Research article

Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity

  • In this paper, we establish generalized fractional versions of Hermite-Hadamard inequalities for exponentially (α, h - m)-convex functions, exponentially (h - m)-convex functions and exponentially (α, m)-convex functions. These inequalities arise when using the generalized fractional integral operators containing Mittag-Leffler function via a monotonically increasing function. The presented results hold at the same time for various kinds of convexities and well-known fractional integral operators. Moreover, the established inequalities reproduce several known results which are part of the existing literature.

    Citation: Hengxiao Qi, Muhammad Yussouf, Sajid Mehmood, Yu-Ming Chu, Ghulam Farid. Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity[J]. AIMS Mathematics, 2020, 5(6): 6030-6042. doi: 10.3934/math.2020386

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  • In this paper, we establish generalized fractional versions of Hermite-Hadamard inequalities for exponentially (α, h - m)-convex functions, exponentially (h - m)-convex functions and exponentially (α, m)-convex functions. These inequalities arise when using the generalized fractional integral operators containing Mittag-Leffler function via a monotonically increasing function. The presented results hold at the same time for various kinds of convexities and well-known fractional integral operators. Moreover, the established inequalities reproduce several known results which are part of the existing literature.


    A molecular graph in chemical graph theory is the graphical representation of the structural formula of a chemical compound in which the vertices represent atoms and edges represent chemical bond between those atoms. A topological index of a molecular graph G is a real number which characterizes the topology of G. Also it is invariant under graph automorphism. Topological indices have been widely used in Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR) studies. It has application in many folds, to name a few areas, biochemistry, nanotechnology, pharmacology. Bond energy is a measure of bond strength of a chemical compound. The distance between two atoms is considered as the bond length between them. The higher the bond energy, the smaller is the bond length between those atoms. The recently introduced 2-degree based topological invariants, analogous to novel graph invariants (Zagreb indices), namely leap Zagreb indices, may be applied in studying such bond energy between atoms in a molecular graph of a chemical compound.

    Throughout this paper, G=(V,E) represents a connected molecular graph with the vertex set V(G) and the edge set E(G). Let the number of vertices and edges of G be n and m respectively. The degree of a vertex v in G is the number of vertices adjacent to v in G and denoted by deg(v:G). The 2-degree (or the second-degree) of a vertex v in G is the number of vertices which are at distance two from v in G and denoted by d2(v:G). The Zagreb indices, namely, the first and second Zagreb indices, are the most important and oldest molecular structure descriptors. These indices have been studied extensively in the field of Mathematical Chemistry [3,4,5]. Recently, the concept of Forgotten topological index also known as F-index have attracted many researchers which results in over 100 research articles related to F-index. A.M.Naji et al. [13] have recently introduced and studied some properties of a new topological invariant called Leap Zagreb indices. They are defined as follows:

    Definition 1. (ⅰ) The first leap Zagreb index LM1(G) of a graph G is equal to the sum of squares of the second degrees of the vertices, LM1(G)=uV(G)d2(u)2.

    (ⅱ) The second leap Zagreb index LM2(G) of a graph G is equal to the sum of the products of the second degrees of pairs of adjacent vertices, LM2(G)=uvE(G)d2(u)d2(v).

    (ⅲ) The third leap Zagreb index LM3(G) of a graph G is equal to the sum of the products of the degree with the second degree of every vertex in G, LM3(G)=uV(G)deg(u)d2(u)

    Subsequently, Z. Shao et al. [18] generalized the results of Naji et al.[13] for trees and unicyclic graphs and determined upper and lower bounds on leap Zagreb indices and characterized extremal graphs. Basavanagoud et al.[2] computed exact values for first and second leap hyper Zagreb indices of some nano structures. V. R. Kulli [7,8,9] introduced and studied various leap indices. Shiladhar et al.[17] computed leap Zagreb indices of wind mill graphs. Most recently, Naji et al.[14] have studied some properties of leap graphs.

    Azari et al.[1] found formulae for first and second Zagreb indices of bridge and chain graphs. Nilanjan De [15,16] computed F-index and hyper Zagreb index of bridge and chain graphs. Jerline et al. [6] obtained exact values for harmonic index of bridge and chain graphs. E. Litta et al. [10] worked on modified Zagreb indices of bridge graphs. Mohanad Ali et al. [11] computed F-leap index of some special classes of bridge and chain graphs. Zhang et al.[12] worked on Edge-Version Atom-Bond Connectivity and Geometric Arithmetic Indices of generalized bridge molecular graphs. Motivated by their results, we compute exact values for the first and third leap Zagreb indices of bridges and chain graphs. Also we discuss some applications related to these indices in the last section of this paper. First, we recall the definitions of bridge and chain graphs from [1] as follows:

    Definition 2. Let {Gi}di=1 be a set of finite pairwise disjoint graphs with distinct vertices viV(Gi). The bridge graph B1=B1(G1,G2,,Gd;v1,v2,v3,,vd) of {Gi}di=1 with respect to the vertices {vi}di=1 as shown in Figure 1, is the graph obtained from the graphs G1,G2,,Gd by connecting the vertices vi and vi+1 by an edge for all i=1,2,,d1.

    Figure 1.  The bridge graph B1.

    Definition 3. The bridge graph B2=B2(G1,G2,,Gd;v1,w1,v2,w2,,vd,wd) of {Gi}di=1 with respect to the vertices {vi,wi}di=1 as shown in Figure 2, is the graph obtained from the graphs G1,G2,G3,,Gd by connecting the vertices wi and vi+1 by an edge for all i=1,2,,d1.

    Figure 2.  The bridge graph B2.

    Definition 4. The chain graph C=C(G1,G2,,Gd;v1,w1,v2,w2,,vd,wd) of {Gi}di=1 with respect to the vertices {vi,wi}di=1 as shown in Figure 3, is the graph obtained from the graphs G1,G2,,Gd by identifying the vertices wi and vi+1 for all i=1,2,,d1.

    Figure 3.  The chain graph C.

    The following lemma gives the 2-degree of any arbitrary vertex in the bridge graph B1.

    Lemma 5. Let G1,G2,,Gd be d5 connected graphs. Then the 2-degree of any arbitrary vertex u in the bridge graph B1 formed by these graphs is as follows:

    d2(u:B1)={ν1+μ2+1,ifu=v1νd+μd1+1,ifu=vdν2+μ1+μ3+1,ifu=v2νd1+μd+μd2+1,ifu=vd1νi+μi1+μi+1+2,,ifu=vi,3id2d2(u:G1)+1,ifuNG1(v1)d2(u:Gd)+1,ifuNGd(vd)d2(u:Gi)+2,ifuNGi(vi),2id1d2(u:Gi),ifuV(Gi)NGi[vi],1id, (2.1)

    where νi=d2(vi:Gi) and μi=deg(vi:Gi),1id.

    Next, we compute the first leap Zagreb index of the type-Ⅰ bridge graph B1.

    Let Si=uNGi(vi)d2(u:Gi), 1id.

    Theorem 6. LM1(B1)=di=1LM1(Gi)+d1i=2[(μi1+μi+1+1)2+2νi(μi1+μi+1+1)+4Si+8μi]+2d2i=3νi+2(S1+Sd)+(μ1+μd2μ32μd2)+(μ2+1)(μ2+2ν1+1)+(μd1+1)(μd1+2νd+1)+3d12.

    Proof. By virtue of Lemma 5

    LM1(B1)=uV(B1)d2(u:B1)2=(ν1+μ2+1)2+(νd+μd1+1)2+(ν2+μ1+μ3+1)2+(νd1+μd+μd2+1)2+d2i=3(νi+μi1+μi+1+2)2+uNG1(v1)(d2(u:G1)+1)2+uNGd(vd)(d2(u:Gd)+1)2+d1i=2uNGi(vi)(d2(u:Gi)+2)2+di=1uV(Gi)NGi[vi]d2(u:Gi)2
          =ν21+(μ2+1)2+2ν1(μ2+1)+ν2d+(μd1+1)2+2νd(μd1+1)+ν22+(μ1+μ3+1)2+2ν2(μ1+μ3+1)+ν2d1+(μd+μd2+1)2+2νd1(μd+μd2+1)+d2i=3[(νi+1)2+2(νi+1)(μi1+μi+1+1)+(μi1+μi+1+1)2]+uNG1(v1)[d2(u:G1)2+2d2(u:G1)]+μ1+uNGd(vd)[d2(u:Gd)2+2d2(u:Gd)]+μd+d1i=2uNGi(vi)[d2(u:Gi)2+4d2(u:Gi)]+4d1i=2μi+di=1uV(Gi)NGi[vi]d2(u:Gi)2
    =di=1LM1(Gi)+d1i=2[(μi1+μi+1+1)2+2νi(μi1+μi+1+1)+4Si+8μi]+2d2i=3νi+2(S1+Sd)+(μ1+μd2μ22μ32μd22μd1)+(μ2+1)(μ2+2ν1+1)+(μd1+1)(μd1+2νd+1)+3d12.

    Thus the result follows.

    Corollary 7. If G1=G2==Gd=G in a bridge graph B1, then LM1(B1)=dLM1(G)+(4d6)μ2+(4d8)ν+(12d26)μ+(4d4)(νμ+S)+4d12, where S=uNG(v)d2(u:G).

    Lemma 8. [1] The degree of an arbitrary vertex u of the bridge graph B1,d5 is given by:

    deg(u:B1)={μ1+1,ifu=v1μd+1,ifu=vdμi+2,ifu=vi,2id1deg(u:Gi),ifuV(Gi){vi},1id, (2.2)

    where μi=deg(vi:Gi),1id.

    Next, we compute the third leap Zagreb index of the type-Ⅰ bridge graph B1 Let us denote si=uNGi(vi)deg(u:Gi), 1id.

    Theorem 9. LM3(B1)=di=1LM3(Gi)+(s1+sd)+2d1i=2si+di=1(2νi+6μi)+2di=2(μi1μi)2(μ2+μd1)(ν1+νd)3(μ1+μd)+4d10.

    Proof. By virtue of Lemma 5 and 8

    LM3(B1)=uv(B1)d2(u)deg(u)=(ν1+μ2+1)(μ1+1)+(ν2+μ1+μ3+1)(μ2+2)+(νd+μd1+1)(μd+1)+(νd1+μd+μd2+1)(μd1+2)+d2i=3(νi+μi1+μi+1+2)(μi+2)+uNG1(v1)(d2(u:G1)+1)(deg(u:G1))+uNGd(vd)(d2(u:Gd)+1)(deg(u:Gd))+d1i=2uNGi(vi)(d2(u:Gi)+2)(deg(u:Gi))+di=1uV(Gi)NGi[vi](d2(u:Gi))(deg(u:Gi))
    =(ν1μ1+ν1+μ2μ1+μ2+μ1+1)+(ν2μ2+2ν2+μ1μ2+2μ1+μ3μ2+2μ3+μ2+2)+(νdμd+νd+μd1μd+μd1+μd+1)+(νd1μd1+2νd1+μdμd1+2μd+μd2μd1+2μd2+μd1+2)+d2i=3(νiμi+2νi+μi1μi+2μi1+μi+1μi+2μi+1+2μi+4)+uNG1(v1)(d2(u:G1)deg(u:G1)+deg(u:G1))+uNGd(vd)(d2(u:Gd)deg(u:Gd)+deg(u:Gd))+d1i=2uNGi(vi)(d2(u:Gi)deg(u:Gi)+2deg(u:Gi))+di=1uV(Gi)NGi[vi]d2(u:Gi)deg(u:Gi)

    Thus the result follows.

    Corollary 10. If G1=G2==Gd=G in a bridge graph B1, then LM3(B1)=dLM3(G)+2(d1)(s+ν+μ2)+2μ(3d5)+4d10, where s=uNG(v)deg(u:G).

    For any two nonempty sets A and B, AΔB denotes the symmetric difference of A and B and defined as AΔB=(AB)(BA)=(AB)(AB). First, we obtain the 2-degree of any arbitrary vertex in the type-Ⅱ bridge graph B2 as follows:

    Lemma 11. Let G1,G2,,Gd be d5 triangle free connected graphs. Then 2-degree of any arbitrary vertex u in the bridge graph B2 formed by these graphs is as follows:

    d2(u:B2)={d2(u:G1),ifuV(G1)NG1[w1]d2(u:G1)+1,ifuNG1(w1)d2(u:Gi),ifuV(Gi){NGi[vi]NGi[wi]},2id1d2(u:Gd),ifuV(Gd)NGd[vd]d2(u:Gd)+1,ifuNGd(vd)d2(u:Gi)+1,ifu(NGi(vi)ΔNGi(wi)),2id1d2(u:Gi)+2,ifuNGi(vi)NGi(wi),2id1δi+μi+1,ifu=wi,1id1νi+λi1,ifu=vi,2id. (2.3)

    where νi=d2(vi:Gi),μi=deg(vi:Gi);2id,δi=d2(wi:Gi),λi=deg(wi:Gi);1id1.

    Next, we compute the first leap Zagreb index of type-Ⅱ bridge graph B2.

    Let us denote S1=uNG1(w1)d2(u:G1) and Sd=uNGd(vd)d2(u:Gd)

    Theorem 12. LM1(B2)=di=1LM1(Gi)+2(S1+Sd)+(λ1+μd)+d1i=2uNGi(vi)ΔNGi(wi)[2d2(u:Gi)+1]+4d1i=2uNGi(vi)NGi(wi)[d2(u:Gi)+1]+d1i=1(μ2i+1+2δiμi+1)+di=2(λ2i1+2νiλi1).

    Proof.

    LM1(B2)=uV(B2)d2(u:B2)2=uV(G1)NG1[w1]d2(u:G1)2+d1i=2  uV(Gi){NGi[vi]NGi[wi]}d2(u:Gi)2+uV(Gd)NGd[vd]d2(u:Gd)2+uNG1(w1)(d2(u:G1)+1)2+d1i=2uNGi(vi)ΔNGi(wi)(d2(u:Gi)+1)2+uNGd(vd)(d2(u:Gd)+1)2+d1i=2uNGi(vi)NGi(wi)(d2(u:Gi)+2)2+d1i=1(δi+μi+1)2+di=2(νi+λi1)2
    =LM1(G1)δ21uNG1(w1)d2(u:G1)2+d1i=2[uV(Gi)d2(u:Gi)2uN(vi)N(wi)d2(u:Gi)2ν2iδ2i]+LM1(Gd)ν2duNGd(vd)d2(u:Gd)2+uNG1(w1)d2(u:G1)2+2uNG1(w1)d2(u:G1)+λ1+d1i=2[uNGi(vi)ΔNGi(wi)[d2(u:Gi)2+2d2(u:Gi)+1]]+uNGd(vd)[d2(u:Gd)2+2d2(u:Gd)+1]+d1i=2[uNGi(vi)NGi(wi)[d2(u:Gi)2+4d2(u:Gi)+4]]+d1i=1[δ2i+2δiμi+1+μ2i+1]+di=2[ν2i+2νiλi1+λ2i1]

    Thus,

    LM1(B2)=di=1LM1(Gi)+2(S1+Sd)+(λ1+μd)+d1i=2uNGi(vi)ΔNGi(wi)[2d2(u:Gi)+1]+4d1i=2uNGi(vi)NGi(wi)[d2(u:Gi)+1]+d1i=1(μ2i+1+2δiμi+1)+di=2(λ2i1+2νiλi1).

    Corollary 13. If G1=G2=,Gd=G, in a bridge graph B2, then LM1(B2)=dLM1(G)+λ+μ+2(S+S)+(d2)uNG(v)ΔNG(w)(2d2(u:G)+1)+4(d2)uNG(v)NG(w)(d2(u:G)+1)+(d1)[μ2+λ2]+2(d1)[δμ+νλ], where S=uNG(w)d2(u:G) and S=uNG(v)d2(u:G).

    In what follows next, we compute the third leap Zagreb index of B2.

    Lemma 14. The degree of an arbitrary vertex u of the bridge graph B2, d5 is given by:

    deg(u:B2)={deg(u:G1),ifuV(G1){w1}deg(u:Gd),ifuV(Gd){vd}deg(u:Gi),ifuV(Gi){vi,wi},2id1λi+1,ifu=wi,1id1μi+1,ifu=vi,2id. (2.4)

    where μi=deg(vi:Gi);2id,λi=deg(wi:Gi);1id1.

    Theorem 15. LM3(B2)=di=1LM3(Gi)+uNG1(w1)deg(u:G1)+uNGd(vd)deg(u:Gd)+d1i=2uNGi(wi)NGi(vi)deg(u:Gi)+d1i=2uNGi(vi)NGi(wi)deg(u:Gi)+d1i=2uNGi(vi)NGi(wi)2deg(u:Gi)+d1i=12μi+1λi+d1i=1μi+1+di=2λi1+di=1(δi+νi)ν1δd.

    Proof. By virtue of Lemma 2.7 and 2.10

    LM3(B2)=uV(B2)d2(u)deg(u)=uV(G1)NG1[w1]d2(u:G1)deg(u:G1)+d1i=2uV(Gi){NGi[vi]NGi[wi]}d2(u:Gi)deg(u:Gi)+uV(Gd)NGd[vd]d2(u:Gd)deg(u:Gd)+uNG1(w1)(d2(u:G1)+1)(deg(u:G1))+d1i=2uNGi(wi)NGi(vi)(d2(u:Gi)+1)(deg(u:Gi))+d1i=2uNGi(vi)NGi(wi)(d2(u:Gi)+1)(deg(u:Gi))+uNGd(vd)(d2(u:Gd)+1)(deg(u:Gd))+d1i=2uNGi(vi)NGi(wi)(d2(u:Gi)+2)(deg(u:Gi))+d1i=1(δi+μi+1)(λi+1)+di=2(νi+λi1)(μi+1)

    Thus the result follows.

    Corollary 16. If G1=G2==Gd=G in a bridge graph B2, then LM3(B2)=dLM3(G)+uNG(w)deg(u:G)+uNG(v)deg(u:G)+(d2)(uNG(w)NG(v)deg(u:G)+uNG(v)NG(w)deg(u:G)+uNG(v)NG(w)2deg(u:G))+(d1)(2μλ+μ+λ)+d(δ+ν)(ν+δ).

    In the following lemma, we obtain the 2-degree of any vertex in the chain graph C.

    Lemma 17. Let G1,G2,,Gd, d5 be C3-free connected graphs and let C=C(G1,G2,,Gd;w1,v2,w2,v3,,wd1,vd) be the chain graph formed using these graphs. Then the 2-degree of any vertex u in C is given as follows:

    d2(u:C)={d2(u:G1),ifuV(G1)NG1[w1]d2(u:G1)+μ2,ifuNG1(w1)d2(u:Gd),ifuV(Gd)NGd[vd]d2(u:Gd)+λd1,ifuNGd(vd)d2(u:Gi),ifuV(Gi){NGi[wi]NGi[vi]},2id1d2(u:Gi)+μi+1,ifuNGi(wi)NGi(vi),2id1d2(u:Gi)+λi1,ifuNGi(vi)NGi(wi),2id1d2(u:Gi)+λi1+μi+1,ifuNGi(vi)NGi(wi),2id1δi+νi+1,ifu=wi=vi+1,1id1, (2.5)

    where νi=d2(vi:Gi),μi=deg(vi:Gi),λi=deg(wi:Gi) and δi=d2(wi:Gi) for all 1id.

    Now, we compute the first leap Zagreb index of the chain graph C by applying Lemma 17.

    Theorem 18. For the chain graph C,

    LM1(C)=di=1LM1(Gi)+uNG1(w1)[2μ2d2(u:G1)+μ22]+uNGd(vd)[2λd1d2(u:Gd)+λ2d1]+d1i=2  uNGi(wi)NGi(vi)[2μi+1d2(u:Gi)+μ2i+1]+d1i=2  uNGi(vi)NGi(wi)[2λi1d2(u:Gi)+λ2i1]+2d1i=2  uNGi(vi)NGi(wi)[λi1d2(u:Gi)+μi+1d2(u:Gi)+λi1μi+1]+d1i=2  uNGi(vi)NGi(wi)(λ2i1+μ2i+1)+2d1i=1δiνi+1.

    Proof. By Lemma 17, we have

    LM1(C)=uV(C)d2(u:C)2=uV(G1)NG1[w1]d2(u:G1)2+uNG1(w1)[d2(u:G1)+μ2]2+uV(Gd)NGd[vd]d2(u:Gd)2+uNGd(vd)[d2(u:Gd)+λd1]2+d1i=2  uV(Gi){NGi[vi]NGi[wi]}d2(u:Gi)2+d1i=2  uNGi(wi)NGi(vi)[d2(u:Gi)+μi+1]2+d1i=2  uNGi(vi)NGi(wi)[d2(u:Gi)+λi1]2+d1i=2  uNGi(vi)NGi(wi)[d2(u:Gi)+λi1+μi+1]2+d1i=1[δi+νi+1]2
    =LM1(G1)uNG1(w1)[d2(u:G1)2]δ21+uNG1(w1)[d2(u:G1)2+2d2(u:G1)μ2+μ22]+LM1(Gd)uNGd(vd)d2(u:Gd)2ν2d+uNGd(vd)[d2(u:Gd)2+2λd1d2(u:Gd)+λ2d1]+d1i=2uV(Gi)d2(u:Gi)2d1i=2  uNGi[vi]NGi[wi]d2(u:Gi)2+d1i=2  uNGi(wi)NGi(vi)[d2(u:Gi)2+2μi+1d2(u:Gi)+μ2i+1]+d1i=2  uNGi(vi)NGi(wi)[d2(u:Gi)2+2λi1d2(u:Gi)+λ2i1]+d1i=2uNGi(vi)NGi(wi)[d2(u:Gi)2+2λi1d2(u:Gi)+2μi+1d2(u:Gi)+2λi1μi+1+λ2i1+μ2i+1]+d1i=1[δ2i+ν2i+1]+2d1i=1δiνi+1
    =di=1LM1(Gi)+uNG1(w1)[2μ2d2(u:G1)+μ22]+uNGd(vd)[2λd1d2(u:Gd)+λ2d1]+d1i=2  uNGi(wi)NGi(vi)[2μi+1d2(u:Gi)+μ2i+1]+d1i=2  uNGi(vi)NGi(wi)[2λi1d2(u:Gi)+λ2i1]+2d1i=2  uNGi(vi)NGi(wi)[λi1d2(u:Gi)+μi+1d2(u:Gi)+λi1μi+1]+d1i=2  uNGi(vi)NGi(wi)(λ2i1+μ2i+1)+2d1i=1δiνi+1.

    Corollary 19. In a chain graph C, if G1=G2==Gd=G, then LM1(C)=dLM1(G)+uNG(w)[2μd2(u:G)+μ2]+uNG(v)[2λd2(u:G)+λ2]+(d2)uNG(w)NG(v)[2μd2(u:G)+μ2]+(d2)uNG(v)NG(w)[2λd2(u:G)+λ2]+2(d2)uNG(v)NG(w)[λd2(u:G)+μd2(u:G)+λμ]+(d2)uNG(v)NG(w)(λ2+μ2)+2(d1)δν.

    Lemma 20. Let G1,G2,,Gd, d5 be C3-free connected graphs and let C=C(G1,G2,,Gd;w1,v2,w2,v3,,wd1,vd) be the chain graph formed using these graphs. Then the degree of any vertex u in C is given as follows:

    deg(u:C)={deg(u:G1),ifuV(G1){w1}deg(u:Gd),ifuV(Gd){vd}deg(u:Gi),ifuV(Gi){vi,wi},2id1λi+μi+1,ifu=wi=vi+1,1id1, (2.6)

    where μi=deg(vi:Gi),λi=deg(wi:Gi) for all 1id

    Finally, we compute the third leap Zagreb index of the chain graph C by applying Lemma 17 and 2.16.

    Theorem 21. LM3(C)=di=1LM3(Gi)+uNG1(w1)μ2deg(u:G1)+uNGd(vd)λd1deg(u:Gd)+d1i=2uNGi(wi)NGi(vi)μi+1deg(u:Gi)+d1i=2uNGi(vi)NGi(wi)λi1deg(u:Gi)+d1i=2uNGi(vi)NGi(wi)(λi1deg(u:Gi)+μi+1deg(u:Gi))+d1i=1(δiμi+1+νi+1λi).

    Proof. By virtue of Lemma 17 and 20

    LM3(C)=uV(C)d2(u)deg(u)=uV(G1)NG1[w1]d2(u:G1)deg(u:G1)+uNG1(w1)(d2(u:G1)+μ2)deg(u:G1)+uV(Gd)NGd[vd]d2(u:Gd)deg(u:Gd)+uNGd(vd)(d2(u:Gd)+λd1)deg(u:Gd)+d1i=2uV(Gi){NGi[wi]NGi[vi]}d2(u:Gi)deg(u:Gi)+d1i=2uNGi(wi)NGi(vi)(d2(u:Gi)+μi+1)deg(u:Gi)+d1i=2uNGi(vi)NGi(wi)(d2(u:Gi)+λi1)deg(u:Gi)+d1i=2uNGi(vi)NGi(wi)(d2(u:Gi)+λi1+μi+1)deg(u:Gi)+d1i=1(δi+νi+1)(λi+μi+1).

    Thus the result follows.

    Corollary 22. In a chain graph C, if G1=G2==Gd=G, then LM3(C)=dLM3(G)+uNG(w)μdeg(u:G)+uNG(v)λdeg(u:G)+(d2)(uNG(w)NG(v)μdeg(u:G)+uNG(v)NG(w)λdeg(u:G)+uNG(v)NG(w)(λ+μ)deg(u:G))+(d1)(δμ+νλ).

    In this section, we determine the first and third leap Zagreb indices of some molecular graph structures. Two vertices v and w of a hexagon H (C6) (please refer Figure 4) are said to be in

    Figure 4.  Ortho, meta and para positions of two vertices in a hexagon H.

    (ⅰ) ortho-position, if they are adjacent in H

    (ⅱ) meta-position, if they are distance two in H

    (ⅲ) para-position, if they are distance three in H.

    We connect h5 ortho-hexagons to form a polyphenyl chain denoted by Oh as follows:

    One can observe that the Polyphenyl chain Oh shown in Figure 5 is a B1 type bridge graph. Therefore, from Corollary 7, we get

    LM1(Oh)=hLM1(G)+(4h6)μ2+(4h8)ν+(12h26)μ+(4h4)[νμ+uNG(v)d2(u:G)]+4h12=24h+(4h6)(4)+(4h8)(2)+(12h26)(2)+(4h4)(4)+(4h4)(4)+4h12=108h136.
    Figure 5.  Polyphenyl chain Oh.

    Similarly,

    From Corollary 10, we get

    LM3(Oh)=24h+(2h2)(2)+(2h2)(2)+2(2)(3h5)+2(h1)(2+4)+4h10=60h50

    The polyphenyl chain Mh is formed by connecting h5 meta-hexagons as shown in Figure 6.

    Figure 6.  Polyphenyl chain Mh.

    The polyphenyl chain Ph is formed by connecting h5 para-hexagons as shown in the following Figure 7.

    Figure 7.  Polyphenyl chain Ph.

    It is clear that the Polyphenyl chains Mh and Ph are type-Ⅱ bridge graphs B2.

    Using Corollary 2.9, we get

    LM1(Mh)=hLM1(G)+λ+μ+2uNG(w)d2(u:G)+(h2)[uNG(w)NG(v)(2d2(u:G)+1)]+(h2)uNG(v)NG(w)(2d2(u:G)+1)+4(h2)uNG(v)NG(w)(d2(u:G)+1)+2uNG(v)d2(u:G)+(h1)μ2+2(h1)δμ+2(h1)νλ+(h1)λ2=24h+4+2(4)+(h2)[2(2)+1]+(h2)[2(2)+1]+4(h2)(2+1)+2(4)+(h1)(4)+4(h1)(4)+(h1)(4)

    Thus LM1(Mh)=70h48.

    Similarly, by Corollary 13, we have

    LM1(Ph)=24h+4+2(4)+(h2)[2(4)+2]+(h2)(8+2)+4(h2)(0)+2(4)+(h1)(4)+8(h1)+8(h1)+(h1)(4)

    Therefore, LM1(Ph)=68h44.

    Using Corollary 2.12, we get

    LM3(Mh)=24h+8+(h2)8+(h1)12+h(4)4=48h24
    LM3(Ph)=24h+8+(h2)8+(h1)12+4h4=48h24.

    Next, we shall see an application related to the chain graph C. The spiro-chain SPC4(d,3) is a chain graph formed using d5 copies of the cycle C4.

    Here the number 3 in the construction denotes the position of the vertices v and w in the spiro-chain (refer Figure 8).

    Figure 8.  Spiro-chain SPC4(d,3) formed with d5 copies of C4 connected in 3rd position.

    The spiro-chain SPC6(d,4) is a chain graph formed using d5 copies of the cycle C6 or hexagon where the vertices v and w are connected in the 4th position (refer Figure 9).

    Figure 9.  Spiro-chain SPC6(d,4) formed with d5 copies of C6 connected in 4th position.

    By applying Corollary 19, we get

    LM1(SPC4(d,3))=dLM1(G)+uNG(w)[2μd2(u:G)+μ2]+uNG(v)[2λd2(u:G)+λ2]+(d2)uNG(w)NG(v)[2μd2(u:G)+μ2]+(d2)uNG(v)NG(w)[2λd2(u:G)+λ2]+2(d2)uNG(v)NG(w)[λd2(u:G)+μd2(u:G)+λμ]+(d2)uNG(v)NG(w)(λ2+μ2)+2(d1)δν=54d66.

    Similarly, from Corollary 19, we have LM1(SPC6(d,4))=80d56.

    By applying Corollary 22, we get

    LM3(SPC4(d,3))=8d+2(2+2)+2(2+2)+(d2)(16)+(d1)(4)=28d20

    Similarly, from Corollary 22, we have LM3(SPC6(d,4))=48d24.

    We have computed exact values of one of the recent topological invariants namely first and third leap Zagreb indices for bridge and chain graphs. It is worth mentioning that computing second leap Zagreb index of bridges and chain graphs has not yet addressed and interested researchers may work on it. Also these indices need to be explored for several other interesting graph structures arising from mathematical chemistry and other branches of science.

    The authors wish to thank the referees for their careful reading of the manuscript and valuable suggestions. This work was supported in part by the National Key Technologies R & D Program of China under Grant 2017YFB0802300, 2018YFB0904205, in part by the Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province under Grant MSSB-2020-12.

    The authors declare that no competing interests exist.



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