Research article Special Issues

Fixed points of non-linear set-valued (α,ϕM)-contraction mappings and related applications

  • Received: 12 December 2021 Revised: 02 February 2022 Accepted: 11 February 2022 Published: 04 March 2022
  • MSC : 47H10, 54H25

  • The aim of this manuscript is to prove some fixed point results for non-linear set-valued maps with new approach of (α,ϕM)-contraction in complete M-metric space. Also, we prove some fixed point results in ordered M-metric space. As an presented work which are the extension and improves the current study of set-valued mappings. Finally, we also give an non-trivial extensive examples and application to homotopy theory and the existence solution of functional equations to show that our concepts are meaningful and to support our results.

    Citation: Muhammad Tariq, Mujahid Abbas, Aftab Hussain, Muhammad Arshad, Amjad Ali, Hamid Al-Sulami. Fixed points of non-linear set-valued (α,ϕM)-contraction mappings and related applications[J]. AIMS Mathematics, 2022, 7(5): 8861-8878. doi: 10.3934/math.2022494

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  • The aim of this manuscript is to prove some fixed point results for non-linear set-valued maps with new approach of (α,ϕM)-contraction in complete M-metric space. Also, we prove some fixed point results in ordered M-metric space. As an presented work which are the extension and improves the current study of set-valued mappings. Finally, we also give an non-trivial extensive examples and application to homotopy theory and the existence solution of functional equations to show that our concepts are meaningful and to support our results.



    Topological indices have become an important research topic associated with the study of their mathematical and computational properties and, fundamentally, for their multiple applications to various areas of knowledge (see, e.g., [1,2,3]). Within the study of mathematical properties, we will contribute to the study of inequalities and optimization problems associated with topological indices. Our main goals are the Sombor indices, introduced by Gutman in [4].

    In what follows, G=(V(G),E(G)) will be a finite undirected graph, and we will assume that each vertex has at least a neighbor. We denote by dw the degree of the vertex w, i.e., the number of neighbors of w. We denote by uv the edge joining the vertices u and v (or v and u). For each graph G, its Sombor index is

    SO(G)=uvE(G)d2u+d2v.

    In the same paper is also defined the reduced Sombor index by

    SOred(G)=uvE(G)(du1)2+(dv1)2.

    In [5] it is shown that these indices have a good predictive potential.

    Also, the modified Sombor index of G was proposed in [6] as

    mSO(G)=uvE(G)1d2u+d2v. (1.1)

    In addition, two other Sombor indices have been introduced: the first Banhatti-Sombor index [7]

    BSO(G)=uvE(G)1d2u+1d2v (1.2)

    and the α-Sombor index [8]

    SOα(G)=uvE(G)(dαu+dαv)1/α, (1.3)

    here αR{0}. In fact, there is a general index that includes most Sombor indices listed above: the first (α,β)KA index of G which was introduced in [9] as

    KAα,β(G)=KA1α,β(G)=uvE(G)(dαu+dαv)β, (1.4)

    with α,βR. Note that SO(G)=KA2,1/2(G), mSO(G)=KA2,1/2(G), BSO(G)=KA2,1/2(G), and SOα(G)=KAα,1/α(G). Also, we note that KA1,β(G) equals the general sum-connectivity index [10] χβ(G)=uvE(G)(du+dv)β. Reduced versions of SO(G), mSO(G) and KAα,β(G) were also introduced in [4,6,11], e.g., the reduced (α,β)KA index is

    redKAα,β(G)=uvE(G)((du1)α+(dv1)α)β.

    If α<0, then redKAα,β(G) is just defined for graphs without pendant vertices (recall that a vertex is said pendant if its degree is equal to 1).

    Since I. Gutman initiated the study of the mathematical properties of Sombor index in [4], many papers have continued this study, see e.g., [12,13,14,15,16,17,18].

    Our main aim is to obtain new bounds of Sombor indices, and to characterize the graphs where equality occurs. In particular, we have obtained bounds for Sombor indices relating them with the first Zagreb index, the forgotten index and the first variable Zagreb index. Also, we solve some extremal problems for Sombor indices.

    The following inequalities are known for x,y>0:

    xa+ya<(x+y)a2a1(xa+ya)if a>1,2a1(xa+ya)(x+y)a<xa+yaif 0<a<1,(x+y)a2a1(xa+ya)if a<0,

    and the second, third or fifth equality is attained for each a if and only if x=y. These inequalities allow to obtain the following result relating KA indices.

    Theorem 1. Let G be any graph and α,β,λR{0}.Then

    KAαβ/λ,λ(G)<KAα,β(G)2βλKAαβ/λ,λ(G)if β>λ,βλ>0,2βλKAαβ/λ,λ(G)KAα,β(G)<KAαβ/λ,λ(G)if β<λ,βλ>0,KAα,β(G)2βλKAαβ/λ,λ(G)if β<0,λ>0,KAα,β(G)2βλKAαβ/λ,λ(G)if β>0,λ<0,

    and the second, third, fifth or sixth equality is attained for each α,β,λ if and only if all the connected components of G are regular graphs.

    Proof. If a=β/λ, x=dαu and y=dαv, then the previous inequalities give

    dαβ/λu+dαβ/λv<(dαu+dαv)β/λ2β/λ1(dαβ/λu+dαβ/λv)if β/λ>1,2β/λ1(dαβ/λu+dαβ/λv)(dαu+dαv)β/λ<dαβ/λu+dαβ/λvif 0<β/λ<1,(dαu+dαv)β/λ2β/λ1(dαβ/λu+dαβ/λv)if β/λ<0,

    and the second, third or fifth equality is attained if and only if du=dv.

    Hence, we obtain

    (dαβ/λu+dαβ/λv)λ<(dαu+dαv)β2βλ(dαβ/λu+dαβ/λv)λif β/λ>1,λ>0,2βλ(dαβ/λu+dαβ/λv)λ(dαu+dαv)β<(dαβ/λu+dαβ/λv)λif β/λ>1,λ<0,2βλ(dαβ/λu+dαβ/λv)λ(dαu+dαv)β<(dαβ/λu+dαβ/λv)λif 0<β/λ<1,λ>0,(dαβ/λu+dαβ/λv)λ<(dαu+dαv)β2βλ(dαβ/λu+dαβ/λv)λif 0<β/λ<1,λ<0,(dαu+dαv)β2βλ(dαβ/λu+dαβ/λv)λif β<0,λ>0,(dαu+dαv)β2βλ(dαβ/λu+dαβ/λv)λif β>0,λ<0,

    and the equality in the non-strict inequalities is tight if and only if du=dv.

    If we sum on uvE(G) these inequalities, then we obtain (1).

    Remark 2. Note that the excluded case β=λ in Theorem 1 is not interesting, since KAαβ/λ,λ(G)=KAα,β(G) if β=λ.

    The argument in the proof of Theorem 1 also allows to obtain the following result relating reduced KA indices.

    Theorem 3. Let G be any graph and α,β,λR{0}.If α<0 or αβλ<0, we also assume that G does not have pendant vertices.Then

    redKAαβ/λ,λ(G)<redKAα,β(G)2βλredKAαβ/λ,λ(G)if β>λ,βλ>0,2βλredKAαβ/λ,λ(G)redKAα,β(G)<redKAαβ/λ,λ(G)if β<λ,βλ>0,redKAα,β(G)2βλredKAαβ/λ,λ(G)if β<0,λ>0,redKAα,β(G)2βλredKAαβ/λ,λ(G)if β>0,λ<0,

    and the second, third, fifth or sixth equality is attained for each α,β,λ if and only if all the connected components of G are regular graphs.

    If we take β=1/α and μ=1/λ in Theorem 1, we obtain the following inequalities for the α-Sombor index.

    Corollary 4. Let G be any graph and α,μR{0}.Then

    SOμ(G)<SOα(G)21/α1/μSOμ(G)if μ>α,αμ>0,21/α1/μSOμ(G)SOα(G)<SOμ(G)if μ<α,αμ>0,SOα(G)21/α1/μSOμ(G)if α<0,μ>0,

    and the second, third or fifth equality is attained for each α,μ if and only if all the connected components of G are regular graphs.

    Recall that one of the most studied topological indices is the first Zagreb index, defined by

    M1(G)=uV(G)d2u.

    If we take μ=1 in Corollary 4, we obtain the following result.

    Corollary 5. Let G be any graph and αR{0}.Then

    M1(G)<SOα(G)21/α1M1(G)if 0<α<1,21/α1M1(G)SOα(G)<M1(G)if α>1,SOα(G)21/α1M1(G)if α<0,

    and the second, third or fifth equality is attained for each α if and only if all the connected components of G are regular graphs.

    If we take α=2, β=1/2 and λ=1/2 in Theorem 1, we obtain the following inequality relating the modified Sombor and the first Banhatti-Sombor indices.

    Corollary 6. Let G be any graph.Then

    mSO(G)12BSO(G)

    and the bound is tight if and only if all the connected components of G are regular graphs

    In [19,20,21], the first variable Zagreb index is defined by

    Mα1(G)=uV(G)dαu,

    with αR.

    Note that Mα1 generalizes numerous degree–based topological indices which earlier have independently been studied. For α=2, α=3, α=1/2, and α=1, Mα1 is, respectively, the ordinary first Zagreb index M1, the forgotten index F, the zeroth–order Randić index 0R, and the inverse index ID [2,22].

    The next result relates the KAα,β and Mα+11 indices.

    Theorem 7. Let G be any graph with maximum degree Δ, minimum degree δ and m edges, and αR{0}, β>0.Then

    KAα,β(G)(Mα+11(G)+2Δα/2δα/2m2(Δα/2+δα/2))2βif 0<β<1/2,KAα,β(G)(Mα+11(G)+2Δα/2δα/2m2(Δα/2+δα/2))2βm12βif β1/2,

    and the second equality is attained for some α,β if and only if G is a regular graph.

    Proof. If uvE(G) and α>0, then

    2δα/2dαu+dαv2Δα/2.

    If α<0, then the converse inequalities hold. Hence,

    (dαu+dαv2δα/2)(2Δα/2dαu+dαv)0,2(Δα/2+δα/2)dαu+dαvdαu+dαv+2Δα/2δα/2.

    Since

    uvE(G)(dαu+dαv)=uV(G)dudαu=uV(G)dα+1u=Mα+11(G),

    If 0<β<1/2, then 1/(2β)>1 and

    uvE(G)dαu+dαv=uvE(G)((dαu+dαv)β)1/(2β)(uvE(G)(dαu+dαv)β)1/(2β)=KAα,β(G)1/(2β).

    Consequently, we obtain

    KAα,β(G)1/(2β)Mα+11(G)+2Δα/2δα/2m2(Δα/2+δα/2).

    If β1/2, then 2β1 and Hölder inequality gives

    uvE(G)dαu+dαv=uvE(G)((dαu+dαv)β)1/(2β)(uvE(G)(dαu+dαv)β)1/(2β)(uvE(G)12β/(2β1))(2β1)/(2β)=m(2β1)/(2β)KAα,β(G)1/(2β).

    Consequently, we obtain

    KAα,β(G)1/(2β)Mα+11(G)+2Δα/2δα/2m2(Δα/2+δα/2)m(12β)/(2β).

    If G is regular, then

    (Mα+11(G)+2Δα/2δα/2m2(Δα/2+δα/2))2βm12β=(2Δαm+2Δαm22Δα/2)2βm12β=(2Δα/2m)2βm12β=(2Δα)βm=KAα,β(G).

    If the second equality is attained for some α,β, then we have dαu+dαv=2δα or dαu+dαv=2Δα for each uvE(G). Also, the equality in Hölder inequality gives that there exists a constant c such that dαu+dαv=c for every uvE(G). Hence, we have either dαu+dαv=2δα for each edge uv or dαu+dαv=2Δα for each edge uv, and hence, G is regular.

    If we take α=2 and β=1/2 in Theorem 7 we obtain:

    Corollary 8. Let G be any graph with maximum degree Δ and minimum degree δ, and m edges.Then

    SO(G)F(G)+2Δδm2(Δ+δ),

    and the bound is tight if and only if G is regular.

    In order to prove Theorem 10 below we need an additional technical result. A converse of Hölder inequality appears in [23,Theorem 3], which, in the discrete case, can be stated as follows [23,Corollay 2].

    Proposition 9. Consider constants 0<αβ and 1<p,q< with 1/p+1/q=1.If wk,zk0 satisfy αzqkwpkβzqk for 1kn, then

    (nk=1wpk)1/p(nk=1zqk)1/qCp(α,β)nk=1wkzk,

    where

    Cp(α,β)={1p(αβ)1/(2q)+1q(βα)1/(2p),when  1<p<2,1p(βα)1/(2q)+1q(αβ)1/(2p),when  p2.

    If (w1,,wn)0, then the bound is tight if and only if wpk=αzqkfor each 1kn and α=β.

    Recall that a bipartite graph with X and Y partitions is called (a,b)-biregular if all vertices of X have degree a and all vertices of Y have degree b.

    The next result relates several KA indices.

    Theorem 10. Let G be any graph, α,β,μR and p>1.Then

    DppKAα,p(βμ)(G)KAα,pμ/(p1)(G)p1KAα,β(G)pKAα,p(βμ)(G)KAα,pμ/(p1)(G)p1

    where

    Dp={Cp((2δα)p(βμpp1),(2Δα)p(βμpp1))1,if  α(βμpp1)0,Cp((2Δα)p(βμpp1),(2δα)p(βμpp1))1,if  α(βμpp1)<0,

    and Cp is the constant in Proposition 9. The equality in the upper(lower) bound is tight for each α,β,μ,p if G is a biregular graph (with α(βμpp1)0 if and only if G is a regular graph.)

    Proof. Hölder inequality gives

    KAα,β(G)=uvE(G)(dαu+dαv)βμ(dαu+dαv)μ(uvE(G)(dαu+dαv)p(βμ))1/p(uvE(G)(dαu+dαv)pμ/(p1))(p1)/p,KAα,β(G)pKAα,p(βμ)(G)KAα,pμ/(p1)(G)p1.

    If G is a biregular graph with m edges, we obtain

    KAα,p(βμ)(G)KAα,pμ/(p1)(G)p1=(Δα+δα)p(βμ)m((Δα+δα)pμ/(p1)m)p1=(Δα+δα)p(βμ)(Δα+δα)pμmp=((Δα+δα)βm)p=KAα,β(G)p.

    Since

    (dαu+dαv)p(βμ)(dαu+dαv)pμ/(p1)=(dαu+dαv)p(βμpp1),

    if αp(βμpp1)0, then

    (2δα)p(βμpp1)(dαu+dαv)p(βμ)(dαu+dαv)pμ/(p1)(2Δα)p(βμpp1),

    and if αp(βμpp1)<0, then

    (2Δα)p(βμpp1)(dαu+dαv)p(βμ)(dαu+dαv)pμ/(p1)(2δα)p(βμpp1).

    Proposition 9 gives

    KAα,β(G)=uvE(G)(dαu+dαv)βμ(dαu+dαv)μDp(uvE(G)(dαu+dαv)p(βμ))1/p(uvE(G)(dαu+dαv)pμ/(p1))(p1)/p,KAα,β(G)pDppKAα,p(βμ)(G)KAα,pμ/(p1)(G)p1.

    Proposition 9 gives that the equality is tight in this last bound for some α,β,μ,p with α(βμpp1)0 if and only if

    (2δα)p(βμpp1)=(2Δα)p(βμpp1)δ=Δ,

    i.e., G is regular.

    If we take β=0 in Theorem 10 we obtain the following result.

    Corollary 11. Let G be any graph with m edges, α,μR and p>1.Then

    KAα,pμ(G)KAα,pμ/(p1)(G)p1mp.

    The equality in the bound is tight for each α,μ,p if G is a biregular graph.

    If we take α=2, β=0, p=2 and μ=1/4 in Theorem 10 we obtain the following result.

    Corollary 12. Let G be any graph with maximum degree Δ, minimum degree δ and m edges, then

    m2mSO(G)SO(G)(Δ+δ)24Δδm2.

    The equality in the upper bound is tight if and only if G is regular.The equality in the lower bound is tight if G is a biregular graph.

    Note that the following result improves the upper bound in Corollary 5 when α>1.

    Theorem 13. Let G be any graph with minimum degree δ, and α1.Then

    21/α1M1(G)SOα(G)M1(G)(221/α)δ,

    and the equality holds for some α>1 in each bound if and only if G is regular.

    Proof. The lower bound follows from Corollary 5. Let us prove the upper bound.

    First of all, we are going to prove that

    (xα+yα)1/αx+(21/α1)y (2.1)

    for every α1 and xy0. Since (2.1) is direct for α=1, it suffices to consider the case α>1.

    We want to compute the minimum value of the function

    f(x,y)=x+(21/α1)y

    with the restrictions g(x,y)=xα+yα=1, xy0. If (x,y) is a critical point, then there exists λR such that

    1=λαxα1,21/α1=λαyα1,

    and so, (y/x)α1=21/α1 and y=(21/α1)1/(α1)x; this fact and the equality xα+yα=1 imply

    (1+(21/α1)α/(α1))xα=1,x=(1+(21/α1)α/(α1))1/α,y=(21/α1)1/(α1)(1+(21/α1)α/(α1))1/α,f(x,y)=(1+(21/α1)α/(α1))1/α+(21/α1)(21/α1)1/(α1)(1+(21/α1)α/(α1))1/α=(1+(21/α1)α/(α1))1/α+(21/α1)α/(α1)(1+(21/α1)α/(α1))1/α=(1+(21/α1)α/(α1))(α1)/α>1.

    If y=0, then x=1 and f(x,y)=1.

    If y=x, then x=21/α=y and

    f(x,y)=21/α+(21/α1)21/α=1.

    Hence, f(x,y)1 and the bound is tight if and only if y=0 or y=x. By homogeneity, we have f(x,y)1 for every xy0 and the bound is tight if and only if y=0 or y=x. This finishes the proof of (2.1).

    Consequently,

    (dαu+dαv)1/αdu+(21/α1)dv=du+dv(221/α)dv

    for each α1 and dudv. Thus,

    (dαu+dαv)1/αdu+dv(221/α)δ

    for each α1 and uvE(G), and the equality holds for some α>1 if and only if du=dv=δ. Therefore,

    SOα(G)M1(G)(221/α)δ,

    and the equality holds for some α>1 if and only if du=dv=δ for every uvE(G), i.e., G is regular.

    Corollary 14. Let G be any graph with minimum degree δ.Then

    21/2M1(G)SO(G)M1(G)(22)δ,

    and the equality holds in each bound if and only if G is regular.

    The upper bound in Corollary 14 appears in [14,Theorem 7]. Hence, Theorem 13 generalizes [14,Theorem 7].

    A family of topological indices, named Adriatic indices, was put forward in [24,25]. Twenty of them were selected as significant predictors in Mathematical Chemistry. One of them, the inverse sum indeg index, ISI, was singled out in [25] as a significant predictor of total surface area of octane isomers. This index is defined as

    ISI(G)=uvE(G)dudvdu+dv=uvE(G)11du+1dv.

    In the last years there has been an increasing interest in the mathematical properties of this index. We finish this section with two inequalities relating the Sombor, the first Zagreb and the inverse sum indeg indices.

    Theorem 15. Let G be any graph, then

    2(M1(G)2ISI(G))SO(G)>M1(G)2ISI(G)

    and the upper bound is tight if and only if all the connected components of G are regular graphs.

    Proof. It is well-known that for x,y>0, we have

    x2+y2<(x+y)22(x2+y2),x2+y2<x+y2x2+y2,

    and the equality

    d2u+d2vd2u+d2v+2dudv=(du+dv)2

    give

    (du+dv)d2u+d2v+2dudv>(du+dv)2,d2u+d2v+2dudvdu+dv>du+dv,SO(G)+2ISI(G)>M1(G).

    In a similar way, we obtain

    12(du+dv)d2u+d2v+2dudv(du+dv)2,d2u+d2v+22dudvdu+dv2(du+dv),SO(G)+22ISI(G)2M1(G).

    The equality in this last inequality is tight if and only if 2(d2u+d2v)=(du+dv)2 for each edge uv, i.e., du=dv for every uvE(G), and this happens if and only if all the connected components of G are regular graphs.

    We start this section with a technical result.

    Proposition 16. Let G be any graph, u,vV(G) with uvE(G), and α,βR{0} with αβ>0.Then KAα,β(G{uv})>KAα,β(G).If α>0, then redKAα,β(G{uv})>redKAα,β(G).Furthermore, if α<0 and G does not have pendant vertices, then redKAα,β(G{uv})>redKAα,β(G).

    Proof. Let {w1,,wdu} and {w1,,wdv} be the sets of neighbors of u and v in G, respectively. Since αβ>0, the function

    U(x,y)=(xα+yα)β

    is strictly increasing in each variable if x,y>0. Hence,

    KAα,β(G{uv})KAα,β(G)=((du+1)α+(dv+1)α)β++duj=1(((du+1)α+dαwj)β(dαu+dαwj)β)+dvk=1(((dv+1)α+dαwk)β(dαv+dαwk)β)>((du+1)α+(dv+1)α)β>0.

    The same argument gives the results for the redKAα,β index.

    Given an integer number n2, let Γ(n) (respectively, Γc(n)) be the set of graphs (respectively, connected graphs) with n vertices.

    We study in this section the extremal graphs for the KAα,β index on Γc(n) and Γ(n).

    Theorem 17. Consider α,βR{0} with αβ>0, and an integer n2.

    (1) The complete graph Kn is the unique graph that maximizes KAα,β on Γc(n) or Γ(n).

    (2) Any graph that minimizes KAα,β on Γc(n) is a path.

    (3) If n is even, then the union of n/2 paths P2 is the unique graph that minimizes KAα,β on Γ(n).If n is odd, then the union of (n3)/2 paths P2 with a path P3 isthe unique graph that minimizes KAα,β on Γ(n).

    (4) Furthermore, if α,β>0, then the three previous statements hold if we replace KAα,β with redKAα,β.

    Proof. Let G be a graph with order n, minimum degree δ and m edges.

    Items (1) and (2) follow directly from Proposition 16.

    (3) Assume that n is even. It is well known that the sum of the degrees of a graph is equal to twice the number of edges of the graph (handshaking lemma). Thus, 2mnδn. Since αβ>0, the function

    U(x,y)=(xα+yα)β

    is strictly increasing in each variable if x,y>0. Hence, for any graph GΓ(n), we have

    KAα,β(G)=uvE(G)(dαu+dαv)βuvE(G)(1α+1α)β=2βm2βn2=2β1n,

    and the equality is tight in the inequality if and only if du=1 for all uV(G), i.e., G is the union of n/2 path graphs P2.

    Finally, assume that n is odd. Fix a graph GΓ(n). If du=1 for every uV(G), then handshaking lemma gives 2m=n, a contradiction (recall that n is odd). Therefore, there exists a vertex w with dw2. By handshaking lemma we have 2m(n1)δ+2n+1. Recall that the set of neighbors of the vertex w is denoted by N(w). Since U(x,y) is a strictly increasing function in each variable, we obtain

    KAα,β(G)=uN(w)(dαu+dαw)β+uvE(G),u,vw(dαu+dαv)βuN(w)(1α+2α)β+uvE(G),u,vw(1α+1α)β2(1+2α)β+2β(m2)2(1+2α)β+2β(n+122)=2(1+2α)β+2βn32,

    and the bound is tight if and only if du=1 for all uV(G){w}, and dw=2. Hence, G is the union of (n3)/2 path graphs P2 and a path graph P3.

    (4) If α,β>0, then the same argument gives the results for the redKAα,β index.

    We deal now with the optimization problem for redKAα,β when α,β<0.

    Given an integer number n3, we denote by Γwp(n) (respectively, Γwpc(n)) the set of graphs (respectively, connected graphs) with n vertices and without pendant vertices.

    Theorem 18. Consider α,β<0, and an integer n3.

    (1) The cycle graph Cn is the unique graph that minimizes redKAα,β on Γwpc(n).

    (2) The union of cycle graphs are the only graphs that minimize redKAα,β on Γwp(n).

    (3) The complete graph Kn is the unique graph that maximizes redKAα,β on Γwpc(n) or Γwp(n).

    Proof. Let G be a graph with order n, minimum degree δ and m edges. Since a graph without pendant vertices satisfies δ2, handshaking lemma gives 2mnδ2n. Since α,β<0, the function

    U(x,y)=(xα+yα)β

    is strictly increasing in each variable if x,y>0. Hence, for any graph GΓwp(n), we have

    KAα,β(G)=uvE(G)(dαu+dαv)βuvE(G)(2α+2α)β=2(α+1)βm2(α+1)βn,

    and the inequality is tight if and only if du=2 for all uV(G), i.e., the graph G is the union of cycle graphs. If G is connected, then it is the cycle graph Cn.

    Item (3) follows from Proposition 16.

    In this paper, we contributed to the study of inequalities and optimization problems associated with topological indices. In particular, we obtained new lower and upper optimal bounds of general Sombor indices, and we characterized the graphs where equality occurs.

    Specifically, we have obtained inequalities for these indices relating them with other indices: the first Zagreb index, the forgotten index and the first variable Zagreb index. Finally, we solve some extremal problems for general Sombor indices

    We would like to thank the reviewers by their careful reading of the manuscript and their suggestions which have improved the presentation of this work. The research of José M. Rodríguez and José M. Sigarreta was supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB- ´ I00/AEI/10.13039/501100011033), Spain. The research of Jose M. Rodríguez is supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).

    All authors declare no conflicts of interest in this paper.



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