Research article

Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions

  • In this work, we establish inequalities of Hermite-Hadamard-Mercer (HHM) type for convex functions by using generalized fractional integrals. The results of our paper are the extensions and refinements of Hermite-Hadamard (HH) and Hermite-Hadamard-Mercer (HHM) type inequalities. We discuss special cases of our main results and give new inequalities of HH and HHM type for different fractional integrals like, Riemann-Liouville (RL) fractional integrals, k-Riemann-Liouville (k-RL) fractional integrals, conformable fractional integrals and fractional integrals of exponential kernel.

    Citation: Miguel Vivas-Cortez, Muhammad Aamir Ali, Artion Kashuri, Hüseyin Budak. Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions[J]. AIMS Mathematics, 2021, 6(9): 9397-9421. doi: 10.3934/math.2021546

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  • In this work, we establish inequalities of Hermite-Hadamard-Mercer (HHM) type for convex functions by using generalized fractional integrals. The results of our paper are the extensions and refinements of Hermite-Hadamard (HH) and Hermite-Hadamard-Mercer (HHM) type inequalities. We discuss special cases of our main results and give new inequalities of HH and HHM type for different fractional integrals like, Riemann-Liouville (RL) fractional integrals, k-Riemann-Liouville (k-RL) fractional integrals, conformable fractional integrals and fractional integrals of exponential kernel.



    Graph theory has provided the researcher with various useful tools, such as graph labeling, locating numbers and topological indices. Graph theory subject has many applications and implementations in different research subjects like chemistry, medicine and engineering. A graph can be recognized by a numeric value, a polynomial, a sequence of numbers or a matrix. The representation of the chemical compound in terms of diagram, known as its molecular graph, in which its atoms and the chemical bonding between them represent the nodes and edges, respectively. Recently, a new subject caught attention of the researchers was introduced, which is the combination of chemistry, information science and mathematics is called Cheminformatics, which studies QSAR/QSPR relationship, bioactivity and characterization of chemical compounds [1].

    The topological index is a numeric value related with chemical compositions maintaining the correlation of chemical structures with many physico-chemical properties, chemical reactivity or biological activity. Topological indices are prepared on the grounds of the transformation of a chemical network into a number that describes the topology of the chemical network. Some of the main types of topological indices of graphs are distance-based topological indices, degree-based topological indices, and counting-related topological indices. Recently, numerous researchers have found topological indices for the study of fundamental properties of molecular graph or network. These networks have very motivating topological properties which have been considered in different characteristics in [2,3,4,5,6,7,8,9].

    Let G=(V,E) be a simple connected graph, with V be the vertex set and E be the edge set of graph G, with order |V|=p, size |E|=q. The number of edges incident with a vertex ω is known as the degree of ω, denoted by ζ(ω). The reverse vertex degree (R(ω)) was introduced by Kulli [10] defined as: R(ω)=1ζ(ω)+Δ, where Δ denoted the maximum degree of the given graph. Let ER(ω),R(μ) represents the edge partition of the given graph based on reverse degree of end vertices of an edge ωμE and |ER(ω),R(μ)| represents its cardinality. There are detailed variations of topological indices mainly distance-based and degree-based indices, see [11,12,13,14,15,16]. Milan Randic [30] was the first who defined the degree-based indices and its reverse Randic index is defined as:

    RRα(G)=ωμE(G)(R(ω)×R(μ))α,α=12,12,1,1. (1.1)

    Estrada et al. presented the atom bond connectivity (ABC) index in [18] and the reverse atom bond connectivity (RABC) is defined as:

    RABC(G)=ωμE(G)R(ω)+R(μ)2R(ω)×R(μ) (1.2)

    Vukicevic and Furtula defined the geometric arithmetic (GA) index in [19] and the reverse geometric arithmetic (RGA) is presented as:

    RGA(G)=ωμE(G)2R(ω)×R(μ)R(ω)+R(μ) (1.3)

    Gutman et al. [20,21] defined the first and second Zagreb and its reverse indices as:

    RM1(G)=ωμE(G)(R(ω)+R(μ)) (1.4)
    RM2(G)=ωμE(G)(R(ω)×R(μ)) (1.5)

    Shirdel et al. [22] introduced hyper Zagreb index. We defined the reverse hyper Zagreb index as:

    RHM(G)=ωμE(G)(R(ω)+R(μ))2 (1.6)

    Furtula and Gutman [23] accomplished the forgotten index and its reverse forgotten index as:

    RF(G)=ωμE(G)((R(ω))2+(R(μ))2) (1.7)

    Augmented Zagreb index was introduced by Furtula et al. [24] and the reverse augmented Zagreb index as:

    RAZI(G)=ωμE(G)(R(ω)×R(μ)R(ω)+R(μ)2)3 (1.8)

    Ranjini et al. [25] introduced the first redefined, second redefined and third redefined Zagreb indices. The reverse first redefined, second redefined and third redefined Zagreb indices are defined as:

    RReZ1(G)=ωμE(G)R(ω)+R(μ)R(ω)×R(μ) (1.9)
    RReZ2(G)=ωμE(G)R(ω)×R(μ)R(ω)+R(μ) (1.10)
    RReZ3(G)=ωμE(G)(R(ω)+R(μ))(R(ω)×R(μ)) (1.11)

    For latest results on topological indices see [26,27,28,29,31,32,33,34,35]. In this paper, we compute the exact results for all the above reverse indices.

    With the help of complete graphs of order 3 (K3), Chen et al. [36] assembled a hexagonal mesh. In terms of chemistry, these K3 graphs are also called oxide graphs. The Figure 1 is obtained by joining these K3 graphs. Two dimensional mesh graph HX(2) (see Figure 1 (a)), is obtained by joining six K3 graphs and three dimensional mesh graph HX(3) (see Figure 1 (b)) is obtained by putting K3 graphs around all side of HX(2). Furthermore, repeating the same process by putting the t K3 graph around each hexagon, we obtained the tth hexagonal mesh. To be noted that the one dimensional hexagonal mesh graph does not exist.

    Figure 1.  Hexagonal meshes: (a) HX(2) and (b) HX(3).

    Simonraj et al. [37] created the new network which is named as third type of hex-derived networks. The graphically construction algorithm for third type of hexagonal hex-derived network HHDN3(t) (see Figure 2), triangular hex-derived network THDN3(t) (see Figure 3) and rectangular hex-derived network RHDN3(t) (see Figure 4) are defined in [38,39] and they determined some topological indices of these new derived networks. Some networks such as hexagonal, honeycomb, and grid networks, for instance, endure closeness to atomic or molecular lattice configurations. Related research that applies this theory and which could get additional advantages from the visions of the new research is found in [40,41,42,43,44,45,46].

    Figure 2.  Third type of hexagonal hex-derived network HHDN3(t) for t=4.
    Figure 3.  Third type of triangular hex-derived network THDN3(t) for t=7.
    Figure 4.  Third type of rectangular hex-derived network RHDN3(t) for t=4.

    Let Γ1=HHDN3(t) be the third type of hexagonal hex-derived network which is shown in Figure 2, where t4. The graph Γ1 has 21t239t+19 vertices from which 18t236t+18 vertices of reverse degree 15, 4 vertices of reverse degree 12, 6t12 vertices of reverse degree 9 and 3t29t+9 vertices of reverse degree 1. There are 63t2123t+60 number of edges of Γ1 is partitioned into nine classes based on their reverse degrees which are given in Eq (3.1).

    |ER(ω),R(μ)(Γ1)|={9t233t+30,for; R(ω)=1,R(μ)=112t24,for; R(ω)=9,R(μ)=16t18,for; R(ω)=9,R(μ)=96,for; R(ω)=12,R(μ)=112,for; R(ω)=12,R(μ)=936t2108t+84,for; R(ω)=15,R(μ)=136t72,for; R(ω)=15,R(μ)=924,for; R(ω)=15,R(μ)=1218t236t+18,for; R(ω)=15,R(μ)=15 (3.1)

    In the next theorem, we will calculate the reverse Randic index, reverse Zagreb indices, reverse forgotten index for Γ1 graph.

    Theorem 3.1. Let Γ1 be the third type of hexagonal hex-derived network, then

    RRα(Γ1)=[9+36(15)α+18(225)α]t2+[33+12(9)α+6(81)α108(15)α+36(135)α36(225)α]t+3024(9)α18(81)α+6(12)α+12(108)α+84(15)α72(135)α+24(180)α+18(225)α

    RM1(Γ1)=1134t21782t+630

    RM2(Γ1)=4599t24299t366

    RHM(Γ1)=25452t236300t+11922

    RF(Γ1)=16254t227702t+12654

    Proof. Let Γ1 be the third type of hexagonal hex-derived network which is shown in Figure 2. The order of hexagonal hex derived network Γ1 is p=|Γ1|=21t239t+19 and size is q=63t2123t+60. The edge partitioned of Γ1 based on their reverse degrees are shown in Eq (3.1). Reverse Randic index can be calculated by using Eq (3.1). Thus, from Eq (1.1), it follows,

    RRα(Γ1)=(1)α|E1,1(Γ1)|+(9)α|E9,1(Γ1)|+(81)α|E9,9(Γ1)|+(12)α|E12,1(Γ1)|+(108)α|E12,9(Γ1)|+(15)α|E15,1(Γ1)|+(135)α|E15,9(Γ1)|+(180)α|E15,12(Γ1)|+(225)α|E15,15(Γ1)|.

    =(9t233t+30)+(9)α(12t24)+(81)α(6t18)+(12)α(6)+(108)α(12)+(15)α(36t2108t+84)+(135)α(36t72)+(180)α(24)+(225)α(18t236t+18).

    =[9+36(15)α+18(225)α]t2+[33+12(9)α+6(81)α108(15)α+36(135)α36(225)α]t+3024(9)α18(81)α+6(12)α+12(108)α+84(15)α72(135)α+24(180)α+18(225)α. Put α=1 and after some calculation, we get reverse second Zagreb index RM2(Γ1) as:

    RM2(Γ1)=4599t24299t366. (3.2)

    Using the Eq (1.4), we can determine the reverse first Zagreb index RM1(Γ1) as:

    RM1(Γ1)=2×|E1,1(Γ1)|+10×|E9,1(Γ1)|+18×|E9,9(Γ1)|+13×|E12,1(Γ1)|+21×|E12,9(Γ1)|+16×|E15,1(Γ1)|+24×|E15,9(Γ1)|+27×|E15,12(Γ1)|+30×|E15,15(Γ1)|.

    By putting the values of from equation (3.1) and after simplification, we obtain:

    RM1(Γ1)=1134t21782t+630. (3.3)

    Using the Eq (1.6), we can determine the reverse hyper Zagreb index RHM(Γ1) as:

    RHM(Γ1)=4×|E1,1(Γ1)|+100×|E9,1(Γ1)|+324×|E9,9(Γ1)|+169×|E12,1(Γ1)|+441×|E12,9(Γ1)|+256×|E15,1(Γ1)|+576×|E15,9(Γ1)|+729×|E15,12(Γ1)|+900×|E15,15(Γ1)|.

    After simplification, we get

    RHM(Γ1)=25452t236300t+11922

    Using the Eq (1.7), we can determine the reverse forgotten index RF(Γ1)) as:

    RF(Γ1))=2×|E1,1(Γ1)|+82×|E9,1(Γ1)|+162×|E9,9(Γ1)|+145×|E12,1(Γ1)|+225×|E12,9(Γ1)|+226×|E15,1(Γ1)|+306×|E15,9(Γ1)|+369×|E15,12(Γ1)|+450×|E15,15(Γ1)|.

    After simplification, we get

    RF(Γ1)=16254t227702t+12654

    In the next theorem, we will calculate the reverse atom bond connectivity index, reverse geometric arithmetic index for Γ1 graph.

    Theorem 3.2. Let Γ1 be the third type of hexagonal hex-derived network, then

    RABC(Γ1)=(122105+1275)t2+(82+83362105+433052475)t8162+33+2573+28210583305+45+1275

    RGA(Γ1)=(27+9152)t2+(27959152)t+785+79239115152+3253.

    Proof. The reverse atom bond connectivity (RABC(Γ1)), can be determined by using Eq (1.2) and Eq (3.1), as follows:

    RABC(Γ1)=0×|E1,1(Γ1)|+89×|E9,1(Γ1)|+1681×|E9,9(Γ1)|+1112×|E12,1(Γ1)|+19108×|E12,9(Γ1)|+1415×|E15,1(Γ1)|+22135×|E15,9(Γ1)|+25180×|E15,12(Γ1)|+28225×|E15,15(Γ1)|.

    After some simplification, we get

    RABC(Γ1)=(122105+1275)t2+(82+83362105+433052475)t8162+33+2573+28210583305+45+1275.

    The reverse geometric arithmetic (RGA(Γ1)), can be determined by using Eq (1.3) and Eq (3.1), as follows:

    RGA(Γ1)=|E1,1(Γ1)|+2910×|E9,1(Γ1)|+28118×|E9,9(Γ1)|+21213×|E12,1(Γ1)|+210821×|E12,9(Γ1)|+21516×|E15,1(Γ1)|+213524×|E15,9(Γ1)|+218027×|E15,12(Γ1)|+222530×|E15,15(Γ1)|.

    After some simplification, we get

    RGA(Γ1)=(27+9152)t2+(27959152)t+785+79239115152+3253. In the next theorems, we will calculate the reverse redefined Zagreb indices for Γ1 graph.

    Theorem 3.3. Let Γ1 be the third type of hexagonal hex-derived network, then

    RReZ1(Γ1)=294t252474t15+362930

    RReZ2(Γ1)=693t242949t20939071820

    RReZ3(Γ1)=130158t2142518t+24828

    Proof. Reverse redefined Zagreb indices can be calculated by using Eq (3.1), the RReZ1(Γ1) by using Eq (1.9) as follows:

    RReZ1(Γ1)=2×|E1,1(Γ1)|+109×|E9,1(Γ1)|+1881×|E9,9(Γ1)|+1312×|E12,1(Γ1)|+21108×|E12,9(Γ1)|+1615×|E15,1(Γ1)|+24135×|E15,9(Γ1)|+27108×|E15,12(Γ1)|+30225×|E15,15(Γ1)|.

    After some simplification, we get

    RReZ1(Γ1)=294t252474t15+362930.

    The RReZ2(Γ1) can be determined by using Eq (1.10) as follows:

    RReZ2(Γ1)=12×|E1,1(Γ1)|+910×|E9,1(Γ1)|+8118×|E9,9(Γ1)|+1213×|E12,1(Γ1)|+10821×|E12,9(Γ1)|+1516×|E15,1(Γ1)|+13524×|E15,9(Γ1)|+10827×|E15,12(Γ1)|+22530×|E15,15(Γ1)|.

    After some simplification, we get

    RReZ2(Γ1)=693t242949t20939071820.

    The RReZ3(Γ1) can be calculated by using Eq (1.11) as follows:

    RReZ3(Γ1)=2×|E1,1(Γ1)|+90×|E9,1(Γ1)|+1458×|E9,9(Γ1)|+156×|E12,1(Γ1)|+2268×|E12,9(Γ1)|+240×|E15,1(Γ1)|+3240×|E15,9(Γ1)|+2916×|E15,12(Γ1)|+6750×|E15,15(Γ1)|.

    After some simplification, we get

    RReZ3(Γ1)=130158t2142518t+24828.

    Let Γ2=THDN3(t) be the third type of triangular hex-derived network which is shown in Figure 3, where t4. The graph Γ2 has 7t211t+62 vertices. There are 21t239t+182 number of edges of Γ2 is partitioned into six classes based on their reverse degrees which are given in Eq (4.1). Now we calculated reverse degree based indices such as: reverse Randic index RRα, reverse atom bond connectivity index RABC, reverse geometric arithmetic index RGA, first reverse Zagreb index RM1, second reverse Zagreb index RM2, reverse forgotten index RF, reverse hyper Zagreb index RHM and reverse redefined Zagreb indices for Γ2 graph.

    |ER(ω),R(μ)(Γ2)|={3t2221t2+18,for; R(ω)=1,R(μ)=16t18,for; R(ω)=9,R(μ)=13t6,for; R(ω)=9,R(μ)=96t230t+36,for; R(ω)=15,R(μ)=118t30,for; R(ω)=15,R(μ)=93t26t+9,for; R(ω)=15,R(μ)=15 (4.1)

    In the next theorem, we will calculate the reverse Randic index, reverse Zagreb indices, reverse forgotten index for Γ2 graph.

    Theorem 4.1. Let Γ2 be the third type of triangular hex-derived network, then

    RRα(Γ2)=(32+6(15)α+3(225)α)t2+(212+6(9)α+3(81)α30(15)α+18(135)α6(225)α)t+1818(9)α6(81)α+36(15)α30(135)α+9(225)α.

    RM1(Γ2)=189t2135t126

    RM2(Γ2)=1533t22+1833t22115

    RHM(Γ2)=4242t21182t3636

    RF(Γ2)=2709t23015t+594

    Proof. Let Γ2 be the third type of triangular hex-derived network which is shown in Figure 3. The order of triangular hex derived network Γ2 is p=|Γ2|=7t211t+62 and size is q=21t239t+182. The edge partitioned of Γ2 based on their reverse degrees are shown in Eq (4.1). Reverse Randic index can be calculated by using Eq (4.1). Thus, from Eq (1.1), it follows,

    RRα(Γ2)=(1)α|E1,1(Γ2)|+(9)α|E9,1(Γ2)|+(81)α|E9,9(Γ2)|+(15)α|E15,1(Γ2)|+(135)α|E15,9(Γ2)|+(225)α|E15,15(Γ2)|.

    After simplification, we get

    RRα(Γ2)=(32+6(15)α+3(225)α)t2+(212+6(9)α+3(81)α30(15)α+18(135)α6(225)α)t+1818(9)α6(81)α+36(15)α30(135)α+9(225)α.

    Put α=1 and after some calculation, we get reverse second Zagreb index RM2(Γ2) as:

    RM2(Γ2)=1533t22+1833t22115 (4.2)

    Using the Eq (1.4), we can determine the reverse first Zagreb index RM1(Γ2) as:

    RM1(Γ2)=2×|E1,1(Γ1)|+10×|E9,1(Γ1)|+18×|E9,9(Γ1)|+16×|E15,1(Γ1)|+24×|E15,9(Γ1)|+30×|E15,15(Γ1)|.

    By putting the values of from Eq (4.1) and after simplification, we obtain:

    RM1(Γ2)=189t2135t126 (4.3)

    Using the Eq (1.6), we can determine the reverse hyper Zagreb index RHM(Γ2) as:

    RHM(Γ2)=4×|E1,1(Γ2)|+100×|E9,1(Γ2)|+324×|E9,9(Γ2)|+256×|E15,1(Γ2)|+576×|E15,9(Γ2)|+900×|E15,15(Γ2)|.

    After simplification, we get

    RHM(Γ2)=4242t21182t3636.

    Using the Eq (1.7), we can determine the reverse forgotten index RF(Γ2)) as:

    RF(Γ2))=2×|E1,1(Γ2)|+82×|E9,1(Γ2)|+162×|E9,9(Γ2)|+226×|E15,1(Γ2)|+306×|E15,9(Γ2)|+450×|E15,15(Γ2)|.

    After simplification, we get

    RF(Γ2)=2709t23015t+594.

    In the next theorem, we will calculate the reverse atom bond connectivity index, reverse geometric arithmetic index for Γ2 graph.

    Theorem 4.2. Let Γ2 be the third type of triangular hex-derived network, then

    RABC(Γ2)=(22105+275)t2+(42+432210+23305475)t83122+12210523303+675

    RGA(Γ2)=(92+3154)t2+(9910+3154)t+515315.

    Proof. The reverse atom bond connectivity (RABC(Γ2)), can be determined by using Eq (1.2) and Eq (4.1), as follows:

    RABC(Γ2)=0×|E1,1(Γ2)|+89×|E9,1(Γ2)|+1681×|E9,9(Γ2)|+1415×|E15,1(Γ2)|+22135×|E15,9(Γ2)|+28225×|E15,15(Γ2)|.

    After some simplification, we get

    RABC(Γ2)=(22105+275)t2+(42+432210+23305475)t83122+12210523303+675.

    The reverse geometric arithmetic (RGA(Γ2)), can be determined by using Eq (1.3) and Eq (4.1), as follows:

    RGA(Γ2)=|E1,1(Γ2)|+2910×|E9,1(Γ2)|+28118×|E9,9(Γ2)|+21516×|E15,1(Γ2)|+213524×|E15,9(Γ2)|+222530×|E15,15(Γ2)|.

    After some simplification, we get

    RGA(Γ2)=(92+3154)t2+(9910+3154)t+515315.

    In the next theorems, we will calculate the reverse redefined Zagreb indices for Γ2 graph.

    Theorem 4.3. Let Γ2 be the third type of triangular hex-derived network, then

    RReZ1(Γ2)=49t25649t15+73415

    RReZ2(Γ2)=231t28+1671t40101710

    RReZ3(Γ2)=21693t2+15513t38142.

    Proof. Reverse redefined Zagreb indices can be calculated by using Eq (4.1), the RReZ1(Γ2) by using Eq (1.9) as follows:

    RReZ1(Γ2)=2×|E1,1(Γ2)|+109×|E9,1(Γ2)|+1881×|E9,9(Γ2)|+1615×|E15,1(Γ2)|+24135×|E15,9(Γ2)|+30225×|E15,15(Γ2)|.

    After some simplification, we get

    RReZ1(Γ2)=49t25649t15+73415.

    The RReZ2(Γ2) can be determined by using Eq (1.10) as follows:

    RReZ2(Γ2)=12×|E1,1(Γ2)|+910×|E9,1(Γ2)|+8118×|E9,9(Γ2)|+1516×|E15,1(Γ2)|+13524×|E15,9(Γ2)|+22530×|E15,15(Γ2)|.

    After some simplification, we get

    RReZ2(Γ2)=231t28+1671t40101710.

    The RReZ3(Γ2) can be calculated by using Eq (1.11) as follows:

    RReZ3(Γ2)=2×|E1,1(Γ2)|+90×|E9,1(Γ2)|+1458×|E9,9(Γ2)|+240×|E15,1(Γ2)|+3240×|E15,9(Γ2)|+6750×|E15,15(Γ2)|.

    After some simplification, we get

    RReZ3(Γ2)=21693t2+15513t38142.

    In this section, we calculate certain reverse degree based topological indices of the third type of rectangular hex-derived network, RHDN3(t,w) of dimension t=w. Now we calculated reverse degree based indices such as: reverse Randic index RRα, reverse atom bond connectivity index RABC, reverse geometric arithmetic index RGA, first reverse Zagreb index RM1, second reverse Zagreb index RM2, reverse forgotten index RF, reverse augmented Zagreb index RAZI, reverse hyper Zagreb index RHM and reverse redefined Zagreb indices for Γ3 graph.

    |ER(ω),R(μ)(Γ3)|={3t216t+21,for; R(ω)=1,R(μ)=18t20,for; R(ω)=9,R(μ)=14t10,for; R(ω)=9,R(μ)=92,for; R(ω)=12,R(μ)=14,for; R(ω)=12,R(μ)=912t248t+48,for; R(ω)=15,R(μ)=124t44,for; R(ω)=15,R(μ)=98,for; R(ω)=15,R(μ)=126t212t+10,for; R(ω)=15,R(μ)=15 (5.1)

    In the next theorem, we will calculate the reverse Randic index, reverse Zagreb indices, reverse forgotten index for Γ3 graph.

    Theorem 5.1. Let Γ3 be the third type of rectangular hex-derived network, then

    RRα(Γ3)=(3+12(15)α+6(225)α)t2+(16+8(9)α+4(81)α48(15)α+24(135)α12(225)α)t+2120(9)α10(81)α+2(12)α+4(108)α+48(15)α44(135)α+8(180)α+10(225)α

    RM1(Γ3)=378t2432t

    RM2(Γ3)=1533t2+200t2043

    RHM(Γ3)=8484t27232t1278

    RF(Γ3)=5418t27632t+2808.

    Proof. Let Γ3 be the third type of rectangular hex-derived network which is shown in Figure 4. The order of hexagonal hex derived network Γ3 is p=|Γ1|=7t212t+6 and size is q=21t240t+19. The edge partitioned of Γ3 based on their reverse degrees are shown in Eq (5.1). Reverse Randic index can be calculated by using Eq (5.1). Thus, from Eq (1.1), it follows,

    RRα(Γ3)=(1)α|E1,1(Γ3)|+(9)α|E9,1(Γ3)|+(81)α|E9,9(Γ3)|+(12)α|E12,1(Γ3)|+(108)α|E12,9(Γ3)|+(15)α|E15,1(Γ3)|+(135)α|E15,9(Γ3)|+(180)α|E15,12(Γ3)|+(225)α|E15,15(Γ3)|.

    After Simplification, we get

    RRα(Γ3)=(3+12(15)α+6(225)α)t2+(16+8(9)α+4(81)α48(15)α+24(135)α12(225)α)t+2120(9)α10(81)α+2(12)α+4(108)α+48(15)α44(135)α+8(180)α+10(225)α.

    Put α=1 and after some calculation, we get reverse second Zagreb index RM2(Γ3) as:

    RM2(Γ3)=1533t2+200t2043. (5.2)

    Using the Eq (1.4), we can determine the reverse first Zagreb index RM1(Γ3) as:

    RM1(Γ3)=2×|E1,1(Γ3)|+10×|E9,1(Γ3)|+18×|E9,9(Γ3)|+13×|E12,1(Γ3)|+21×|E12,9(Γ3)|+16×|E15,1(Γ3)|+24×|E15,9(Γ3)|+27×|E15,12(Γ3)|+30×|E15,15(Γ3)|.

    By putting the values of from Eq (5.1) and after simplification, we obtain:

    RM1(Γ3)=378t2432t. (5.3)

    Using the Eq (1.6), we can determine the reverse hyper Zagreb index RHM(Γ3) as:

    RHM(Γ3)=4×|E1,1(Γ3)|+100×|E9,1(Γ3)|+324×|E9,9(Γ3)|+169×|E12,1(Γ3)|+441×|E12,9(Γ3)|+256×|E15,1(Γ3)|+576×|E15,9(Γ3)|+729×|E15,12(Γ3)|+900×|E15,15(Γ3)|.

    After simplification, we get

    RHM(Γ3)=8484t27232t1278.

    Using the Eq (1.7), we can determine the reverse forgotten index RF(Γ3)) as:

    RF(Γ3))=2×|E1,1(Γ3)|+82×|E9,1(Γ3)|+162×|E9,9(Γ3)|+145×|E12,1(Γ3)|+225×|E12,9(Γ3)|+226×|E15,1(Γ3)|+306×|E15,9(Γ3)|+369×|E15,12(Γ3)|+450×|E15,15(Γ3)|.

    After simplification, we get

    RF(Γ3)=5418t27632t+2808.

    In the next theorem, we will calculate the reverse atom bond connectivity index, reverse geometric arithmetic index for Γ3 graph.

    Theorem 5.2. Let Γ3 be the third type of rectangular hex-derived network, then

    RABC(Γ3)=(42105+475)t2+(1623+169162105+833015875)t4094023+333+2579+1621054433045+453+473.

    RGA(Γ3)=(9+3152)t296t5+9+264391515+3259.

    Proof. The reverse atom bond connectivity (RABC(Γ3)), can be determined by using Eq (1.2) and Eq (5.1), as follows:

    RABC(Γ3)=0×|E1,1(Γ3)|+89×|E9,1(Γ3)|+1681×|E9,9(Γ3)|+1112×|E12,1(Γ3)|+19108×|E12,9(Γ3)|+1415×|E15,1(Γ3)|+22135×|E15,9(Γ3)|+25180×|E15,12(Γ3)|+28225×|E15,15(Γ3)|.

    After some simplification, we get

    RABC(Γ3)=(42105+475)t2+(1623+169162105+833015875)t4094023+333+2579+1621054433045+453+473.

    The reverse geometric arithmetic (RGA(Γ3)), can be determined by using Eq (1.3) and Eq (5.1), as follows:

    RGA(Γ3)=|E1,1(Γ3)|+2910×|E9,1(Γ3)|+28118×|E9,9(Γ3)|+21213×|E12,1(Γ3)|+210821×|E12,9(Γ3)|+21516×|E15,1(Γ3)|+213524×|E15,9(Γ3)|+218027×|E15,12(Γ3)|+222530×|E15,15(Γ3)|.

    After some simplification, we get

    RGA(Γ3)=(9+3152)t296t5+9+264391515+3259.

    In the next theorems, we will calculate the reverse redefined Zagreb indices for Γ3 graph.

    Theorem 5.3. Let Γ3 be the third type of rectangular hex-derived network, then

    RReZ1(Γ3)=98t253184t45+597790

    RReZ2(Γ3)=231t24+86t528460273

    RReZ3(Γ3)=43386t28240t31614.

    Proof. Reverse redefined Zagreb indices can be calculated by using Eq (5.1), the RReZ1(Γ3) by using Eq (1.9) as follows:

    RReZ1(Γ3)=2×|E1,1(Γ3)|+109×|E9,1(Γ3)|+1881×|E9,9(Γ3)|+1312×|E12,1(Γ3)|+21108×|E12,9(Γ3)|+1615×|E15,1(Γ3)|+24135×|E15,9(Γ3)|+27108×|E15,12(Γ3)|+30225×|E15,15(Γ3)|.

    After some simplification, we get

    RReZ1(Γ3)=98t253184t45+597790.

    The RReZ2(Γ1) can be determined by using equation (1.10) as follows:

    RReZ2(Γ3)=12×|E1,1(Γ3)|+910×|E9,1(Γ3)|+8118×|E9,9(Γ3)|+1213×|E12,1(Γ3)|+10821×|E12,9(Γ3)|+1516×|E15,1(Γ3)|+13524×|E15,9(Γ3)|+10827×|E15,12(Γ3)|+22530×|E15,15(Γ3)|.

    After some simplification, we get

    RReZ2(Γ3)=231t24+86t528460273.

    The RReZ3(Γ3) can be calculated by using Eq (1.11) as follows:

    RReZ3(Γ3)=2×|E1,1(Γ3)|+90×|E9,1(Γ3)|+1458×|E9,9(Γ3)|+156×|E12,1(Γ3)|+2268×|E12,9(Γ3)|+240×|E15,1(Γ3)|+3240×|E15,9(Γ3)|+2916×|E15,12(Γ3)|+6750×|E15,15(Γ3)|.

    After some simplification, we get

    RReZ3(Γ3)=43386t28240t31614.

    In this article, we have calculated the exact solutions of reverse degree-based topological descriptors for hex-derived networks of third type. Hex-derived network has a variety of useful applications in pharmacy, electronics, and networking. We obtained the reverse degree-based indices such as reverse Randic index, reverse atom bond connectivity index, reverse geometric arithmetic index, reverse Zagreb indices, reverse redefined Zagreb indices for hex derived networks. These results may be helpful for people working in computer science and chemistry who encounter hex-derived networks.

    The authors declare that there is no conflict of financial interests regarding the publication of this paper.



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