Elsevier

Applied Mathematics and Computation

Volume 354, 1 August 2019, Pages 411-421
Applied Mathematics and Computation

Relationship between the rank and the matching number of a graph

https://doi.org/10.1016/j.amc.2019.02.055Get rights and content

Abstract

Given a simple graph G, let A(G) be its adjacency matrix and α′(G) be its matching number. The rank of G, written as r(G), refers to the rank of A(G). In this paper, some relations between the rank and the matching number of a graph are studied. Firstly, it is proved that 2d(G)r(G)2α(G)No, where d(G) and No are, respectively, the dimension of cycle space and the number of odd cycles of G. Secondly, sharp lower bounds on both r(G)α(G) and r(G)/α′(G) are determined. All the corresponding extremal graphs are characterized, respectively.

Introduction

We will start with introducing some background information that will lead to our main results. Some important previously established facts will also be presented.

All graphs considered in this paper are finite, undirected and simple. Let G=(VG,EG) be a graph with the vertex set VG={v1,v2,,vn} and the edge set EG. Then Gvi,Gvivj denote the graph obtained from G by deleting vertex vi ∈ VG, or edge vivj ∈ EG, respectively (this notation is naturally extended if more than one vertex or edge is deleted). For an induced subgraph H of G with vi ∈ VGVH, the induced subgraph of G with vertex set VH∪{vi} is simply written as H+vi. A vertex of a graph G is called a pendant vertex if it is a vertex of degree one in G, whereas a vertex of G is called a quasi-pendant vertex if it is adjacent to a pendant vertex in G. Denote by Pn, Cn and Kn a path, a cycle and a complete graph of order n, respectively. We follow the notation and terminology in [12] except otherwise stated.

Denote by d(G) the dimension of the cycle space of G, that is d(G)=|EG||VG|+ω(G), where ω(G) is the number of components of G. Two distinct edges in a graph G are independent if they do not have a common endvertex in G. A set of pairwise independent edges of G is called a matching, while a matching with the maximum cardinality is a maximum matching of G. The matching number of G, denoted by α′(G), is the cardinality of a maximum matching of G. A simple graph G is called acyclic if it contains no cycles, whereas it is called a k-cyclic graph if it is connected with d(G)=k. In particular, a k-cyclic graph is called a unicyclic graph (resp., bicyclic graph, tricyclic graph) if k=1 (resp., k=2,k=3). A graph is called an empty graph if it has no edges. We call v a cut-vertex of G if ω(Gv)>ω(G).

The adjacency matrix A(G) of G is an n × n matrix whose (i, j)-entry equals to 1 if vertices vi and vj are adjacent and 0 otherwise. The eigenvalues of A(G) are said to be the eigenvalues of G. The rank of G, denoted by r(G), is the rank of A(G). The nullity of G, denoted by η(G), is the number of zero eigenvalues of A(G). It is obvious that r(G)+η(G)=|VG|. A matrix is said to be nonsingular if its nullity equals to zero.

The rank of graphs has been studied intensively. Cheng and Liu [10] characterized graphs with rank 2 or 3; Chang et al. [6], [7], respectively, determined graphs with rank 4 or 5. Tan and Tan [32] computed the rank of multicyclic graphs. Bevis et al. [3] investigated the changes of the rank of a graph after vertex addition. Recently, more and more work on skew rank of oriented graphs appeared. One may be referred to [8], [19], [20], [23], [24], [36] and the references with in.

The study on nullity of graphs also attracts a number of researchers’ attention. Zhu et al. [38] characterized all the unicyclic graphs with extremal nullity. Hu et al. [18] determined the bicyclic graphs with maximum nullity. Cheng and Liu [11] identified the tricyclic graphs with maximum nullity. Fan and Qian [13] investigated the nullity set of bipartite graphs of order n, and characterized the bipartite graphs with nullity n4 and the regular bipartite graphs with nullity n6. The nullity of k-cyclic graphs were, respectively, considered in [25], [31]. The nullity of graphs with pendant vertices and pendant trees were studied in [22] and [15] respectively. For more properties and applications about the nullity or the rank of graphs, we refer the readers to [9], [21], [26], [30], [33], [35], [37] and the references therein.

The r(G) over the real field is a natural parameter associated with G, and there has been substantial interest in relating this to other graph theoretic parameters. Godsil and Royle [14] studied the relationship between the rank and the chromatic number of a graph. It is natural and interesting to consider the relation between the rank and some other parameters of graphs. In this paper, the relation between the rank and the matching number of a graph is considered. We determine sharp upper and lower bounds on r(G)2α(G) of the graph G. Furthermore, sharp lower bounds are determined for both r(G)α(G) and r(G)/α′(G). All the corresponding extremal graphs are identified, respectively.

Let G be a graph with pairwise vertex-disjoint cycles, and let CG denote the set of all cycles in G. By shrinking each cycle of G (that is, contracting each cycle to a single vertex) we obtain an acyclic graph TG from G. More definitely, the vertex set VTG=UWC, where U consists of all vertices of G that do not lie on any cycle and WC consists of all the vertices each of which is obtained by shrinking a cycle in CG; Two vertices in U are adjacent in TG if and only if they are adjacent in G, a vertex u ∈ U is adjacent to a vertex vCWC if and only if u is adjacent (in G) to a vertex in the cycle C, and vertices vC1,vC2 are adjacent in TG if and only if there exists an edge in G joining a vertex of C1CG to a vertex of C2CG.

It is clear that TG is always acyclic. Observe that the graph TGWC (obtained from TG by deleting vertices in WC and the incident edges) is the same as the graph obtained from G by deleting all the vertices on cycles and their incident edges, the resultant graph is denoted by ΓG. For example, in Fig. 1, TG is obtained from G by shrinking each cycle to a single vertex, and ΓG is obtained from G by deleting all the vertices on cycles and their incident edges. Wang and Wong [34] obtained the following bounds of the matching number.

Theorem 1.1 [34]

Let G be a graph of rank r(G). Then(r(G)d(G))/2α(G)(r(G)+2d(G))/2,where d(G) is the dimension of the cycle space of G.

Theorem 1.1 can be equivalently rewritten in the following form.

Theorem 1.2 [34]

Let G be a graph of rank r(G). Then2α(G)2d(G)r(G)2α(G)+d(G),where d(G) is the dimension of the cycle space of G.

Song et al. [29] characterized all graphs attaining the lower bound in (1.1), which reads as follows.

Theorem 1.3 [29]

For all graphs G, r(G)=2α(G)2d(G) holds if and only if the following three conditions are all satisfied:

  • (i)

    the cycles (if any) of G are pairwise vertex-disjoint;

  • (ii)

    the order of each cycle (if any) of G is a multiple of 4;

  • (iii)

    α(TG)=α(ΓG).

Rula et al. [28] characterized all graphs attaining the upper bound in (1.1), which reads as follows.

Theorem 1.4 [28]

For all graphs G, r(G)=2α(G)+d(G) holds if and only if the following three conditions are all satisfied:

  • (i)

    the cycles (if any) of G are pairwise vertex-disjoint;

  • (ii)

    each cycle of G (if any) is an odd cycle;

  • (iii)

    α(TG)=α(ΓG).

In this paper, we investigate the relationship between the rank and the matching number of a graph. Combining Theorems 1.1 and 1.3, we have the following:

Theorem 1.5

Let G be a simple graph with the dimension of cycle space d(G) and the number of odd cycles No. Then2d(G)r(G)2α(G)No.Moreover, the upper bound is best possible when the graph G is composed of the union of No disjoint odd cycles, whereas the equality on the left of (1.2) holds if and only if all the following conditions hold for G:

  • (i)

    the cycles (if any) of G are pairwise vertex-disjoint;

  • (ii)

    the order of each cycle (if any) of G is a multiple of 4;

  • (iii)

    α(TG)=α(ΓG).

Remark 1

The lower bound in (1.2) is obtained by Wang and Wong [34] and the corresponding extremal graphs were determined by Song et al. [29]. Here we identify the corresponding extremal graphs just by a new method, which is inspired by Wong et al. [36].

Let G be the graph as depicted in Fig. 1. Then G satisfies Theorem 1.5(i)–(iii) (α(TG)=α(ΓG)=2) and r(G)2α(G)=2d(G) holds, where r(G)=16,α(G)=12 and d(G)=4.

Note that that r(G)+η(G)=|VG|. If G is unicyclic, then d(G)=1. Then the following result is a direct consequence of Theorem 1.5, which can be found in [17].

Corollary 1.6 [17]

Let G be an n-vertex unicyclic graph with Cq the unique cycle contained in it. Thenn2α(G)1η(G)n2α(G)+2.The equality on the right hand holds if and only if q is a multiple of 4 and α(TG)=α(GVCq).

In our next results we establish sharp lower bounds on r(G)α(G) and r(G)/α′(G), respectively.

Theorem 1.7

Let G be a nonempty graph. Thenr(G)α(G)1d(G).The equality holds if and only if G consists of one Sn and some isolated vertices or G consists of one C4 and some isolated vertices, where Sn is an n-vertex star.

Theorem 1.8

Let G be a nonempty graph. Thenr(G)/α(G)21+d(G)with equality if and only if G is acyclic or G consists of one C4 and some isolated vertices.

In this section, we give some preliminary results, which will be used to prove our main results. The following two lemmas follow immediately from the definitions of the rank and the matching number, respectively.

Lemma 1.9

Let G be a simple graph.

  • (i)

    If H is an induced subgraph of G, then r(H) ≤ r(G);

  • (ii)

    If G1,G2,,Gt are all the connected components of G, then r(G)=i=1tr(Gi);

  • (iii)

    r(G) ≥ 0 with equality if and only if G is an empty graph.

Lemma 1.10

Let G be a simple graph. Then α(G)1α(Gv)α(G) for any v ∈ VG.

Lemma 1.11 [12]

Let Pn and Cn be a path and a cycle on n vertices, respectively. Then

  • (i)

    r(Pn)=n if n is even, and r(Pn)=n1 if n is odd;

  • (ii)

    r(Cn)=n2 if n is a multiple of 4, and r(Cn)=n otherwise.

Lemma 1.12 [23]

Let F be a simple acyclic graph with matching number α′(F). Then r(F)=2α(F).

Lemma 1.13 [10]

Let y be a pendant vertex of G,   x is the unique neighbor of y. Then r(G)=r(Gxy)+2.

The following is a result which is similar as that of Lemma 1.13; see [27].

Lemma 1.14 [27]

Let y be a pendant vertex of G and x the unique neighbor of y. Then α(G)=α(Gx)+1=α(Gxy)+1.

The following two lemmas (formulated in equivalent form) come from [16].

Lemma 1.15 [16]

Let x be a cut vertex of a graph G and H be a component of Gx. If r(H)=r(H+x), then r(G)=r(H)+r(GH).

Lemma 1.16 [16]

Let x be a cut vertex of a graph G and H be a component of Gx. If r(H)=r(H+x)2, then r(G)=r(Gx)+2.

Lemma 1.17 [3]

Let G be a graph with x ∈ VG. Then r(G)2r(Gx)r(G).

The following result characterizes the relation of the dimensions of the cycle space between G and a subgraph obtained from G by deleting one vertex. Its proof follows immediately from the definition of d(G).

Lemma 1.18 [36]

Let G be a graph with x ∈ VG. Then

  • (i)

    d(G)=d(Gx) if x lies outside any cycle of G;

  • (ii)

    d(Gx)d(G)1 if x lies on a cycle;

  • (iii)

    d(Gx)d(G)2 if x is a common vertex of distinct cycles.

The following result characterizes the relationship of the ranks between an acyclic graph and its subgraph.

Lemma 1.19 [27]

Let F be a simple acyclic graph with at least one edge. Then

  • (i)

    r(F˜)<r(F), where F˜ is the subgraph obtained from F by deleting all the pendants of F.

  • (ii)

    If r(FD)=r(F) for DVF, then there is a pendant vertex v such that vD.

By Lemmas 1.12 and 1.19, the next corollary follows immediately.

Corollary 1.20

Let F be a simple acyclic graph with at least one edge. Then

  • (i)

    α(F˜)<α(F);

  • (ii)

    If α(F)=α(FD) for DVF, then there is a pendant vertex v such that vD.

Recall that the characteristic polynomial of a graph G is defined asϕ(G;x)=det(xIA(G))=xn+a1xn1+a2xn2++an.A graph is called an elementary graph if it is either a K2 or a cycle Cq, q ≥ 3. We call N a basic graph if all of its components are elementary graphs.

Lemma 1.21 [12]

Let ω(N) be the number of components of N and c(N) the number of cycles in N. Then the coefficient ai defined in (1.4) isai=NNi(1)ω(N)·2c(N),i=1,2,,n,where Ni denotes the set of basic graphs with i vertices of G.

Section snippets

Proof for Theorem 1.5

In this section we give the proof of Theorem 1.5 according to the following steps. We first show that the inequalities in (1.2) hold. Then we show the upper bound in (1.2) is best possible. Finally, we characterize all the simple graphs G which attain the lower bound in (1.2).

Proof of Theorem 1.5

First we prove that r(G)2α(G)+No. Let ϕ(G;x)=i=0naixni be the characteristic polynomial of G. Then it follows from Lemma 1.21 that ai=0 for any i>2α(G)+No since G contains no basic graphs with i vertices if i>2α(G)+No

Proofs for Theorems 1.7 and 1.8

In this section, we give the proofs of Theorems 1.7 and 1.8. Here we only give the proof of Theorem 1.7. By a similar discussion, we may also show that Theorem 1.8 holds, which will be omitted here.

Proof of Theorem 1.7

Note that for a given non-empty graph G, it must contain K2 as its induced subgraph. By Lemma 1.9(i), we have r(G)r(K2)=2.

Together with (1.2), we can get2r(G)2α(G)2d(G)+r(G)22d(G).Hence we have r(G)α(G)1d(G), as desired.

Now we give the sufficient and necessary conditions for the equality

Concluding remark

It is well-known that the AutoGraphiX system determines classes of extremal or near-extremal graphs with a variable neighborhood search heuristic. As part of a larger study [1], the AutographiX2 (AGX2) [2], [4], [5] system was used to study the following type of problems. For each pair of invariants i1(G) and i2(G), eight bounds of the following form were considered:b̲α·i1(G)β·i2(G)b¯,where ⊕ denotes one of the operations +,,×,/,   α, β are two constants, while b and b¯ are, respectively,

Acknowledgments

The authors would like to express their sincere gratitude to the referee for his or her careful reading and insightful suggestions, which led to a number of improvements to this paper.

References (38)

  • J.M. Guo et al.

    On the nullity and the matching number of unicyclic graphs

    Linear Algebra Appl.

    (2009)
  • S.B. Hu et al.

    On the nullity of bicyclic graphs

    Linear Algebra Appl.

    (2008)
  • W.J. Luo et al.

    On the relationship between the skew-rank of an oriented graph and the rank of its underlying graph

    Linear Algebra Appl.

    (2018)
  • S.C. Li

    On the nullity of graphs with pendent vertices

    Linear Algebra Appl.

    (2008)
  • X.B. Ma et al.

    The nullity of k-cyclic graphs of ∞-type

    Linear Multilinear Algebra

    (2015)
  • X.B. Ma et al.

    Skew-rank of an oriented graph in terms of matching number

    Linear Algebra Appl.

    (2016)
  • S. Rula et al.

    The extremal graphs with respect to their nullity

    J. Inequal. Appl.

    (2016)
  • Y.Z. Song et al.

    An upper bound for the nullity of a bipartite graph in terms of its maximum degree

    Linear Multilinear Algebra

    (2016)
  • X.Z. Tan et al.

    The rank of multicyclic graphs

    Chin. Ann. Math. Ser. A

    (2014)
  • Cited by (0)

    S.L. acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 11671164, 11271149).

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