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Research article

The bounds of the energy and Laplacian energy of chain graphs

  • Received: 26 November 2020 Accepted: 24 February 2021 Published: 26 February 2021
  • MSC : 05C50, 05C09, 05C92

  • Let G be a simple connected graph of order n with m edges. The energy ε(G) of G is the sum of the absolute values of all eigenvalues of the adjacency matrix A. The Laplacian energy is defined as LE(G)=ni=1|μi2mn|, where μ1,μ2,,μn are the Laplacian eigenvalues of a graph G. In this article, we obtain some upper and lower bounds on the energy and Laplacian energy of chain graph. Finally, we attain the maximal Laplacian energy among all connected bicyclic chain graphs by comparing algebraic connectivity.

    Citation: Yinzhen Mei, Chengxiao Guo, Mengtian Liu. The bounds of the energy and Laplacian energy of chain graphs[J]. AIMS Mathematics, 2021, 6(5): 4847-4859. doi: 10.3934/math.2021284

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  • Let G be a simple connected graph of order n with m edges. The energy ε(G) of G is the sum of the absolute values of all eigenvalues of the adjacency matrix A. The Laplacian energy is defined as LE(G)=ni=1|μi2mn|, where μ1,μ2,,μn are the Laplacian eigenvalues of a graph G. In this article, we obtain some upper and lower bounds on the energy and Laplacian energy of chain graph. Finally, we attain the maximal Laplacian energy among all connected bicyclic chain graphs by comparing algebraic connectivity.



    In 2008, Bhattacharya et al. [5] and Bell et al. [4] discovered that bipartite chain graphs whose largest eigenvalues within the connected bipartite graph is maximal, and named therein as double nested graphs. After that, many scholars began to study some mathematical properties of chain graphs. Andelic et al. provide that some upper and lower bounds on index of chain graphs [3]. And Alazemi et al. proved that any chain graph has its least positive eigenvalue greater than 12 [2]. Hence Zhang et al. proposed that upper bounds on Laplacian spectral radius of chain graphs [13]. Das et al. studied the energy and Laplacian energy of chain graphs [8]. In this paper, we further study some bounds of energy and Laplacian energy of chain graphs.

    We consider finite undirected connected graphs without loops and multiple edges. Let G be a such graph with vertex set V(G)={v1,v2,,vn} and edge set E(G), where |E(G)|=m. Let di be the degree of the vertex vi for i=1,2,,n. The minimum vertex degrees of G are denoted by δ(G). Let NG(vi) be the adjacent set of the vertex vi, then di=|NG(vi)|. If G has distinct vertices vi and vj with NG(vi)=NG(vj), then vi and vj are duplicates and (vi,vj) is a duplicate pair.

    Let A(G) be the adjacency matrix of G, and rank(G) be the rank of the adjacency matrix A(G). Let λ1λ2λn the eigenvalues of A(G). We denote S(G)={λ1,λ2,,λn} as the spectrum of G. The energy of graph G is defined as [11]

    ε(G)=ni=1|λi|.

    For its basic properties and application, including various lower and upper bounds, see the [17], the recent paper [1,7,8,11,12,20] and the references cited therein.

    The Laplacian matrix of graph G is defined as L(G)=D(G)A(G), where D(G) is the diagonal matrix of vertex degrees. The matrix L(G) has non-negative eigenvalues μ1μ2μn1μn=0, and the Laplacian spectrum of graph G be denoted by LS(G)={μ1,μ2,,μn}. The Laplacian energy of G is defined as [10]

    LE(G)=ni=1|μi2mn|.

    It can also be defined as

    LE(G)=2Sσ(G)4mσn, (1.1)

    where σ(1σn) be the largest positive integer such that μσ2mn and Sk(G)=ki=1μi.

    For its basic properties, including various lower and upper bounds, see [7,8,10,18,19] and the references cited therein. The Laplacian energy found applications not only in theoretical organic chemistry [12,21], but also in image processing [22] and information theory [16].

    In the class of bipartite graphs of fixed order and size those having maximal spectral radius of adjacency/Laplacian/signless Laplacian matrix are chain graphs. Thus, they can be significant in modeling some bipartite networks with large spectral radius. Their applications involve ecological networks, in which graphs with nested properties are considered [14] and are used in some applications for economic network modeling.

    We now introduce the structure of a (connected) chain graph. The vertex set of any chain graph consists of two color classes, which are U and V. Both of them are divided into h non-empty units U1,U2,,Uh and V1,V2,,Vh, respectively. All the vertices in Us are joined by edges to all vertices in h+1sk=1Vk, for s=1,2,,h. Therefore, if uiUs+1 and ujUs, then NG(ui)NG(uj), or if viVt+1 and vjvt, then NG(vi)NG(vj).

    If ns=|Us| and ms=|Vs| for s=1,2,,h, then G is denoted by G(m1,,mh;n1,,nh), as shown in Figure 1. And

    m=m1hi=1ni+m2h1i=1ni++mhn1=hi=1aimi,
    m=n1hi=1mi+n2h1i=1mi++nhm1=hi=1bini,
    Figure 1.  Structure of G(m1,,mh;n1,,nh).

    where

    ai=h+1ik=1nk,bi=h+1ik=1mk.

    Moreover,

    n=hk=1mk+hk=1nk.

    The second smallest Laplacian eigenvalue of a graph is well known as the algebraic connectivity. It has been proved that the second smallest Laplacian eigenvalue μn1=0 if and only if G is disconnected. The algebraic connectivity is often applied in theoretical chemistry, control theory, combinatorial optimization and other fields [15].

    As usual, Kn, Kp,q(p+q=n) and K1,n1, denote, respectively, the complete graph, the complete bipartite graph and the star on n vertices. For other undefined notations and terminology from graph theory, the readers are referred to [6].

    The paper is organized as follows. In Section 2, we list some previously known results. In Section 3, we get some upper and lower bounds on ε(G) of a chain graph G. In Section 4, we establish an upper bound on LE(G) of the chain graphs in terms of vertex cover number. In Section 5, we attain the maximal Laplacian energy of the bicyclic chain graph G by comparing the algebraic connectivity.

    This section lists some known results to be used in this paper.

    Lemma 2.1. [8] Let B be a p×p real symmetric matrix and Bk be its leading k×k submatrix. Then for i=1,2,,k,

    λpi+1(B)λki+1(Bk)λki+1(B),

    where λi(B) is the i-th largest eigenvalue of B.

    Lemma 2.2. [9] Let G be a graph with vertices {v1,v2,,vk}V(G) having same set of adjacent vertices, then G has at least k1 equal eigenvalues 0.

    Lemma 2.3. [18] Let GKn. Then μn1δ(G).

    Lemma 2.4. [10] Let A and B be real symmetric matrices of order n. Then for any 1kn,

    ki=1λi(A+B)ki=1λi(A)+ki=1λi(B),

    where λi(M) denotes the i-th largest eigenvalue of the matrix M.

    Lemma 2.5. [1] If G is a connected bipartite graph of rank r, then

    ε(G)(r+1)25.

    Lemma 2.6. [11] If G is a connected bipartite graph of rank r, then

    LE(G)2(ε(G)r).

    Lemma 2.7. [8] Let GG(m1,,mh;n1,,nh) be a chain graph of order n. Then

    ε(G)2n1,

    with equation holds if and only if GK1,n1.

    Lemma 2.8. [8] Let G be a graph with vertex set V(G)={v1,v2,,vn}. If G has k1 duplicate pairs (vi,vi+1), where i=1,2,,k1, then G has at least k1 equal Laplacian eigenvalues and they are all equal to the cardinality of the neighbor set.

    Theorem 3.1. Let GG(m1,,mh;n1,,nh) be a chain graph of order n. Then

    ε(G)2hm (3.1)

    with equation holds if and only if GKn1,m1, where n1+m1=n.

    Proof. By Lemma 2.2, the eigenvalue 0 with multiplicity hi=1(ni+mi2) of A(G), and the remaining eigenvalues are the eigenvalues of the following matrix,

    C=(0000m1m2mh1mh0000m1m2mh100000m1m2000000m1000n1n2nh1nh0000n1n2nh100000n1n2000000n10000000).

    Let λ1λ2λ2h be the eigenvalues of C. Then

    ε(G)=2hi=1|λi|.

    Since G be a bipartite graph, we have λi and λi are eigenvalues of G. Thus we have

    ε(G)=2hi=1λi.

    Obviously,

    2hi=1λ2i=Tr(C2)=2hi=1hi+1j=1mjni=2m,

    that is,

    hi=1λ2i=hi=1hi+1j=1mjni=m.

    So

    ε(G)=2hi=1λ2i+21i<jhλiλj2hi=1λ2i+hi=1(h1)λ2i=2hhi=1λ2i=2hm.

    First we assume that h=1. Then GKn1,m1, where n1+m1=n. So S(G)={±m1n1,0,,0} and ε(G)=2m1n1=2m. Hence the equation holds in (3.1).

    Next we assume that h2. By the definition of chain graph, G(1,1;1,1), that is, P4 is an induced subetaaph of G. By Lemma 2.1, we get λ2(G)λ2(P4)>0. Since G is connected, by Perron-Frobenius theorem we have λ1(G)>λ2(G). Hence the inequality 21i<jhλiλjhi=1(h1)λ2i is strict. This completes the proof.

    Theorem 3.2. Let GG(m1,,mh;n1,,nh) be a chain graph of order n. Then

    ε(G)(2h+1)25. (3.2)

    Proof. By calculating the matrix C in the proof of Theorem 3.1, we get

    det(C)=(1)hhi=1mini0.

    Therefore, all the eigenvalues of matrix C are non-zero. Hence r(G)=2h. Using Lemma 2.5, we can get result in (3.2).

    In this section, we give an upper bound on LE(G) of chain graphs in terms of vertex cover number. Also, the lower bound follows from a known lower bound for Laplacian energy of any graph in terms of rank and energy.

    Theorem 4.1. Let GG(m1,,mh;n1,,nh) be a chain graph of order n, and a1b1. Then

    LE(G){2(m+b1)4mn,if2mnb1,2b1(n2)2m+8mn,if2mn<b1, (4.1)

    with equation holds if and only if GK1,n1.

    Proof. Let Γ={v11,v12,,v1m1,v21,v22,,v2m2,,vh1,vh2,,vhmh} be a vertex cover set of the graph G, where vij is the j-th vertex in Vi. Hence {vi1,vi2,,vimi}Vi. We can assume that Gij are spanning subetaaphs of G such that V(G)=V(Gi1)=V(Gi2)==V(Gimi), and the edge set of Gij is defined as

    E(Gij)={vijUk:UkNG(vij)}.

    Since |NG(vi1)|=|NG(vi2)|==|NG(vimi)|=ai,

    Gij=K1,ai(nai1)K1,

    we have

    E(Kmi,ai)=E(Gi1)E(Gi2)E(Gim1),

    so

    L(Kmi,ai)=L(Gi1)+L(Gi2)++L(Gim1),i=1,2,,h.

    By Figure 1,

    E(G)=E(Km1,a1)E(Km2,a2)E(Kmi,ai),

    then we can see easily that

    L(G)=L(Km1,a1)+L(Km2,a2)++L(Kmi,ai).

    Note that

    Sk(Gi1)=Sk(Gi2)==Sk(Gimi)ai+k,

    where Sk(G) is the sum of the k largest Laplacian eigenvalues of graph G.

    By Lemma 2.4, we get

    Sk(G)m1Sk(G11)+m2Sk(G21)++mhSk(Gh1)m1(a1+k)+m2(a2+k)++mh(ah+k)=hi=1miai+khi=1mi=m+kb1.

    So from (1.1), we get

    LE(G)=2Sσ(G)4mσn2(m+σb1)4mσn=2m+2σ(b12mn).

    Since G is connected, 1σn1. So it suffices to consider the following two cases.

    Case1. 2mnb1.

    Then we have

    LE(G)2m+2b14mn=2(m+b1)4mn.

    Case2. 2mn<b1.

    By Lemma 2.3, we get μn1δ(G)2mn. Thus it must be 1σn2. Hence

    LE(G)2m+2(n2)(b12mn)=2b1(n2)2m+8mn.

    Next we prove that the equality holds.

    If GK1,n1, we get b1=m1=1,n1=n1, and S(G)={0,1n2,n}. Then

    LE(K1,n1)=ni=1|μi2mn|=2n4(n1)n=2(m+b1)4mn.

    Theorem 4.2. Let GG(m1,,mh;n1,,nh) be a chain graph of order n. Then

    LE(G)4(n1h). (4.2)

    Proof. By Theorem 3.2, we get r(G)=2h. Using Lemmas 2.6 and 2.7, we get result in (4.2).

    Let G be a connected bicyclic chain graph. We have m=n+1, and h=2 or h=3. If h=2, then GG(1,1;3,n5) or GG(1,2;2,n5). If h=3, then GG(1,2,k3;1,1,nk2), where 4kn3 (Figure 2). In this section, we will attain the maximal Laplacian energy of all connected bicyclic chain graphs.

    Figure 2.  Graphs G(1,1;3,n5), G(1,2;2,n5) and G(1,2,k3;1,1,nk2).

    Lemma 5.1. Let G be a connected bicyclic chain graph (n8).

    (1) If GG(1,1;3,n5), then LE(G)=6+2(n4)(n+1)n2μn1.

    (2) If GG(1,2;2,n5), then LE(G)=10+2(n6)(n+1)n2μn1.

    (3) If GG(1,2,k3;1,1,nk2), where 4kn3, then LE(G)=10+2(n6)(n+1)n2μn1.

    Proof. (1) Let GG(1,1;3,n5). By Lemma 2.8, we conclude that 2,2,1,1,,1n6 are the Laplacian eigenvalues of G and the remaining Laplacian eigenvalues of G are satisfying the equation f1(x)=0, where f1(x) is the characteristic polynomial of the matrix

    A1=(n2035n033011201001),

    that is, f1(x)=x(x3(4+n)x2+(5n2)x3n).

    Let h1(x)=x3(4+n)x2+(5n2)x3n. Then we obtain h1(0)=3n<0, h1(1)=n5>0, h1(2)=3n12>0, h1(n1)=3<0 and limxh1(x)=. Thus the Laplacian eigenvalues of G are μ1,μ2,2,2,1,1,,1n6,μn1,0, where μ1n1, 2μ2n1, μn1<1 and μ1+μ2+μn1=n+4.

    Therefore

    LE(G)=ni=1|μi2(n+1)n|=6+2(n4)(n+1)n2μn1. (5.1)

    (2) Let GG(1,2;2,n5). By Lemma 2.8, we conclude that 3,2,1,1,,1n6 are the Laplacian eigenvalues of G and the remaining Laplacian eigenvalues of G are satisfying the equation f2(x)=0, where f2(x) is the characteristic polynomial of the matrix

    A2=(n3025n022012301001),

    that is, f2(x)=x(x3(3+n)x2+(5n8)x2n).

    Let h2(x)=x3(3+n)x2+(5n8)x2n. Then we obtain h2(0)=2n<0, h2(1)=2n10>0, h2(3)=4n24>0, h2(n2)=4<0 and limxh2(x)=. Thus the Laplacian eigenvalues of G are μ1,μ2,3,2,1,1,,1n6,μn1,0, where μ1n2, 3μ2n2, μn1<1 and μ1+μ2+μn1=n+3.

    Therefore

    LE(G)=ni=1|μi2(n+1)n|=10+2(n6)(n+1)n2μn1. (5.2)

    (3) Let GG(1,2,k3;1,1,nk2). When 4kn2, by Lemma 2.8, we conclude that 2,1,1,,1n7 are the Laplacian eigenvalues of G and the remaining laplacian eigenvalues of G are satisfying equation f3(x)=0, where f3(x) is the characteristic polynomial of the matrix

    A3=(nk00112+kn020110001100123kk00120030100001),

    that is

    f3(x)=x(x1)(x4(n+6)x3+(kn+5nk2+10)x2(4kn+5n4k2+12)x+6n). (5.3)

    Let g(x)=x4(n+6)x3+(kn+5nk2+10)x2(4kn+5n4k2+12)x+6n. Then we obtain g(0)=6n>0, g(1)=3k23kn+5n7<0, g(2)=4(k2)(2+kn)<0, g(k)=(k2)(k3)(2kn)0. Since when n is odd, g(x) is same for k=n2 and k=n2, we take a smaller value k=n2. g(nk)=(2+kn)(2kn)(n+3+k)0 and limxg(x)=. Thus the Laplacian eigenvalues of G are μ1,μ2,μ3,2,1,1,,1n7,μn1,0, where μ1nk, kμ2nk, 2<μ3<k, μn1<1.

    Since ni=1μi=2m=2(n+1)=2n+2, we get μ1+μ2+μ3+μn1=n+6, that is, μ1+μ2+μ3=n+6μn1.

    Therefore

    LE(G)=ni=1|μi2(n+1)n|=10+2(n6)(n+1)n2μn1. (5.4)

    When n2<k<n3, letting k=nk in the Eq (5.3) we get the same characteristic polynomial, so it is equal to the Laplacian energy when 4kn2.

    When k=n3, f3(x)=x(x1)(x3)(x3(3+n)x2+(5n8)x2n), so it is equal to the Laplacian energy of G(1,2;2,5).

    This completes the proof.

    Lemma 5.2. Let Gn,kG(1,2,k3;1,1,nk2), where 4kn2. Then μn1(Gn,k)μn1(G(1,2,n23;1,1,n22)), with equation holds if and only if k=n2. In particular, if n is odd, then μn1(G(1,2,n23;1,1,n22))=μn1(G(1,2,n24;1,1,n21)).

    Proof. If k=n2, then μn1(Gn,k)=μn1(G(1,2,n23;1,1,n22)). By Lemma 5.1, we obtain that μ1,μ2,μ3,μn1 are the roots of the equation P(Gn,k,x)=0, where

    P(Gn,k,x)=x4(n+6)x3+(kn+5nk2+10)x2(4kn+5n4k2+12)x+6n,

    and μ1nk, kμ2nk, 2<μ3<k, μn1<1.

    We need to prove that

    μn1(Gn,k)>μn1(G(1,2,n23;1,1,n22)),for4kn21.

    Since

    P(Gn,k+1,x)P(Gn,k,x)=x(x4)(n2k1),for0<x<1,

    we get P(Gn,k+1,x)P(Gn,k,x)0. Hence P(Gn,k+1,x)P(Gn,k,x). So when n is odd and k=n21, the equation holds.

    Thus we have μn1(Gn,k)>μn1(Gn,k+1), that is,

    μn1(Gn,4)>μn1(Gn,5)>>μn1(Gn,n21)μn1(Gn,n2). (5.5)

    Hence μn1(Gn,k)>μn1(Gn,n2)=μn1(G(1,2,n23;1,1,n22)).

    This completes the proof.

    Lemma 5.3. Let G be a bicyclic graph of order n8. Then μn1(G(1,2;2,n5))>μn1(G(1,2,n23;1,1,n22)).

    Proof. When k=3, we get P(Gn,k,x)=f2(x), that is μn1(Gn,3)=μn1(G(1,2;2,n5)).

    By Lemma 5.2, we have P(Gn,k+1,x)P(Gn,k,x), and P(Gn,4,x)P(Gn,3,x) still hold.

    By inequation (5.5), we obtain

    μn1(Gn,3)>μn1(Gn,4)>>μn1(Gn,n21)μn1(Gn,n2).

    Hence μn1(G(1,2;2,n5))>μn1(Gn,n2)=μn1(G(1,2,n23;1,1,n22)) for n8.

    Lemma 5.4. Let G be a bicyclic graph of order n8. Then μn1(G(1,1;3,n5))μn1(G(1,2,n23;1,1,n22))>2n.

    Proof. For n=8 and n=9, it can be verified by using Maple.

    Let n=8, μn1(G(1,1;3,n5))=0.8377 and μn1(G(1,2,n23;1,1,n22))=0.5858. Then μn1(G(1,1;3,n5))μn1(G(1,2,n23;1,1,n22))=0.2519>14, so the conclusion is correct.

    Let n=9, μn1(G(1,1;3,n5))=0.8169 and μn1(G(1,2,n23;1,1,n22))=0.5344. Then μn1(G(1,1;3,n5))μn1(G(1,2,n23;1,1,n22))=0.2825>29, so the conclusion is correct.

    Next we prove when n10, the inequality holds.

    By Lemma 5.3, we get μn1(G(1,2;2,n5))μn1(G(1,2,n23;1,1,n22)), so we can prove μn1(G(1,1;3,n5))μn1(G(1,2;2,n5))>2n. Let α=μn1(G(1,1;3,n5)), β=μn1(G(1,2;2,n5)). Then it is satisfying

    h1(x)=x3(4+n)x2+(5n2)x3nandh1(α)=0.
    h2(x)=x3(3+n)x2+(5n8)x2nandh2(β)=0.

    By the implicit function existence theorem and Figure 3, when GG(1,1;3,n5), the relation between the decreases of α and the increase of n, and h1(x) is monotonically increasing on the interval [0,1]. Hence h1(0.81)=3.713+0.39n>0, h1(0.69)=2.9560.26n<0, so 0.69<α<0.81.

    Figure 3.  h1(x) (thin line) and h2(x) (thick line).

    Similarly, h2(0.58)=5.454+0.56n>0, h2(0.43)=3.9150.035n<0, so 0.43<β<0.58. Therefore, αβ>0.11>219, that is, when n19, hence the conclusion is correct.

    When 10n18, αβ>2n is obvious. The results are shown in Table 1.

    Table 1.  The correlation between αβ and 2n.
    n α β αβ 2n
    10 0.8107 0.5735 0.2372 0.200
    11 0.7899 0.5566 0.2333 0.182
    12 0.7804 0.5438 0.2366 0.167
    13 0.7728 0.5332 0.2396 0.154
    14 0.7666 0.5248 0.2418 0.143
    15 0.7612 0.5176 0.2436 0.133
    16 0.7566 0.5116 0.2450 0.125
    17 0.7526 0.5064 0.2462 0.118
    18 0.7491 0.5020 0.2471 0.111

     | Show Table
    DownLoad: CSV

    So we conclude that when n8,

    μn1(G(1,1;3,n5))μn1(G(1,2,n23;1,1,n22))>2n.

    Theorem 5.1. Let G be a connected bicyclic chain graph of order n8. Then G(1,2,n23;1,1,n22) attains the maximal Laplacian energy. In particular, when n is odd, LE(G(1,2,n23;1,1,n22))=LE(G(1,2,n24;1,1,n21)).

    Proof. By Lemma 5.1, we can attain the maximal Laplacian energy by comparing μn1 in equations (5.1), (5.2) and (5.4). It is obvious that LE(G(1,2;2,n5))<LE(G(1,2,n23;1,1,n22)). In particular, when n is odd, LE(G(1,2,n23;1,1,n22))=LE(G(1,2,n24;1,1,n21)). So

    LE(G(1,2,n23;1,1,n22))LE(G(1,1;3,n5))=10+2(n6)(n+1)n2μn1(G(1,2,n23;1,1,n22))62(n4)(n+1)n+2μn1(G(1,1;3,n5))=2(μn1(G(1,1;3,n5))μn1(G(1,2,n23;1,1,n22)))4n.

    Hence by Lemma 5.4, LE(G(1,2,n23;1,1,n22))LE(G(1,1;3,n5))>0, that is, LE(G(1,2,n23;1,1,n22))>LE(G(1,1;3,n5)). In conclusion, we get G(1,2,n23;1,1,n22) has the maximal Laplacian energy among all connected bicyclic chain graphs (n8).

    In this paper, we introduced the definition of chain graph. We obtain some bounds on ε(G) of the chain graphs. Since the rank of the chain graphs is 2h, we can get some bounds on ε(G) and LE(G) of the chain graphs. We present the upper bound on LE(G) of the chain graphs in terms of vertex cover number. In order to attain the maximal Laplacian energy of bicyclic chain graphs, we compare algebraic connectivity of each kind of bicyclic chain graphs. The problem is still open to discuss what chain graphs give the maximal Laplacian energy for given n and whether it is still related to algebraic connectivity.

    This work was supported by National Nature Science Foundation of China (Grant No. 61774137). The authors express their sincere thanks to the anonymous referee for many valuable comments and suggestions.

    The authors declare that they have no conflict of interest in this paper.



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