsPtsts+t−1s+t−1Y(s−1)(t−1) 3.1.19Wn,n≥4n33Yn−3 3.1.20Ns4ss+2s+2Y3s−2 3.1.21Hs(half-graph)2sssYs 3.1.23Km∪K1,k,k≥2,m≥3m+km−1m−1Nk+1 3.1.24unicyclicgraphG|G|22N|G|−2 3.1.25block-clique|G||G|−b(G)|G|−b(G)Nb(G) (1-chordal)graphG, b(G)=#blocksofG 3.1.27Cn,n≥5nn−3n−3Y3 3.1.28,3.1.29T,Tatree,nn−3n−3Y3 |T|=n,n≥4, T6=K1,n−1 3.1.30complementofan
n|H|=3, n−1|H|≥4, l=2, n−3|H|≥5, l=1, n−4|H|≥6, l=0
n|H|=3, n−1|H|≥4, l=2, n−3|H|≥5, l=1, n−4|H|≥6, l=0
Y
0|H|=3, 1|H|≥4, l=2, 3|H|≥5, l=1, 4|H|≥6, l=0
2-tree,n≥3, l=#dominating vertices 1 Notethat3=M+(ML8)<Z+(ML8)=4.
Table3.1(Continued) Summaryofpositivesemidefinitezeroforcingnumber,minimumrank,andmaximumnullityresults result#GorderZ+(G)M+(G)M(G)=Z(G)=mr+(G) M+(G)=Z+(G)? 3.1.31L(Kn)n(n−1) 2n(n−1) 2−n+2n(n−1) 2−n+2Yn−2 3.1.33L(G),Ghasan−2 Hamiltonianpath, |G|=n≥2 3.1.34L(T),Tatreeand|T|−1l−1l−1Y|T|−l l=#pendent verticesofT [1]Kt◦Ks,t≥2st+tst−1st−1Yt+1 3.1.35Ct◦Ksst+tst−t+2st−t+2N,s=12t−2 Y,s>1 3.1.36W(1,s)(s-helm)2s+133N2s−2 3.1.37S2k(spider)2k33N2k−3 [3]Ct(Kr),t≥3,r≥3(r−1)t(r−2)t(r−2)tYt 3.2.9St(Kr),t≥3,r≥4(r−1)t+1(r−3)t+1(r−3)t+1N2t 3.2.10St(K3),t≥32t+1t+1t+1Yt 3.2.11Ct(H0),t≥34tttN2 3t 3.2.12Ct(H1),t≥34tttN33t [3]Ct(H2),t≥34t2t2tY2t 3.2.13Ct(Cr),t≥3,r≥4(r−1)tttN(r−2)t 2ForG=Ct(H0),M(G)=M+(G)=Z+(G),Z(G)maydifferforoddt. 3 ForG=Ct(H1),M(G)=M+(G)=Z+(G),Z(G)maydifferforoddt.
BIBLIOGRAPHY
[1] AIM Minimum Rank – Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler, S.
M. Cioab˘a, D. Cvetkovi´c, S. M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi´c, H. van der Holst, K. Vander Meulen, A. Wangsness). Zero forcing sets and the minimum rank of graphs. Linear Algebra and its Applications, 428: 1628–1648, 2008.
[2] Webpage for the 2006 American Institute of Mathematics workshop “Spectra of families of matrices described by graphs, digraphs, and sign patterns,” available at http://aimath.org/pastworkshops/matrixspectrum.html. This webpage has links to the AIM minimum rank graph catalog: Families of graphs (available directly at http://aimath.org/pastworkshops/catalog2.html).
[3] E. Almodovar, L. DeLoss, L. Hogben, K. Hogenson, K. Murphy, T. Peters and C. Ramrez.
Minimum rank, maximum nullity and zero forcing number for selected graph families.
Involve a journal of mathematics, 3.4: 371–392, 2010.
[4] F. Barioli, W. Barrett, S. Fallat, H. T. Hall, L. Hogben, B. Shader, P. van den Driessche, H. van der Holst. Parameters related to tree-width, zero forcing, and maximum nullity of a graph. To appear in Journal of Graph Theory.
[5] F. Barioli, W. Barrett, S. Fallat, H. T. Hall, L. Hogben, B. Shader, P. van den Driessche, H. van der Holst. Zero forcing parameters and minimum rank problems. Linear Algebra and its Applications, 433: 401–411, 2010.
[6] F. Barioli, W. Barrett, S. Fallat, H. T. Hall, L. Hogben, H. van der Holst. On the graph complement conjecture for minimum rank. Linear Algebra and its Applications, in press, 2011.
[7] F. Barioli, S. Fallat, L. Hogben. A variant on the graph parameters of Colin de Verdi`ere:
Implications to the minimum rank of graphs. Electronic Journal of Linear Algebra. 13:
387–404, 2005.
[8] F. Barioli, S. Fallat, L. Hogben. Computation of minimal rank and path cover number for graphs. Linear Algebra and its Applications. 392: 289–303, 2004.
[9] F. Barioli, S. Fallat, L. Mitchell, S. Narayan. Minimum semidefinite rank of outerplanar graphs and the tree cover number. Electronic Journal of Linear Algebra. 22: 10–21, 2011.
[10] W. Barrett, H. van der Holst, R. Loewy. Graphs whose minimal rank is two. Electronic Journal of Linear Algebra. 11: 258–280, 2004.
[11] M. Behzad, G. Chartrand. Introduction to the Theory of Graphs, Allyn and Bacon, Inc., Boston, 1971.
[12] M. Booth, P. Hackney, B. Harris, C. R. Johnson, M. Lay, T. D. Lenker, L. H. Mitchell, S. K. Narayan, A. Pascoe, B. D. Sutton (2011): On the minimum semidefinite rank of a simple graph, Linear and Multilinear Algebra, 59:5, 483–506.
[13] M. Booth, P. Hackney, B. Harris, C. R. Johnson, M. Lay, L. H. Mitchell, S. K. Narayan, A. Pascoe, K. Steinmetz, B. D. Sutton, W. Wang. On the minimum rank among positive semidefinite matrices with a given graph. SIAM Journal of Matrix Analysis and Applica-tions, 30: 731–740, 2008.
[14] S. Butler, L. DeLoss, J. Grout, H, T, Hall, J, LaGrange, T. McKay, J. Smith, G.
Tims. Minimum Rank Library (Sage programs for calculating bounds on the mini-mum rank of a graph, and for computing zero forcing parameters). Available at http:
//sage.cs.drake.edu/home/pub/67/. For more information contact Jason Grout at [email protected].
[15] Y. Colin de Verdi`ere, Multiplicities of eigenvalues and tree-width of graphs, J. Combin.
Theory, Ser. B 74 (1998) 121–146.
[16] Y. Colin de Verdi`ere, On a new graph invariant and a criterion for planarity, in: Graph Structure Theory, Contemporary Mathematics, vol. 147, American Mathematical Society, Providence, 1993, pp. 137–147.
[17] L. Dealba, J. Grout, L. Hogben, R. Mikkelson, K. Rasmussen. Universally optimal matrices and field independence of the minimum rank of a graph. Electronic Journal of Linear Algebra. 18: 403–419, 2009.
[18] J. Ekstrand, C. Erickson, H. T. Hall, D. Hay, L. Hogben, R. Johnson, N. Kingsley, S. Os-borne, T. Peters, J. Roat, A. Ross, D. Row, N. Warnberg, M. Young. Positive semidefinite zero forcing. Under review.
[19] S. Fallat, L. Hogben. The minimum rank of symmetric matrices described by a graph: A survey. Linear Algebra and its Applications. 426: 558–582, 2007.
[20] C. Godsil, G. Royle. Algebraic Graph Theory, Springer-Verlag, New York, 2001.
[21] P. Hackney, B. Harris, M. Lay, L. H. Mitchell, S. K. Narayan, A. Pascoe. Linearly in-dependent vertices and minimum semidefinite rank. Linear Algebra and its Applications.
431: 1105–1115, 2009.
[22] L. Hogben. Orthogonal representations, minimum rank, and graph complements. Linear Algebra and its Applications. 428: 2560–2568, 2008.
[23] L. Hogben, H. van der Holst. Forbidden minors for the class of graphs G with ξ(G) ≤ 2.
Linear Algebra and its Applications. 423: 42–52, 2007.
[24] H. van der Holst. Graphs whose positive semi-definite matrices have nullity at most two.
Linear Algebra and its Applications. 375: 1–11, 2003.
[25] H. van der Holst. On the maximum positive semi-definite nullity and the cycle matroid of graphs. Electronic Journal of Linear Algebra. 18: 192–201, 2009.
[26] H. van der Holst. Three-connected graphs whose maximum nullity is at most three. Linear Algebra and its Applications. 429: 625–632, 2008.
[27] L.-H. Huang, G. J. Chang, H.-G. Yeh. A note on universally optimal matrices and field independence of the minimum rank of a graph. Linear Algebra and its Applications. 433:
585–594, 2010.
[28] L.-H. Huang, G. J. Chang, H.-G. Yeh. On minimum rank and zero forcing sets of a graph.
Linear Algebra and its Applications. 432: 2961–2973, 2010.
[29] L.-Y. Hsieh. On minimum rank matrices having prescribed graph, Ph.D. thesis, University of Wisconsin-Madison, 2001.
[30] C.R. Johnson, R. Loewy, P.A. Smith. The graphs for which maximum multiplicity of an eigenvalue is two. <http://arxiv.org/pdf/math.CO/0701562>.
[31] A. Kotlov. Minors and strong products. European Journal of Combinatorics. 22: 511–512, 2001.
[32] L. Lov´asz, M. Saks, and A. Schrijver. Orthogonal representations and connectivity of graphs. Linear Algebra and its Applications. 114/115: 439–454, 1989.
[33] L. Lov´asz, M. Saks, and A. Schrijver. A correction: “Orthogonal representations and connectivity of graphs”. Linear Algebra and its Applications. 313: 101–105, 2000.
[34] O. Melnikov, V. Sarvanov, R. Tyshkevich, V. Yemelichev, and I. Zverovich. Exercises in Graph Theory, Kluwer Academic Publishers, Dordrecht, 1998.
[35] L. Mitchell, S. Narayan, and A. Zimmer. Lower bounds in minimum rank problems. Linear Algebra and its Applications, 432: 430-440, 2010.
[36] P.M. Nylen. Minimum-rank matrices with prescribed graph. Linear Algebra and its Appli-cations. 248: 303–316, 1996.
[37] R. C. Read and R. J. Wilson. An Atlas of Graphs, Oxford University Press, New York, 1998.
[38] A. Ross. Positive semidefinite zero forcing, Master’s creative component, Iowa State Uni-versity, 2011.
[39] J. Sinkovic. Maximum nullity of outerplanar graphs and the path cover number. Linear Algebra and its Applications, 432: 2052–2060, 2010.
[40] S. ˇSpacapan. Connectivity of Cartesian products of graphs. Applied Mathematics Letters.
21: 682–685, 2008.