- Email: [email protected]
The multiplicity of an arbitrary eigenvalue of a graph in terms of cyclomatic number and number of pendant vertices
Linear Algebra and its Applications 584 (2020) 257–266
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
The multiplicity of an arbitrary eigenvalue of a graph in terms of cyclomatic number and number of pendant vertices Long Wang a,1 , Liangli Wei b,∗ , Yongrong Jin b a
School of Mathematics and Big Data, Anhui University of Science and Technology, Huinan, China b College of Economics and Management, Hefei Normal University, Hefei, China
a r t i c l e
i n f o
Article history: Received 22 August 2019 Accepted 16 September 2019 Available online 19 September 2019 Submitted by R. Brualdi MSC: 05C50 Keywords: Multiplicity of an eigenvalue Nullity Rank
a b s t r a c t Let G be a graph with adjacency matrix A(G). The nullity η(G) of G is the multiplicity of zero as an eigenvalue of A(G), which has received a lot of attention because of its chemical importance. The multiplicity of an arbitrary eigenvalue λ of A(G) is denoted as m(G, λ). In [25], the authors proved that η(G) ≤ 2θ(G) + p(G) for a connected graph G, with equality if and only if G is a cycle of order divisible by 4, where θ(G) = |E(G)| −|V (G)| +1 is the cyclomatic number of G and p(G) is the number of pendant vertices of G. In the present paper, we intend to extend this result from the nullity of G to the multiplicity of an arbitrary eigenvalue of G. Differing from the method in [25], by mainly applying algebraic method we prove the following result: For a connected graph G, m(G, λ) ≤ 2θ(G) +p(G) for an arbitrary eigenvalue λ of G, the equality holds if and only if G is a cycle Cn and λ = 2cos 2kπ n with k = 1, 2, . . . , n2 − 1. © 2019 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail address: [email protected] (L. Wei). Supported by National Natural Science Foundation of China (11701008) and Natural Science Foundation of Anhui Province (1808085QA04). 1
https://doi.org/10.1016/j.laa.2019.09.013 0024-3795/© 2019 Elsevier Inc. All rights reserved.
258
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
1. Introduction Throughout this paper we consider finite undirected simple graphs, i.e., multiple edges and loops are not permitted. Let G = (V (G), E(G)) be a graph with vertex set V (G) and edge set E(G), where |V (G)| and |E(G)| are respectively the order and the size of G. The cyclomatic number of a connected graph G (also known as the dimension of cycle space of G), denoted by θ(G), is defined as θ(G) = |E(G)| − |V (G)| + 1. A connected graph G is called a tree (resp., a unicyclic graph; a bicyclic graph) if θ(G) = 0 (resp., θ(G) = 1; θ(G) = 2). We write u ∼ v to mean two vertices u and v of G are adjacent and by uv we mean the edge joining u and v. The neighbor set of a vertex v in G is defined as NG (v) = {u ∈ V (G) : u ∼ v}, and the degree of v is defined as dG (v) = |NG (v)|. A pendant vertex is a vertex of degree 1 and a quasi-pendant vertex is a vertex adjacent to a pendant vertex. A major vertex is a vertex with degree at least 3. We denote by p(G) the number of pendant vertices in G. For u, v ∈ V (G), the distance between u and v, written as dG (u, v), is the shortest length of paths between u and v. A graph H is called a subgraph of G if V (H) ⊆ V (G) and E(H) ⊆ E(G). Further, H is called an induced subgraph of G if for ∀u, v ∈ V (H), u, v are adjacent in H if and only if they are adjacent in G. If U ⊆ V (G), we denote by G[U ] the induced subgraph of G with vertex set U , and we denote by G − U the induced subgraph of G with vertex set V (G)\U , i.e., G − U = G[V (G) \ U ]. For an induced subgraph H of G and a vertex v outside H, we denote by H + v the induced subgraph of G with vertex set V (H) ∪ {v}. For an edge e of G, we denote by G − e the graph with the same vertex set as G and with the edge set E(G) \ {e}. By Pn and Cn we respectively denote a path and a cycle of order n. The set of all real numbers is written as R. The adjacency matrix A(G) of a graph G of order n is an n × n matrix (au,v ), where au,v = av,u = 1 if u, v are adjacent in G and au,v = av,u = 0 if otherwise. The adjacency spectrum of G is the set of all eigenvalues of A(G). For λ ∈ R, we denote by m(G, λ) the algebraic multiplicity of λ in the adjacency spectrum of G. If λ is not an eigenvalue of G, we put m(G, λ) = 0. Particularly, the nullity of G refers to m(G, 0), which is also denoted by η(G) as ordinary. The rank of a matrix A is written as rk(A). Obviously, η(G) + rk(A(G)) = n if the order of G is n. For the multiplicity of 1 as an eigenvalue of certain matrices of graphs, we particularly mention the following results. I. Faria [13] proved that the multiplicity of 1 as an eigenvalue of the Laplacian matrix L(G) and the signless Laplacian matrix Q(G) are respectively bounded from below by p(G) − q(G), where p(G) and q(G) are the number of pendant vertices and the number of quasi-pendant vertices of G. In [1], it is proved that the above bound is attained if each internal vertex of G is a quasi-pendant vertex. More recently, Cardoso et al. [5] extended the above results to an Aα -eigenvalue of a graph. They bounded the multiplicity of α as an Aα -eigenvalue of G as: mα (G, α) ≥ p(G) − q(G),
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
259
and the bound is attained if each internal vertex of G is a quasi-pendant vertex. This result was further extended to Aα -eigenvalues of signed graphs by Belardo et al. [3]. The main result of this paper has close connection with a well studied topic—nullity of graphs. The nullity of graphs is a classical topic in spectrum theory of graphs because it has application in chemistry. The chemical importance of the nullity of a graph lies in the fact, that within the Hückel molecular orbital model, if the nullity of a molecular graph G is larger than 0, then the corresponding chemical compound is highly reactive and unstable, or nonexistent (see [2] or [11]). A lot of publications ([6], [7], [8], [9], [10], [11], [12], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]) have focused on bounding the nullity of a graph in terms the order, the number of pendant vertices, the matching number of the graph, etc. Among all these literatures, we particularly mention [25], where the authors proved the following result: Proposition 1.1. (Theorem 1.1, [25]) Let G be a connected graph with at least two vertices. Then η(G) ≤ 2θ(G) + p(G), the equality is attained if and only if G is a cycle with size a multiple of 4. In the present paper, we intend to extend the above result from the nullity of G to the multiplicity of an arbitrary eigenvalue of G. Differing from the method in [25], by mainly applying algebraic method we prove the following result: Theorem 1.2. Let G be a connected graph with at least two vertices. Then m(G, λ) ≤ 2θ(G) + p(G) for any λ ∈ R, the equality holds if and only if G is a cycle Cn and n λ = 2cos 2kπ n with k = 1, 2, . . . , 2 − 1. We will give a proof for this result in Section 2. 2. Proof of the main result Let G be a graph of order n. By Rn we denote the vector space of all n × 1 column vectors over the field R of all real numbers. The subspace of Rn with only zero vector is simply written as 0. We denote by In the identity matrix of order n. Sometimes we directly write it as I if the order can be seen from the text. For λ ∈ R, we define V (G, λ) = {α ∈ Rn : A(G)α = λα}, which is called the characteristic space of A(G) with respect to λ. Note that V (G, λ) = 0 if λ is not an eigenvalue of A(G). Recall that m(G, λ) denotes the algebraic multiplicity of λ in the adjacency spectrum of A(G). Since A(G) is diagonalizable, we have m(G, λ) = dim(V (G, λ)) = n − rk(λI − A(G)), even if λ is not an eigenvalue of A(G).
260
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
The following observation indicates that Theorem 1.2 holds for a path or a cycle. kπ • The spectrum of Pn is {2cos n+1 : k = 1, 2, . . . , n} (see page 16, [4]), thus m(Pn , λ) ≤ 1 for any λ ∈ R. • The spectrum of Cn is {2cos 2kπ n : k = 0, 1, 2, . . . , n − 1} (see page 15, [4]), thus m(Cn , λ) ≤ 2 for any λ ∈ R, and m(Cn , λ) = 2 if and only if λ = 2cos 2kπ n , where k = 1, 2, . . . , n2 − 1.
We now study how m(G, λ) changes when we delete a vertex or an edge from the graph. Lemma 2.1. Let G be a graph with a vertex v. Then m(G, λ) ≤ m(G − v, λ) + 1 for any λ ∈ R. Proof. The inequality follows immediately from the Interlacing theory of graphs. 2 Lemma 2.2. Let G be a graph with an edge e. Then m(G, λ) ≤ m(G − e, λ) + 2 for any λ ∈ R. Proof. Suppose the endpoint of e are u and v, and arrange the vertices of G such that the adjacency matrix is ⎛
0 ⎜ A(G) = ⎝ 1 αT
1 0 βT
⎞ α ⎟ β ⎠, A1
where the first and the second rows are respectively indexed by u and v and A1 is the adjacency matrix of G − u − v. Thus ⎛
0 1 ⎜ λI − A(G − e) = λI − A(G) + ⎝ 1 0 0 0
⎞ 0 ⎟ 0 ⎠, 0
which implies that rk(λI − A(G − e)) ≤ rk(λI − A(G)) + 2, and hence, m(G, λ) ≤ m(G − e, λ) + 2.
2
With the help of Lemma 2.1 and Lemma 2.2, we can bound m(G, λ) for some special graphs.
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
261
Lemma 2.3. Let G be the graph obtained from a cycle Cs and a path Pt of order at least 2 by identifying one pendant vertex of Pt with a vertex of Cs . Then m(G, λ) ≤ 2 for any λ ∈ R. Proof. Deleting a 2-degree vertex from the cycle of G that is adjacent to the major vertex, we obtain the path Ps+t−2 . Since m(Ps+t−2 , λ) ≤ 1 and m(G, λ) ≤ m(Ps+t−2 , λ) + 1, we obtain m(G, λ) ≤ 2. 2 Lemma 2.4. Let G be the ∞-type bicyclic graph obtained from two cycles Cs , Ct and a path Pm with m ≥ 1 by identifying one vertex of Cs and one vertex of Ct respectively with one pendant vertex of Pm . Then m(G, λ) ≤ 3 for any λ ∈ R. Proof. Note that if m = 1, then the graph G is obtained from Cs and Ct by identifying one vertex of Cs and one vertex of Ct . Let v be a 2-degree vertex of G lying on Cs and adjacent to a 3-degree vertex of G. Then m(G − v, λ) ≤ 2 (using Lemma 2.3). By Lemma 2.1, we have m(G, λ) ≤ m(G − v, λ) + 1 ≤ 3, as required. 2 Lemma 2.5. Let G be the θ-type bicyclic graph obtained from a pair of vertices u and v, joined by three internally disjoint paths Pk , Pl and Pm with k ≥ l ≥ m ≥ 2. Then m(G, λ) ≤ 3 for any λ ∈ R. Proof. To guarantee G to be simple, l ≥ 3. There exists a 2-degree vertex of G, say w, adjacent to u. Applying Lemma 2.3 or the observation on a cycle, we have m(G −w, λ) ≤ 2. By Lemma 2.1, we have m(G, λ) ≤ m(G − w, λ) + 1 ≤ 3, as required. 2 For α ∈ Rn we denote by αu the component of α corresponding to u ∈ V (G). With respect to a subset U of V (G), we define: Z(U ) = {α ∈ Rn : αu = 0, ∀u ∈ U }. Then Z(U ) is a subspace of Rn with dimension n − |U |. The following lemma is helpful for us to establish an upper bound for the multiplicity of an eigenvalue λ. Lemma 2.6. Let G be a graph of order n and let U be a subset of G. If V (G, λ) ∩Z(U ) = 0, then m(G, λ) ≤ |U |. Proof. Since V (G, λ) ∩ Z(U ) = 0 and dim(Z(U )) = n − |U |, we have dim(V (G, λ)) + dim(Z(U )) = dim(V (G, λ) + Z(U )) ≤ dim(Rn ) = n, from which it follows m(G, λ) ≤ |U |.
2
Lemma 2.6 helps us to study how m(G, λ) changes when certain graph operations are applied.
262
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
Lemma 2.7. Let G be a graph obtained from a path Pm with m ≥ 2 and a disjoint graph H by identifying a pendant vertex of Pm with a vertex of H. Then m(G, λ) ≤ m(H, λ) +1 for any λ ∈ R. Proof. Suppose the vertices of Pm is as v1 ∼ v2 ∼ . . . ∼ vm−1 ∼ vm , where vm is identified with a vertex of H. For α ∈ Rn with n the order of G, let α|H be the vector in Rn−m+1 defined by (α|H )v = αv for each v ∈ V (H). Let f : Rn → Rn−m+1 be the mapping such that f (α) = α|H for all α ∈ Rn . Then f is a linear mapping from Rn to Rn−m+1 . Let W = V (G, λ) ∩ Z(v1 ). We now prove f (W ) ⊆ V (H, λ). If β ∈ W , then βv1 = 0. From A(G)β = λβ it follows that λβv1 − βv2 = λβvi − βvi−1 − βvi+1 = 0, i = 2, 3, . . . , m − 1. Thus we have βv1 = βv2 = . . . = βvm = 0. These equalities together with A(G)β = λβ further give that A(H)(β|H ) = λ(β|H ). Hence, f (W ) ⊆ V (H, λ). If f (β) = 0 for β ∈ W , we can easily conclude β = 0. Thus the restriction of f to W is an injection mapping and therefore, dim(W ) ≤ m(H, λ). Noting the dimension of Z(v1 ) is n − 1, we have m(H, λ) ≥ dim(W ) ≥ dim(V (G, λ)) + dim(Z(v1 )) − n = m(G, λ) − 1.
2
Lemma 2.8. Let G be a graph obtained from a cycle Cs and a disjoint graph H which are joined by a path Pm with m ≥ 1, i.e., identify one pendant vertex of Pm with a vertex of Cs and another with a vertex of H. Then m(G, λ) ≤ m(H, λ) + 2 for any λ ∈ R. Proof. If m = 1, the graph G can be obtained from Cs and H by identifying one vertex of Cs with one vertex of H. Let v be a 2-degree vertex of G lying on the cycle and adjacent to the major vertex on the cycle. Applying Lemma 2.1, we have m(G, λ) ≤ m(G − v, λ) + 1. Lemma 2.7 says m(G − v, λ) ≤ m(H, λ) + 1. Thus we have m(G, λ) ≤ m(H, λ) + 2, as required. 2 Lemma 2.9. Let G be a simple graph obtained from a graph H and a disjoint path Pm with m ≥ 2 by identifying two pendant vertices of Pm respectively with two distinct vertices u, v of H. Then m(G, λ) ≤ m(H, λ) + 2 for any λ ∈ R.
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
263
Proof. To guarantee G to be simple, m must be greater than 2 if u, v are adjacent in H. If m = 2, it has been proved by Lemma 2.2. If m = 3, it can be derived from Lemma 2.1. Suppose that m ≥ 4 and let w be a quasi-pendant vertex of Pm . Lemma 2.1 says m(G, λ) ≤ m(G − w, λ) + 1, and Lemma 2.7 says that m(G − w, λ) ≤ m(H, λ) + 1, which together give the required inequality: m(G, λ) ≤ m(H, λ) + 2. 2 Now, we are ready to give a proof for Theorem 1.2. Proof of Theorem 1.2. We proceed by induction on the order n of G to prove that m(G, λ) ≤ 2θ(G) + p(G) − 1 for any λ ∈ R and for any connected graph G that is not a cycle. If n = 2, the assertion is trivial. Assume the assertion holds for connected graphs of order at least n − 1 and let G be a connected graph of order n ≥ 3 that is not a cycle. We partition the discussion into three parts according to three cases. Case 1. G has a pendant vertex v. If G has no major vertex, then G is a path and the assertion follows from the observation that m(Pn , λ) ≤ 1. Suppose G has major vertices. Let u be the major vertex such that m = dG (v, u) ≤ dG (v, w) for all major vertices w of G. Let H be the induced subgraph of G obtained from G by deleting the path Pm+1 joining v and u. Then G is a graph obtained from H := H + u and Pm+1 by identifying one pendant vertex of Pm+1 with u of H. If H is a cycle then by Lemma 2.3, m(G, λ) ≤ 2 = 2θ(G) + p(G) − 1, as required. If H is not a cycle, the induction hypothesis implies that m(H, λ) ≤ 2θ(H) + p(H) − 1. Noting that θ(H) = θ(G) and p(H) = p(G) − 1 and applying Lemma 2.7, we have m(G, λ) ≤ m(H, λ) + 1 ≤ (2θ(H) + p(H) − 1) + 1 = 2θ(G) + p(G) − 1. Case 2. G has no pendant vertices and distinct cycles of G have no common edges. Let Cm be a pendant cycle of G with only one major vertex v of G and let H = G − Cm + v. If H is a cycle, then we are done by Lemma 2.4. Suppose that H is not a cycle and thus m(H, λ) ≤ 2θ(H) + p(H) − 1, by the induction hypothesis. If dG (v) ≥ 4, then p(H) = 0. Applying Lemma 2.8 and noting that θ(H) = θ(G) − 1 we have the required result by m(G, λ) ≤ m(H, λ) + 2 ≤ (2θ(H) − 1) + 2 = 2θ(G) − 1. Suppose that dG (v) = 3. Obviously, G has at least two major vertices (otherwise, G has pendant vertices). Let u be the major vertex of G which minimizes dG (v, u). Let
264
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
H be the induced subgraph of G obtained by deleting the pendant cycle Cm together with the path from v to u. Then G can be viewed as the graph obtained from Cm and H := H +u joined by the path with two pendant vertices v and u. Applying Lemma 2.8, we have m(G, λ) ≤ m(H, λ) + 2. Noting that θ(H) = θ(G) − 1, p(H) = p(G) = 0 and m(H, λ) ≤ 2θ(H) − 1 (using the induction hypothesis), we have m(G, λ) ≤ m(H, λ) + 2 ≤ 2θ(H) − 1 + 2 ≤ 2θ(G) − 1. Case 3. G has no pendant vertices and a pair of cycles of G has common edges. Let e = u v be a common edge of two cycles Ck and Cl . Let u (resp., v) be a major vertex of G which minimizes dG−e (u , u) (resp., dG−e (v , v)). It should be indicated that u = u if u itself is a major vertex. Let P be the path beginning at u, across u , v , and ending at v and let H = G −P , where P = P −u −v. Then G is the graph obtained from H and P by identifying two pendant vertices of P respectively with two vertices u, v of H. If H is a cycle, then G is a θ-type bicyclic graph and we are done by Lemma 2.5. Suppose that H is not a cycle. Then m(H, λ) ≤ 2θ(H) − 1 (using the induction hypothesis). By Lemma 2.9, m(G, λ) ≤ m(H, λ) + 2. Noting that θ(G) = θ(H) + 1, p(G) = p(H) = 0, we have m(G, λ) ≤ m(H, λ) + 2 ≤ 2θ(H) − 1 + 2 ≤ 2θ(G) − 1. Hence, m(G, λ) ≤ 2θ(G) + p(G) − 1 for any λ ∈ R if G is not a cycle. On the other hand, if G is a cycle, by the observation we know m(G, λ) ≤ 2 = 2θ(G) + p(G) for any n λ ∈ R, and the equality holds if and only if λ = 2cos 2kπ n with k = 1, 2, . . . , 2 − 1. 2 Corollary 2.10. Let G be a connected graph and λ be a real number. Then ⎧ ⎪ ⎨ p(G) − 1, m(G, λ) ≤ p(G) + 1, ⎪ ⎩ p(G) + 3,
if G is a tree with at least one edge; if G is a unicyclic graph with pendant vertices; if G is a bicyclic graph.
Remark. In this paper, by applying a different method we extend the main result in [25] from the nullity of a graph to the multiplicity of an arbitrary eigenvalue. Connected graphs G and the eigenvalue λ satisfying m(G, λ) = 2θ(G) + p(G) are characterized explicitly. Here, another problem seems difficult, however interesting: To characterize the connected graphs G together with the eigenvalue λ such that m(G, λ) = 2θ(G) +p(G) −1. Even if G is assumed to be a tree or a unicyclic graph, this problem is also worth of studying. Declaration of competing interest I declare that there is no competing interest.
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
265
References [1] E. Andrade, D.M. Cardoso, G. Pastén, O. Rojo, On the Faria’s inequality for the Laplacian and signless Laplacian spectra: a unified approach, Linear Algebra Appl. 472 (2015) 81–86. [2] P.W. Atkins, J. de Paula, Physical Chemistry, eighth ed., Oxford University Press, 2006. [3] F. Belardo, M. Brunetti, A. Ciampella, On the multiplicity of α as an Aα (Γ)-eigenvalue of signed graphs with pendant vertices, Discrete Math. 342 (2019) 2223–2233. [4] A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, New York, 2012. [5] D.M. Cardoso, G. Pastén, O. Rojo, On the multiplicity of α as an eigenvalue of Aα (G) of graphs with pendant vertices, Linear Algebra Appl. 552 (2018) 52–70. [6] G.J. Chang, L.H. Huang, H.G. Yeh, A characterization of graphs with rank 4, Linear Algebra Appl. 434 (2011) 1793–1798. [7] G.J. Chang, L.H. Huang, H.G. Yeh, A characterization of graphs with rank 5, Linear Algebra Appl. 436 (2012) 4241–4250. [8] C. Chen, J. Huang, S.C. Li, On the relation between the H-rank of a mixed graph and the matching number of its underlying graph, Linear Multilinear Algebra 66 (2018) 1853–1869. [9] B. Cheng, B. Liu, On the nullity of graphs, Electron. J. Linear Algebra 16 (2007) 60–67. [10] L. Collatz, U. Sinogowitz, Spektren endlicher Grafen, Abh. Math. Semin. Univ. Hambg. 21 (1957) 63–77. [11] D. Cvetković, I. Gutman, The algebraic multiplicity of the number zero in the spectrum of a bipartite graph, Mat. Vesnik (Beograd) 9 (1972) 141–150. [12] Y.Z. Fan, K.S. Qian, On the nullity of bipartite graphs, Linear Algebra Appl. 430 (2009) 2943–2949. [13] I. Faria, Permanental roots and the star degree of a graph, Linear Algebra Appl. 64 (1985) 255–265. [14] S. Fiorini, I. Gutman, I. Sciriha, Trees with maximum nullity, Linear Algebra Appl. 397 (2005) 245–251. [15] S.C. Gong, G.H. Xu, On the nullity of a graph with cut-points, Linear Algebra Appl. 436 (2012) 135–142. [16] J. Guo, W. Yan, Y. Yeh, On the nullity and the matching number of unicyclic graphs, Linear Algebra Appl. 431 (2009) 1293–1301. [17] I. Gutman, I. Sciriha, On the nullity of line graphs of trees, Discrete Math. 232 (2001) 35–45. [18] S.B. Hu, X.Z. Tan, B.L. Liu, On the nullity of bicyclic graphs, Linear Algebra Appl. 429 (2008) 1387–1391. [19] H.H. Li, Y.Z. Fan, L. Su, On the nullity of the line graph of unicyclic graph with depth one, Linear Algebra Appl. 437 (2012) 2038–2055. [20] H.H. Li, L. Su, H.X. Sun, On bipartite graphs which attain minimum rank among bipartite graphs with given diameter, Electron. J. Linear Algebra 23 (2012) 137–150. [21] W. Li, A. Chang, On the trees with maximum nullity, MATCH Commun. Math. Comput. Chem. 56 (2006) 501–508. [22] B. Liu, Y. Huang, S. Chen, On the characterization of graphs with pendant vertices and given nullity, Electron. J. Linear Algebra 18 (2009) 719–734. [23] Y. Lu, L.G. Wang, Q. Zhou, The rank of a signed graph in terms of the rank of its underlying graph, Linear Algebra Appl. 538 (2018) 166–186. [24] W. Luo, J. Huang, S.C. Li, On the relationship between the skew-rank of an oriented graph and the rank of its underlying graph, Linear Algebra Appl. 554 (2018) 205–223. [25] X. Ma, D. Wong, F. Tian, Nullity of a graph in terms of the dimension of cycle space and the number of pendant vertices, Discrete Appl. Math. 215 (2016) 171–176. [26] X. Ma, D. Wong, R. Tian, Skew-rank of an oriented graph in terms of matching number, Linear Algebra Appl. 495 (2016) 242–255. [27] G.R. Omidi, On the nullity of bipartite graphs, Graphs Combin. 25 (2009) 111–114. [28] Y. Song, X. Song, B. Tam, A characterization of graphs with nullity |V (G)| − 2m(G) + 2c(G), Linear Algebra Appl. 465 (2015) 363–375. [29] Y. Song, X. Song, C. Zhang, An upper bound for the nullity of a bipartite graph in terms of its maximum degree, Linear Multilinear Algebra 64 (2016) 1107–1112. [30] X. Tan, B. Liu, On the nullity of unicyclic graphs, Linear Algebra Appl. 408 (2005) 212–220. [31] L. Wang, D. Wong, Bounds for the matching number, the edge chromatic number and the independence number of a graph in terms of rank, Discrete Appl. Math. 166 (2014) 276–281. [32] D. Wong, X. Ma, F. Tian, Relation between the skew-rank of an oriented graph and the rank of its underlying graph, European J. Combin. 54 (2016) 76–86.
266
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
[33] D. Wong, M. Zhu, W. Lv, A characterization of long graphs of arbitrary rank, Linear Algebra Appl. 438 (2013) 1347–1355. [34] Q. Zhou, D. Wong, D. Sun, An upper bound of the nullity of a graph in terms of order and maximum degree, Linear Algebra Appl. 555 (2018) 314–320.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
The multiplicity of an arbitrary eigenvalue of a graph in terms of cyclomatic number and number of pendant vertices Long Wang a,1 , Liangli Wei b,∗ , Yongrong Jin b a
School of Mathematics and Big Data, Anhui University of Science and Technology, Huinan, China b College of Economics and Management, Hefei Normal University, Hefei, China
a r t i c l e
i n f o
Article history: Received 22 August 2019 Accepted 16 September 2019 Available online 19 September 2019 Submitted by R. Brualdi MSC: 05C50 Keywords: Multiplicity of an eigenvalue Nullity Rank
a b s t r a c t Let G be a graph with adjacency matrix A(G). The nullity η(G) of G is the multiplicity of zero as an eigenvalue of A(G), which has received a lot of attention because of its chemical importance. The multiplicity of an arbitrary eigenvalue λ of A(G) is denoted as m(G, λ). In [25], the authors proved that η(G) ≤ 2θ(G) + p(G) for a connected graph G, with equality if and only if G is a cycle of order divisible by 4, where θ(G) = |E(G)| −|V (G)| +1 is the cyclomatic number of G and p(G) is the number of pendant vertices of G. In the present paper, we intend to extend this result from the nullity of G to the multiplicity of an arbitrary eigenvalue of G. Differing from the method in [25], by mainly applying algebraic method we prove the following result: For a connected graph G, m(G, λ) ≤ 2θ(G) +p(G) for an arbitrary eigenvalue λ of G, the equality holds if and only if G is a cycle Cn and λ = 2cos 2kπ n with k = 1, 2, . . . , n2 − 1. © 2019 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail address: [email protected] (L. Wei). Supported by National Natural Science Foundation of China (11701008) and Natural Science Foundation of Anhui Province (1808085QA04). 1
https://doi.org/10.1016/j.laa.2019.09.013 0024-3795/© 2019 Elsevier Inc. All rights reserved.
258
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
1. Introduction Throughout this paper we consider finite undirected simple graphs, i.e., multiple edges and loops are not permitted. Let G = (V (G), E(G)) be a graph with vertex set V (G) and edge set E(G), where |V (G)| and |E(G)| are respectively the order and the size of G. The cyclomatic number of a connected graph G (also known as the dimension of cycle space of G), denoted by θ(G), is defined as θ(G) = |E(G)| − |V (G)| + 1. A connected graph G is called a tree (resp., a unicyclic graph; a bicyclic graph) if θ(G) = 0 (resp., θ(G) = 1; θ(G) = 2). We write u ∼ v to mean two vertices u and v of G are adjacent and by uv we mean the edge joining u and v. The neighbor set of a vertex v in G is defined as NG (v) = {u ∈ V (G) : u ∼ v}, and the degree of v is defined as dG (v) = |NG (v)|. A pendant vertex is a vertex of degree 1 and a quasi-pendant vertex is a vertex adjacent to a pendant vertex. A major vertex is a vertex with degree at least 3. We denote by p(G) the number of pendant vertices in G. For u, v ∈ V (G), the distance between u and v, written as dG (u, v), is the shortest length of paths between u and v. A graph H is called a subgraph of G if V (H) ⊆ V (G) and E(H) ⊆ E(G). Further, H is called an induced subgraph of G if for ∀u, v ∈ V (H), u, v are adjacent in H if and only if they are adjacent in G. If U ⊆ V (G), we denote by G[U ] the induced subgraph of G with vertex set U , and we denote by G − U the induced subgraph of G with vertex set V (G)\U , i.e., G − U = G[V (G) \ U ]. For an induced subgraph H of G and a vertex v outside H, we denote by H + v the induced subgraph of G with vertex set V (H) ∪ {v}. For an edge e of G, we denote by G − e the graph with the same vertex set as G and with the edge set E(G) \ {e}. By Pn and Cn we respectively denote a path and a cycle of order n. The set of all real numbers is written as R. The adjacency matrix A(G) of a graph G of order n is an n × n matrix (au,v ), where au,v = av,u = 1 if u, v are adjacent in G and au,v = av,u = 0 if otherwise. The adjacency spectrum of G is the set of all eigenvalues of A(G). For λ ∈ R, we denote by m(G, λ) the algebraic multiplicity of λ in the adjacency spectrum of G. If λ is not an eigenvalue of G, we put m(G, λ) = 0. Particularly, the nullity of G refers to m(G, 0), which is also denoted by η(G) as ordinary. The rank of a matrix A is written as rk(A). Obviously, η(G) + rk(A(G)) = n if the order of G is n. For the multiplicity of 1 as an eigenvalue of certain matrices of graphs, we particularly mention the following results. I. Faria [13] proved that the multiplicity of 1 as an eigenvalue of the Laplacian matrix L(G) and the signless Laplacian matrix Q(G) are respectively bounded from below by p(G) − q(G), where p(G) and q(G) are the number of pendant vertices and the number of quasi-pendant vertices of G. In [1], it is proved that the above bound is attained if each internal vertex of G is a quasi-pendant vertex. More recently, Cardoso et al. [5] extended the above results to an Aα -eigenvalue of a graph. They bounded the multiplicity of α as an Aα -eigenvalue of G as: mα (G, α) ≥ p(G) − q(G),
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
259
and the bound is attained if each internal vertex of G is a quasi-pendant vertex. This result was further extended to Aα -eigenvalues of signed graphs by Belardo et al. [3]. The main result of this paper has close connection with a well studied topic—nullity of graphs. The nullity of graphs is a classical topic in spectrum theory of graphs because it has application in chemistry. The chemical importance of the nullity of a graph lies in the fact, that within the Hückel molecular orbital model, if the nullity of a molecular graph G is larger than 0, then the corresponding chemical compound is highly reactive and unstable, or nonexistent (see [2] or [11]). A lot of publications ([6], [7], [8], [9], [10], [11], [12], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]) have focused on bounding the nullity of a graph in terms the order, the number of pendant vertices, the matching number of the graph, etc. Among all these literatures, we particularly mention [25], where the authors proved the following result: Proposition 1.1. (Theorem 1.1, [25]) Let G be a connected graph with at least two vertices. Then η(G) ≤ 2θ(G) + p(G), the equality is attained if and only if G is a cycle with size a multiple of 4. In the present paper, we intend to extend the above result from the nullity of G to the multiplicity of an arbitrary eigenvalue of G. Differing from the method in [25], by mainly applying algebraic method we prove the following result: Theorem 1.2. Let G be a connected graph with at least two vertices. Then m(G, λ) ≤ 2θ(G) + p(G) for any λ ∈ R, the equality holds if and only if G is a cycle Cn and n λ = 2cos 2kπ n with k = 1, 2, . . . , 2 − 1. We will give a proof for this result in Section 2. 2. Proof of the main result Let G be a graph of order n. By Rn we denote the vector space of all n × 1 column vectors over the field R of all real numbers. The subspace of Rn with only zero vector is simply written as 0. We denote by In the identity matrix of order n. Sometimes we directly write it as I if the order can be seen from the text. For λ ∈ R, we define V (G, λ) = {α ∈ Rn : A(G)α = λα}, which is called the characteristic space of A(G) with respect to λ. Note that V (G, λ) = 0 if λ is not an eigenvalue of A(G). Recall that m(G, λ) denotes the algebraic multiplicity of λ in the adjacency spectrum of A(G). Since A(G) is diagonalizable, we have m(G, λ) = dim(V (G, λ)) = n − rk(λI − A(G)), even if λ is not an eigenvalue of A(G).
260
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
The following observation indicates that Theorem 1.2 holds for a path or a cycle. kπ • The spectrum of Pn is {2cos n+1 : k = 1, 2, . . . , n} (see page 16, [4]), thus m(Pn , λ) ≤ 1 for any λ ∈ R. • The spectrum of Cn is {2cos 2kπ n : k = 0, 1, 2, . . . , n − 1} (see page 15, [4]), thus m(Cn , λ) ≤ 2 for any λ ∈ R, and m(Cn , λ) = 2 if and only if λ = 2cos 2kπ n , where k = 1, 2, . . . , n2 − 1.
We now study how m(G, λ) changes when we delete a vertex or an edge from the graph. Lemma 2.1. Let G be a graph with a vertex v. Then m(G, λ) ≤ m(G − v, λ) + 1 for any λ ∈ R. Proof. The inequality follows immediately from the Interlacing theory of graphs. 2 Lemma 2.2. Let G be a graph with an edge e. Then m(G, λ) ≤ m(G − e, λ) + 2 for any λ ∈ R. Proof. Suppose the endpoint of e are u and v, and arrange the vertices of G such that the adjacency matrix is ⎛
0 ⎜ A(G) = ⎝ 1 αT
1 0 βT
⎞ α ⎟ β ⎠, A1
where the first and the second rows are respectively indexed by u and v and A1 is the adjacency matrix of G − u − v. Thus ⎛
0 1 ⎜ λI − A(G − e) = λI − A(G) + ⎝ 1 0 0 0
⎞ 0 ⎟ 0 ⎠, 0
which implies that rk(λI − A(G − e)) ≤ rk(λI − A(G)) + 2, and hence, m(G, λ) ≤ m(G − e, λ) + 2.
2
With the help of Lemma 2.1 and Lemma 2.2, we can bound m(G, λ) for some special graphs.
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
261
Lemma 2.3. Let G be the graph obtained from a cycle Cs and a path Pt of order at least 2 by identifying one pendant vertex of Pt with a vertex of Cs . Then m(G, λ) ≤ 2 for any λ ∈ R. Proof. Deleting a 2-degree vertex from the cycle of G that is adjacent to the major vertex, we obtain the path Ps+t−2 . Since m(Ps+t−2 , λ) ≤ 1 and m(G, λ) ≤ m(Ps+t−2 , λ) + 1, we obtain m(G, λ) ≤ 2. 2 Lemma 2.4. Let G be the ∞-type bicyclic graph obtained from two cycles Cs , Ct and a path Pm with m ≥ 1 by identifying one vertex of Cs and one vertex of Ct respectively with one pendant vertex of Pm . Then m(G, λ) ≤ 3 for any λ ∈ R. Proof. Note that if m = 1, then the graph G is obtained from Cs and Ct by identifying one vertex of Cs and one vertex of Ct . Let v be a 2-degree vertex of G lying on Cs and adjacent to a 3-degree vertex of G. Then m(G − v, λ) ≤ 2 (using Lemma 2.3). By Lemma 2.1, we have m(G, λ) ≤ m(G − v, λ) + 1 ≤ 3, as required. 2 Lemma 2.5. Let G be the θ-type bicyclic graph obtained from a pair of vertices u and v, joined by three internally disjoint paths Pk , Pl and Pm with k ≥ l ≥ m ≥ 2. Then m(G, λ) ≤ 3 for any λ ∈ R. Proof. To guarantee G to be simple, l ≥ 3. There exists a 2-degree vertex of G, say w, adjacent to u. Applying Lemma 2.3 or the observation on a cycle, we have m(G −w, λ) ≤ 2. By Lemma 2.1, we have m(G, λ) ≤ m(G − w, λ) + 1 ≤ 3, as required. 2 For α ∈ Rn we denote by αu the component of α corresponding to u ∈ V (G). With respect to a subset U of V (G), we define: Z(U ) = {α ∈ Rn : αu = 0, ∀u ∈ U }. Then Z(U ) is a subspace of Rn with dimension n − |U |. The following lemma is helpful for us to establish an upper bound for the multiplicity of an eigenvalue λ. Lemma 2.6. Let G be a graph of order n and let U be a subset of G. If V (G, λ) ∩Z(U ) = 0, then m(G, λ) ≤ |U |. Proof. Since V (G, λ) ∩ Z(U ) = 0 and dim(Z(U )) = n − |U |, we have dim(V (G, λ)) + dim(Z(U )) = dim(V (G, λ) + Z(U )) ≤ dim(Rn ) = n, from which it follows m(G, λ) ≤ |U |.
2
Lemma 2.6 helps us to study how m(G, λ) changes when certain graph operations are applied.
262
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
Lemma 2.7. Let G be a graph obtained from a path Pm with m ≥ 2 and a disjoint graph H by identifying a pendant vertex of Pm with a vertex of H. Then m(G, λ) ≤ m(H, λ) +1 for any λ ∈ R. Proof. Suppose the vertices of Pm is as v1 ∼ v2 ∼ . . . ∼ vm−1 ∼ vm , where vm is identified with a vertex of H. For α ∈ Rn with n the order of G, let α|H be the vector in Rn−m+1 defined by (α|H )v = αv for each v ∈ V (H). Let f : Rn → Rn−m+1 be the mapping such that f (α) = α|H for all α ∈ Rn . Then f is a linear mapping from Rn to Rn−m+1 . Let W = V (G, λ) ∩ Z(v1 ). We now prove f (W ) ⊆ V (H, λ). If β ∈ W , then βv1 = 0. From A(G)β = λβ it follows that λβv1 − βv2 = λβvi − βvi−1 − βvi+1 = 0, i = 2, 3, . . . , m − 1. Thus we have βv1 = βv2 = . . . = βvm = 0. These equalities together with A(G)β = λβ further give that A(H)(β|H ) = λ(β|H ). Hence, f (W ) ⊆ V (H, λ). If f (β) = 0 for β ∈ W , we can easily conclude β = 0. Thus the restriction of f to W is an injection mapping and therefore, dim(W ) ≤ m(H, λ). Noting the dimension of Z(v1 ) is n − 1, we have m(H, λ) ≥ dim(W ) ≥ dim(V (G, λ)) + dim(Z(v1 )) − n = m(G, λ) − 1.
2
Lemma 2.8. Let G be a graph obtained from a cycle Cs and a disjoint graph H which are joined by a path Pm with m ≥ 1, i.e., identify one pendant vertex of Pm with a vertex of Cs and another with a vertex of H. Then m(G, λ) ≤ m(H, λ) + 2 for any λ ∈ R. Proof. If m = 1, the graph G can be obtained from Cs and H by identifying one vertex of Cs with one vertex of H. Let v be a 2-degree vertex of G lying on the cycle and adjacent to the major vertex on the cycle. Applying Lemma 2.1, we have m(G, λ) ≤ m(G − v, λ) + 1. Lemma 2.7 says m(G − v, λ) ≤ m(H, λ) + 1. Thus we have m(G, λ) ≤ m(H, λ) + 2, as required. 2 Lemma 2.9. Let G be a simple graph obtained from a graph H and a disjoint path Pm with m ≥ 2 by identifying two pendant vertices of Pm respectively with two distinct vertices u, v of H. Then m(G, λ) ≤ m(H, λ) + 2 for any λ ∈ R.
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
263
Proof. To guarantee G to be simple, m must be greater than 2 if u, v are adjacent in H. If m = 2, it has been proved by Lemma 2.2. If m = 3, it can be derived from Lemma 2.1. Suppose that m ≥ 4 and let w be a quasi-pendant vertex of Pm . Lemma 2.1 says m(G, λ) ≤ m(G − w, λ) + 1, and Lemma 2.7 says that m(G − w, λ) ≤ m(H, λ) + 1, which together give the required inequality: m(G, λ) ≤ m(H, λ) + 2. 2 Now, we are ready to give a proof for Theorem 1.2. Proof of Theorem 1.2. We proceed by induction on the order n of G to prove that m(G, λ) ≤ 2θ(G) + p(G) − 1 for any λ ∈ R and for any connected graph G that is not a cycle. If n = 2, the assertion is trivial. Assume the assertion holds for connected graphs of order at least n − 1 and let G be a connected graph of order n ≥ 3 that is not a cycle. We partition the discussion into three parts according to three cases. Case 1. G has a pendant vertex v. If G has no major vertex, then G is a path and the assertion follows from the observation that m(Pn , λ) ≤ 1. Suppose G has major vertices. Let u be the major vertex such that m = dG (v, u) ≤ dG (v, w) for all major vertices w of G. Let H be the induced subgraph of G obtained from G by deleting the path Pm+1 joining v and u. Then G is a graph obtained from H := H + u and Pm+1 by identifying one pendant vertex of Pm+1 with u of H. If H is a cycle then by Lemma 2.3, m(G, λ) ≤ 2 = 2θ(G) + p(G) − 1, as required. If H is not a cycle, the induction hypothesis implies that m(H, λ) ≤ 2θ(H) + p(H) − 1. Noting that θ(H) = θ(G) and p(H) = p(G) − 1 and applying Lemma 2.7, we have m(G, λ) ≤ m(H, λ) + 1 ≤ (2θ(H) + p(H) − 1) + 1 = 2θ(G) + p(G) − 1. Case 2. G has no pendant vertices and distinct cycles of G have no common edges. Let Cm be a pendant cycle of G with only one major vertex v of G and let H = G − Cm + v. If H is a cycle, then we are done by Lemma 2.4. Suppose that H is not a cycle and thus m(H, λ) ≤ 2θ(H) + p(H) − 1, by the induction hypothesis. If dG (v) ≥ 4, then p(H) = 0. Applying Lemma 2.8 and noting that θ(H) = θ(G) − 1 we have the required result by m(G, λ) ≤ m(H, λ) + 2 ≤ (2θ(H) − 1) + 2 = 2θ(G) − 1. Suppose that dG (v) = 3. Obviously, G has at least two major vertices (otherwise, G has pendant vertices). Let u be the major vertex of G which minimizes dG (v, u). Let
264
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
H be the induced subgraph of G obtained by deleting the pendant cycle Cm together with the path from v to u. Then G can be viewed as the graph obtained from Cm and H := H +u joined by the path with two pendant vertices v and u. Applying Lemma 2.8, we have m(G, λ) ≤ m(H, λ) + 2. Noting that θ(H) = θ(G) − 1, p(H) = p(G) = 0 and m(H, λ) ≤ 2θ(H) − 1 (using the induction hypothesis), we have m(G, λ) ≤ m(H, λ) + 2 ≤ 2θ(H) − 1 + 2 ≤ 2θ(G) − 1. Case 3. G has no pendant vertices and a pair of cycles of G has common edges. Let e = u v be a common edge of two cycles Ck and Cl . Let u (resp., v) be a major vertex of G which minimizes dG−e (u , u) (resp., dG−e (v , v)). It should be indicated that u = u if u itself is a major vertex. Let P be the path beginning at u, across u , v , and ending at v and let H = G −P , where P = P −u −v. Then G is the graph obtained from H and P by identifying two pendant vertices of P respectively with two vertices u, v of H. If H is a cycle, then G is a θ-type bicyclic graph and we are done by Lemma 2.5. Suppose that H is not a cycle. Then m(H, λ) ≤ 2θ(H) − 1 (using the induction hypothesis). By Lemma 2.9, m(G, λ) ≤ m(H, λ) + 2. Noting that θ(G) = θ(H) + 1, p(G) = p(H) = 0, we have m(G, λ) ≤ m(H, λ) + 2 ≤ 2θ(H) − 1 + 2 ≤ 2θ(G) − 1. Hence, m(G, λ) ≤ 2θ(G) + p(G) − 1 for any λ ∈ R if G is not a cycle. On the other hand, if G is a cycle, by the observation we know m(G, λ) ≤ 2 = 2θ(G) + p(G) for any n λ ∈ R, and the equality holds if and only if λ = 2cos 2kπ n with k = 1, 2, . . . , 2 − 1. 2 Corollary 2.10. Let G be a connected graph and λ be a real number. Then ⎧ ⎪ ⎨ p(G) − 1, m(G, λ) ≤ p(G) + 1, ⎪ ⎩ p(G) + 3,
if G is a tree with at least one edge; if G is a unicyclic graph with pendant vertices; if G is a bicyclic graph.
Remark. In this paper, by applying a different method we extend the main result in [25] from the nullity of a graph to the multiplicity of an arbitrary eigenvalue. Connected graphs G and the eigenvalue λ satisfying m(G, λ) = 2θ(G) + p(G) are characterized explicitly. Here, another problem seems difficult, however interesting: To characterize the connected graphs G together with the eigenvalue λ such that m(G, λ) = 2θ(G) +p(G) −1. Even if G is assumed to be a tree or a unicyclic graph, this problem is also worth of studying. Declaration of competing interest I declare that there is no competing interest.
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
265
References [1] E. Andrade, D.M. Cardoso, G. Pastén, O. Rojo, On the Faria’s inequality for the Laplacian and signless Laplacian spectra: a unified approach, Linear Algebra Appl. 472 (2015) 81–86. [2] P.W. Atkins, J. de Paula, Physical Chemistry, eighth ed., Oxford University Press, 2006. [3] F. Belardo, M. Brunetti, A. Ciampella, On the multiplicity of α as an Aα (Γ)-eigenvalue of signed graphs with pendant vertices, Discrete Math. 342 (2019) 2223–2233. [4] A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, New York, 2012. [5] D.M. Cardoso, G. Pastén, O. Rojo, On the multiplicity of α as an eigenvalue of Aα (G) of graphs with pendant vertices, Linear Algebra Appl. 552 (2018) 52–70. [6] G.J. Chang, L.H. Huang, H.G. Yeh, A characterization of graphs with rank 4, Linear Algebra Appl. 434 (2011) 1793–1798. [7] G.J. Chang, L.H. Huang, H.G. Yeh, A characterization of graphs with rank 5, Linear Algebra Appl. 436 (2012) 4241–4250. [8] C. Chen, J. Huang, S.C. Li, On the relation between the H-rank of a mixed graph and the matching number of its underlying graph, Linear Multilinear Algebra 66 (2018) 1853–1869. [9] B. Cheng, B. Liu, On the nullity of graphs, Electron. J. Linear Algebra 16 (2007) 60–67. [10] L. Collatz, U. Sinogowitz, Spektren endlicher Grafen, Abh. Math. Semin. Univ. Hambg. 21 (1957) 63–77. [11] D. Cvetković, I. Gutman, The algebraic multiplicity of the number zero in the spectrum of a bipartite graph, Mat. Vesnik (Beograd) 9 (1972) 141–150. [12] Y.Z. Fan, K.S. Qian, On the nullity of bipartite graphs, Linear Algebra Appl. 430 (2009) 2943–2949. [13] I. Faria, Permanental roots and the star degree of a graph, Linear Algebra Appl. 64 (1985) 255–265. [14] S. Fiorini, I. Gutman, I. Sciriha, Trees with maximum nullity, Linear Algebra Appl. 397 (2005) 245–251. [15] S.C. Gong, G.H. Xu, On the nullity of a graph with cut-points, Linear Algebra Appl. 436 (2012) 135–142. [16] J. Guo, W. Yan, Y. Yeh, On the nullity and the matching number of unicyclic graphs, Linear Algebra Appl. 431 (2009) 1293–1301. [17] I. Gutman, I. Sciriha, On the nullity of line graphs of trees, Discrete Math. 232 (2001) 35–45. [18] S.B. Hu, X.Z. Tan, B.L. Liu, On the nullity of bicyclic graphs, Linear Algebra Appl. 429 (2008) 1387–1391. [19] H.H. Li, Y.Z. Fan, L. Su, On the nullity of the line graph of unicyclic graph with depth one, Linear Algebra Appl. 437 (2012) 2038–2055. [20] H.H. Li, L. Su, H.X. Sun, On bipartite graphs which attain minimum rank among bipartite graphs with given diameter, Electron. J. Linear Algebra 23 (2012) 137–150. [21] W. Li, A. Chang, On the trees with maximum nullity, MATCH Commun. Math. Comput. Chem. 56 (2006) 501–508. [22] B. Liu, Y. Huang, S. Chen, On the characterization of graphs with pendant vertices and given nullity, Electron. J. Linear Algebra 18 (2009) 719–734. [23] Y. Lu, L.G. Wang, Q. Zhou, The rank of a signed graph in terms of the rank of its underlying graph, Linear Algebra Appl. 538 (2018) 166–186. [24] W. Luo, J. Huang, S.C. Li, On the relationship between the skew-rank of an oriented graph and the rank of its underlying graph, Linear Algebra Appl. 554 (2018) 205–223. [25] X. Ma, D. Wong, F. Tian, Nullity of a graph in terms of the dimension of cycle space and the number of pendant vertices, Discrete Appl. Math. 215 (2016) 171–176. [26] X. Ma, D. Wong, R. Tian, Skew-rank of an oriented graph in terms of matching number, Linear Algebra Appl. 495 (2016) 242–255. [27] G.R. Omidi, On the nullity of bipartite graphs, Graphs Combin. 25 (2009) 111–114. [28] Y. Song, X. Song, B. Tam, A characterization of graphs with nullity |V (G)| − 2m(G) + 2c(G), Linear Algebra Appl. 465 (2015) 363–375. [29] Y. Song, X. Song, C. Zhang, An upper bound for the nullity of a bipartite graph in terms of its maximum degree, Linear Multilinear Algebra 64 (2016) 1107–1112. [30] X. Tan, B. Liu, On the nullity of unicyclic graphs, Linear Algebra Appl. 408 (2005) 212–220. [31] L. Wang, D. Wong, Bounds for the matching number, the edge chromatic number and the independence number of a graph in terms of rank, Discrete Appl. Math. 166 (2014) 276–281. [32] D. Wong, X. Ma, F. Tian, Relation between the skew-rank of an oriented graph and the rank of its underlying graph, European J. Combin. 54 (2016) 76–86.
266
L. Wang et al. / Linear Algebra and its Applications 584 (2020) 257–266
[33] D. Wong, M. Zhu, W. Lv, A characterization of long graphs of arbitrary rank, Linear Algebra Appl. 438 (2013) 1347–1355. [34] Q. Zhou, D. Wong, D. Sun, An upper bound of the nullity of a graph in terms of order and maximum degree, Linear Algebra Appl. 555 (2018) 314–320.