Review

Nanoparticles in biomedical implants: Pioneering progress in healthcare

  • In the quickly evolving world of healthcare today, new technologies are constantly altering the way we approach medical treatments. Biomedical implants are specialist devices designed to enhance physiological functions. Heart pacemakers and artificial joints are merely a few instances of how they could enhance, assist, or replace different bodily parts. These implants have significantly improved our health and enjoyment of life. In this discipline, nanoparticles are now creating new opportunities. These tiny particles are endowed with extraordinary properties that could improve implant functionality. They can decrease side effects, increase the effectiveness of therapy, and facilitate more harmonious interactions between implants and human bodies. We explored the connection between implants and nanoparticles and showed how both might enhance universal healthcare. This review provides a roadmap for where this technology is headed and how it can help us design safer, more effective implants that enhance people's lives by examining the most recent research and discoveries. It is an intriguing look at healthcare in the future that demonstrates how nanoparticles are reshaping the biomedical implant industry.

    Citation: David Raymond, Induni Nayodhara Weerarathna, John Kessellie Jallah, Praveen Kumar. Nanoparticles in biomedical implants: Pioneering progress in healthcare[J]. AIMS Bioengineering, 2024, 11(3): 391-438. doi: 10.3934/bioeng.2024019

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  • In the quickly evolving world of healthcare today, new technologies are constantly altering the way we approach medical treatments. Biomedical implants are specialist devices designed to enhance physiological functions. Heart pacemakers and artificial joints are merely a few instances of how they could enhance, assist, or replace different bodily parts. These implants have significantly improved our health and enjoyment of life. In this discipline, nanoparticles are now creating new opportunities. These tiny particles are endowed with extraordinary properties that could improve implant functionality. They can decrease side effects, increase the effectiveness of therapy, and facilitate more harmonious interactions between implants and human bodies. We explored the connection between implants and nanoparticles and showed how both might enhance universal healthcare. This review provides a roadmap for where this technology is headed and how it can help us design safer, more effective implants that enhance people's lives by examining the most recent research and discoveries. It is an intriguing look at healthcare in the future that demonstrates how nanoparticles are reshaping the biomedical implant industry.



    Combinatorics is the study of how items are arranged by predetermined principles. Combinatorial approaches are required to count, enumerate, or describe potential solutions. Many difficult real-world situations require finding an optimal solution in a finite solution space. The literature contains a large number of issues that could be solved in polynomial time, even though some practical problems, such as determining the shortest or cheapest roundtrips, internet data packet routing, planning, scheduling, and timetabling, appear to be NP-complete. Several combinatorial strategies can be used to achieve this, including metric dimension (MD), fault-tolerant MD, and DMD. The networks considered in this study are simple (no multiple loops permitted), finite (both edge and vertex sets are finite), linked (a path exists between any two pairs of vertices), and undirected (the edges are not oriented). Consider a network G of order n with the edge set EG and vertex set VG. For a vertex νVG, the representation (l-tuple) r(ν)=(dτ|1τl), where dτ=dτ(ν,ντ), is said to be the distance vector of the vertex ν. The distance vector between two vertices νu could be the same. In this case, the question of determining which vertices have unique distance vectors arises. This difficulty led to the definition of the resolving sets, introduced by Slater [1] in 1975 and Harary and Melter [2] in 1976.

    Definition 1.1. 1) The vertex xVG determines (resolves) the vertices v,wVG if d(v,x)d(w,x).

    2) Let WGVG, where WG={yτ|1τl};(ln) is an ordered set and zVG, then the notation r(z|WG) is the distance vector of the vertex z in association with WG is the l-tuple (d(z,yτ))lτ=1, also called the vector of metric coordinates.

    3) The set WG is said to be a resolving set for network G, if the l-tuple (d(z,yτ))lτ=1 is unique with respect to z.

    4) A metric basis of network G is a resolving set with minimal number of entries. The number of entries in a metric basis is denoted by dim(G) and is called the MD of G.

    The MD has a wide range of potential applications in science, sociology, and engineering. In this context, we will explore the diverse applications of the MD within different scientific fields. Several disciplines, such as telecommunications [3], establishing routing protocols geographically [4], combinatorial optimization [5], and robotics [6] have benefited from the rise and diversity of MD applications. Another area where the MD has significant implications is network discovery and verification [3]. Many applications in chemistry are derived from the vertex-edge relation in networks and its equivalence to the atom and bond relation [7]. Many strategies for the Mastermind [8] games are built using the resolving sets, and the MD of hamming networks is linked to various coin-weighing difficulties explained in [9,10]. It is natural to investigate the mathematical features of this parameter due to its significance in other scientific fields. And after that, we look at some studies on the mathematical importance of this graph-theoretic parameter.

    Several network families of mathematical importance have been studied from the perspective of MD. The field of MD has seen significant advancements, which we will discuss below. The chorded cycle, kayak paddle networks [11], and Mobius ladders [12] are examples of families for which MD has been estimated. Bailey and others have examined the MD of specific families of distance-regular networks, such as Grassmann networks [13], Kneser networks, and Johnson networks [14].

    Cayley networks and Cayley digraphs, which are examples of networks generated by finite groups with group-theoretic importance, have been studied from the MD perspective [15,16]. There has also been research into the MD and resolving sets of product networks, such as the categorical product of networks [17] and the cartesian product of graphs [18]. Kratica et al. [19] (and, respectively, Imran et al. [20]) investigated the MD of rotationally symmetric convex polytopes (and, respectively, convex polytopes created by wheel-related networks). Siddiqui et al. [21] investigated certain infinite families derived from wheel networks for the MD problem.

    Burdett et al. [22] determined various domination numbers for flower snarks, including independent domination, 2-domination, total domination, connected domination, upper domination, secure domination, and weak Roman domination numbers. In [23], the exact results of the mixed MD on two special classes of networks, namely flower snarks and wheel-related networks, were presented. Another study by Girisha et al. [24] characterized the MD and distance matrix of flower snark and sunflower networks. Additionally, due to computational complexity in finding exact values, Liu et al. provided bounds for the partition dimension of certain convex polytope structures in their work [25]. Lastly, Nawaz et al. [26] discussed the edge MD of various structures of benzodiazepines, which are types of drugs used for the treatment of depression.

    Caceres et al. [27] responded to the question of whether the MD is a finite number. They demonstrated this by constructing infinite families of networks having infinite MDs. The computational difficulty of the MD problem was addressed by Garey and Johnson [28], just like many other graph-theoretic characteristics. The bounds of the MD have been examined for generalized Petersen networks by Shao et al. (refer to [29]). All connected networks possessing MDs n1, n2, and 1 were determined by Chartrand et al. [7]. Raza et al. [30] examined the fault-tolerant MD of infinite sets of regular networks, categorizing them accordingly. Baca et al. [31] and Tomescu et al. [32] studied the concept of resolving sets for the regular bipartite networks and the family of necklace networks, respectively. For more details, see [33,34].

    MD and its associated parameters are widely used among researchers today because of the relevance of MD in the identification of nodes in networks. The concept of the DRS and DMD of the network, which is a generalization of the MD of the network, is a distinctive and significant parameter. The term "DRS" was defined by Caceres et al. [18] as follows:

    Definition 1.2. 1) The vertices e1,e2G (where |G|2) are said to doubly resolve the vertices f1,f2 of network G if d(e1,f1)d(e1,f2)d(e2,f1)d(e2,f2).

    2) An ordered set DG={eτ|1τl} of G is a DRS of G if there does not exist a pair of vertices in G that has the same difference in their associated metric coordinates with respect to DG.

    3) In other words, all coordinates of the vector r(f1|DG)r(f2|DG) cannot be the same for any pair of distinct vertices f1,f2VG. i.e. r(f1|DG)r(f2|DG)γI, where I is the unit vector (1,1,,1) and γ is an integer.

    A DRS having a minimal cardinal number is called a MDRS. The cardinal number of an MDRS is the DMD of G, denoted by ψ(G). Moreover, a DRS is a resolving set as well, and ψ(G)dim(G). It was shown in [35] (and, respectively, in [6]) that finding ψ(G) (and, respectively, dim(G)) is an NP-hard task. The task of identifying MDRSs for necklace networks and certain convex polytopes was first addressed in [36] and [19]. Furthermore, authors in [37,38] demonstrated that the DMD remains constant for certain convex polytope structures. For various families of Harary networks, Ahmad et al. calculated the MDRSs (see [39] for more information).

    In [40], it was shown that the dim(G) and ψ(G) of some families of circulant networks were the same in all cases. The problem of determining the MDRS is discussed for many network families; for example, the MDRSs of prisms and Hamming networks were investigated in [41] and [42], respectively. Chen and Wang [43] conducted a study to identify the MDRSs for trees, cycles, wheels, and k-edge-augmented trees. In another study [44], Chen et al. were the first to present explicit approximations of upper and lower bounds for the MDRS problem.

    An investigation into the MDRSs and MD of certain network families was conducted by Liu et al. [45]. In a recent study, Ahmad et al. [46] investigated the MDRSs for the chordal ring networks. Moreover, in [47], the MDRSs of layer-sun networks and their line networks have been explored in detail. The DMD and MDRSs for certain classes of convex polytopes were found by Pan et al. [48]. Jannesari [49] characterized all networks G having the DMD 2 by using 2-connected subgraphs of network G. Recently, the generalized solutions for the DRS problem of several families of networks were found by Ahmad et al. [50].

    The reason for studying doubly resolving sets (DRSs) is their potential to improve the accuracy and efficiency of source detection in different network scenarios. Our study aimed to explore the theoretical principles of certain network properties using the flower snark network as a model due to its distinct and well-defined structural characteristics. The results of the study are significant as they present calculated double metric dimensions (DMDs) for complex network models such as flower snarks and quasi-flower snarks. These findings not only demonstrate the real-world application of DRSs in intricate networks but also provide valuable insights into their structural traits. This comprehension can guide future work in the theory of networks and its practical implementations, leading to more effective source detection techniques in various real-world networks.

    MDRSs play a crucial role in identifying diffusion sources in intricate networks. Locating the origins of an outbreak in such networks is an exciting yet challenging task. For instance, consider the scenario where the only available information about the epidemic is from a subset of nodes referred to as sensors. These sensors can indicate if and when they have been infected with the virus. The question then arises: What is the minimum number of sensors required to identify the source of the outbreak accurately? Fortunately, a well-known network feature called DMD offers a solution to this problem (for details, see [51,52]).

    Let us look at a few real-world instances. The DMD in an n-node star network is n1. DRSs cannot be formed from sets that do not include all of the external nodes. On the other hand, the DMD for a path network is 2. The two nodes, one at each end of the path, are sensors always sufficient to identify the source. Using these instances, it is clear that the DMD and the difficulty in finding the source largely depend on the network topology. A path-like network makes it far easier to find the source of an outbreak than a star-like network [52].

    The results on minimal resolving sets for the flower snark Jm and quasi-flower snarks Gm were determined by Imran et al. [53] and Naeem et al. [54] and are given in the following theorems:

    Theorem 1.3. [53] Let Jm be the flower snark. Then, for every odd positive integer m5, we have dim(Jm)=3.

    Theorem 1.4. [54] Let Gm be the quasi-flower snark. Then, for every positive integer m4, we have dim(Gm)={3,if m is odd4,otherwise.

    The results presented above determine the lower bound on the DMD of flower snarks Jm and quasi-flower snarks Gm. Following is how the main contribution of the article is structured:

    ● We compute the DMD for flower snarks Jm and quasi-flower snarks Gm in Sections 2 and 3.

    ● The utilization of the research in the context of the source of diffusion in a complex network is presented in Section 4.

    ● Finally, in Section 5, we demonstrate that the DMD for flower snarks depends on the order of the network, and the DMD for quasi-flower snarks is independent of the order of the network.

    In this section, we will concentrate on determining the DMD of flower snarks Jm.

    The flower snark Jm, depicted in Figure 1, is a cubic network comprising 4m vertices and 6m edges. The vertex set for the flower snarks Jm is VJm = {cτ,fτ,gτ,hτ: 0τm1}, and the edge set is EJm = {cτcτ+1,cτfτ,fτgτ,fτhτ,gτgτ+1,hτhτ+1: 0τm2} {cm1c0,cm1fm1,fm1gm1,fm1hm1,gm1g0,hm1h0,g0hm1,gm1h0}.

    Figure 1.  Flower snark J5.

    As shown in Figure 1, the inner cycle vertices are labeled as {cτ : 0τm1}, the set of central vertices are labeled as {fτ : 0τm1}, and the outer cycle vertices are labeled as {gτ : 0τm1} {hτ : 0τm1}.

    MDRSs for Jm are to be found in this section. It follows from Theorem 1.3 that ψ(Jm)3, for all m6. We shall also show, for all m6:

     ψ(Jm)={4,if m is odd;5,if m is even.

    Define the vertex subset Sτ(f0)={gVJm:d(f0,g)=τ}VJm having distance τ from f0. The Table 1 is a simple formulation for Sτ(f0) and will be used to compute the distances between each pair of vertices in VJm.

    Table 1.  Sτ(f0) for Jm.
    m τ Sτ(f0)
    1 {c0, g0, h0}
    2 {c1, cm1, g1, gm1, h1, hm1}
    3τm2 {cτ1, cmτ+1, fτ2, fmτ+2, gτ1, gmτ+1, hτ1, hmτ+1}
    even m+22 {cm2, fm22, fm+22, gm2, hm2}
    m+42 {fm2}
    odd m+12 {cm12, cm+12, fm32, fm+32, gm12, gm+12, hm12, hm+12}
    m+32 {fm12, fm+12}

     | Show Table
    DownLoad: CSV

    The following facts can be demonstrated as a result of the symmetry of Jm, where m6:

    d(fτ,fυ)=d(f0,f|τυ|) if 0|τυ|m1; d(cτ,cυ)=d(f0,c|τυ|)1 if 0|τυ|m1;

    d(cτ,fυ)=d(f0,c|τυ|) if 0|τυ|m1;

    d(cτ,gυ)=d(cτ,hυ)=d(f0,c|τυ|)+1 if 0|τυ|m1;

    d(fτ,gυ)=d(fτ,hυ)=d(f0,g|τυ|) if 0|τυ|m1.

    If m is an odd integer, then

    d(gτ,hυ)={d(f0,c|τυ|)+1,if 0|τυ|m12;d(f0,c|τυ|)1,if m+12|τυ|m1.d(gτ,gυ)=d(hτ,hυ)={d(f0,g|τυ|)1,if 0|τυ|m12;d(f0,g|τυ|)+1,if m+12|τυ|m1.

    If m is an even integer, then

    d(gτ,hυ)={d(f0,c|τυ|)+1,if 0|τυ|m22;d(f0,c|τυ|)1,if m2|τυ|m1.d(gτ,gυ)=d(hτ,hυ)={d(f0,g|τυ|)1,if 0|τυ|m2;d(f0,g|τυ|)+1,if m+22|τυ|m1.

    Therefore, we can recalculate the distances between each pair of vertices in VJm if d(f0,g) is known for each gVJm.

    Lemma 2.1. For any even positive integer m6, ψ(Jm)>4.

    Proof. To prove the DMD ψ(Jm)>4, we must show that every subset DJmVJm of cardinality 4 is not a DRS for Jm. Table 2 demonstrates the non-doubly resolved pairs of vertices from the set VJm as well as all possible forms of subset DJm. Let us compute that any two vertices of the set {cτ,cυ,cω,f0;0τ<υ<ωm1} do not doubly resolve the vertices f0,g0. For example, it is clear that d(f0,f0)=0,d(f0,g0)=1. Now, for 0τm22,1υm22,2ωm22, we have

    Table 2.  Non-doubly resolving pairs of vertices for Jm, even positive integer m6.
    DJm Non-doubly resolved pairs
    {cτ,cυ,cω,f0}, 0τ<υ<ωm1 {f0,g0}
    {cτ,cυ,f0,fω}, 0τ<υm1, 1ωm1 {g0,h0}
    {cτ,f0,fυ,fω}, 0τm1, 1υ<ωm1 {g0,h0}
    {cτ,f0,fυ,gω} {cτ,f0,fυ,hω}, 0τm1, 1υm1, 0ωm22 {gm+2ω2,hm+2ω2}
    {cτ,f0,fυ,gω} {cτ,f0,fυ,hω}, 0τm1, 1υm1, m2ωm2 {g2ωm2,h2ωm2}
    {cτ,f0,fυ,gω} {cτ,f0,fυ,hω}, 0τm1, 1υm1, ω=m1 {gm22,hm22}
    {cτ,cυ,f0,gω}, 0τ<υm1, 0ωm22 {f0,h0}
    {cτ,cυ,f0,gω}, 0τ<υm1, m2ωm2 {g2ωm2,h2ωm2}
    {cτ,cυ,f0,gω}, 0τ<υm1, ω=m1 {f0,g0}
    {cτ,cυ,f0,hω}, 0τ<υm1, 0ωm22 {f0,g0}
    {cτ,cυ,f0,hω}, 0τ<υm1, m2ωm2 {g2ωm2,h2ωm2}
    {cτ,cυ,f0,hω}, 0τ<υm1, ω=m1 {f0,h0}
    {f0,fτ,fυ,fω}, 1τ<υ<ωm1 {c0,g0}
    {f0,gτ,gυ,hω}, 0τ<υm1, 0ωm1 {c0,f0}
    {f0,fτ,fυ,gω}, 1τ<υm1, 0ωm22 {c0,h0}
    {f0,fτ,fυ,gω}, 1τ<υm1, ω=m2 {g0,h0}
    {f0,fτ,fυ,gω}, 1τ<υm1, m+22ωm1 {c0,g0}
    {f0,fτ,fυ,hω}, 1τ<υm1, 0ωm22 {c0,g0}
    {f0,fτ,fυ,hω}, 1τ<υm1, ω=m2 {g0,h0}
    {f0,fτ,fυ,hω}, 1τ<υm1, m+22ωm1 {c0,h0}
    {f0,gτ,gυ,gω} {f0,hτ,hυ,hω}, 0τ<υ<ωm1 {c0,f0}
    {f0,gτ,hυ,hω}, 0τm1, 0υ<ωm1 {c0,f0}

     | Show Table
    DownLoad: CSV

    d(cτ,f0)=d(f0,cτ)=τ+1,

    d(cτ,g0)=d(f0,cτ)+1=τ+2,

    d(cυ,f0)=d(f0,cυ)=υ+1,

    d(cυ,g0)=d(f0,cυ)+1=υ+2,

    d(cω,f0)=d(f0,cω)=ω+1,

    d(cω,g0)=d(f0,cω)+1=ω+2.

    Also, for m2τm3,m2υm2,m2ωm1, we have

    d(cτ,f0)=d(f0,cτ)=mτ+1,

    d(cτ,g0)=d(f0,cτ)+1=mτ+2,

    d(cυ,f0)=d(f0,cυ)=mυ+1,

    d(cυ,g0)=d(f0,cυ)+1=mυ+2,

    d(cω,f0)=d(f0,cω)=mω+1,

    d(cω,g0)=d(f0,cω)+1=mω+2.

    So, d(cτ,f0)d(cτ,g0)=d(cυ,f0)d(cυ,g0)=d(cω,f0)d(cω,g0)=d(f0,f0)d(f0,g0)=1, that is, the set {cτ,cυ,cω,f0;0τ<υ<ωm1} is not a DRS of Jm. Accordingly, we can investigate all the possible types of subset DJm shown in Table 2 and verify their associated non-doubly resolved pairs of vertices.

    Lemma 2.2. For any odd positive integer m6, ψ(Jm)>3.

    Proof. To prove the DMD ψ(Jm)>3, we must show that every subset DJmVJm of cardinality 3 is not a DRS for Jm. Table 3 demonstrates the non-doubly resolved pairs of vertices from the set VJm as well as all possible forms of subset DJm. Let us compute that any two vertices of the set {f0,gτ,hυ;0τ,υm1} do not doubly resolve the vertices c0,f0. For example, it is clear that d(c0,f0)=1,d(f0,f0)=0. Now, for 0τ,υm12, we have

    Table 3.  Non-doubly resolving pairs of vertices for Jm, odd positive integer m6.
    DJm Non-doubly resolved pairs
    {cτ,cυ,f0}, 0τ<υm1 {f0,g0}
    {cτ,f0,fυ}, 0τm1, 1υm1 {g0,h0}
    {f0,fτ,fυ}, 1τ<υm1 {c0,g0}
    {f0,fτ,gυ}, 1τm1, 0υm32 {c0,h0}
    {f0,fτ,gυ}, 1τm1, m12υm1 {cυ1,hυ1}
    {f0,fτ,hυ}, 1τm1, 0υm32 {c0,g0}
    {f0,fτ,hυ}, 1τm1, m12υm1 {cυ1,gυ1}
    {f0,gτ,gυ} {f0,hτ,hυ}, 0τ<υm1 {c0,f0}
    {f0,gτ,hυ}, 0τ,υm1 {c0,f0}

     | Show Table
    DownLoad: CSV

    d(c0,gτ)=d(f0,cτ)+1=τ+2,

    d(f0,gτ)=d(f0,gτ)=τ+1,

    d(c0,hυ)=d(f0,cυ)+1=υ+2,

    d(f0,hυ)=d(f0,gυ)=υ+1.

    Also, for m+12τ,υm1, we have

    d(c0,gτ)=d(f0,cτ)+1=mτ+2,

    d(f0,gτ)=d(f0,gτ)=mτ+1,

    d(c0,hυ)=d(f0,cυ)+1=mυ+2,

    d(f0,hυ)=d(f0,gυ)=mυ+1.

    So, d(c0,gτ)d(f0,gτ)=d(c0,hυ)d(f0,hυ)=d(c0,f0)d(f0,f0)=1, that is, the set {f0,gτ,hυ;0τ,υm1} is not a DRS of Jm. Accordingly, we can investigate all the possible types of subset DJm shown in Table 3 and verify their associated non-doubly resolved pairs of vertices.

    Lemma 2.3. ψ(Jm)=5, even integers m6.

    Proof. It suffices to identify a DRS having order 5 to demonstrate that ψ(Jm)=5, if m6 is an even integer. Now, by using the sets Sτ(f0) from Table 1, the metric coordinate vectors are demonstrated in Table 4 for each vertex of Jm with respect to the set DJm={c0,c1,cm2,g0,gm1}.

    Table 4.  Metric coordinate vectors for Jm if m6 is even.
    τ Sτ(f0) DJm={c0,c1,cm2,g0,gm1}
    0 f0 (1,2,m+22,1,2)
    1 c0 (0,1,m2,2,3)
    g0 (2,3,m+42,0,3)
    h0 (2,3,m+42,2,1)
    2 c1 (1,0,m22,3,4)
    cm1 (1,2,m22,3,2)
    g1 (3,2,m+22,1,4)
    gm1 (3,4,m+22,3,0)
    h1 (3,2,m+22,3,2)
    hm1 (3,4,m+22,1,2)
    3τm2 cτ1 (τ1,τ2,m2τ+22,τ+1,τ+2)
    cmτ+1 (τ1,τ,m2τ+22,τ+1,τ)
    fτ2 (τ1,τ2,m2τ+62,τ1,τ)
    fmτ+2 ={(τ+1,τ,m2τ+62,τ1,τ+2), if τ<m2;(m+22,m2,3,m22,m2), if τ=m2.
    gmτ+1 (τ+1,τ+2,m2τ+62,τ+1,τ2)
    hτ1 (τ+1,τ,m2τ+62,τ+1,τ)
    hmτ+1 (τ+1,τ+2,m2τ+62,τ1,τ)
    m+22 cm2 (m2,m22,0,m+42,m+22)
    fm22 (m2,m22,2,m2,m+22)
    fm+22 (m2,m+22,2,m2,m22)
    gm2 (m+42,m+22,2,m2,m22)
    hm2 (m+42,m+22,2,m2,m+22)
    m+42 fm2 (m+22,m2,1,m+22,m2)

     | Show Table
    DownLoad: CSV

    Now, consider Table 4, where the vector of f0Sτ(f0) has its first metric coordinate equal to 1. We may also check that no two vertices a1,a2Sτ(f0) exist such as r(a1,DJm)r(a2,DJm)=0I, where I is the unit vector and τ{1,2,,m+42}. Furthermore, no two vertices a1Sτ(f0) and a2Sυ(f0) exist such as r(a1,DJm)r(a2,DJm)=τυ, for some τ,υ{1,2,,m+42} and τυ. Finally, the set DJm={c0,c1,cm2,g0,gm1} is the MDRS. As a result, Lemma 2.3 holds true.

    Lemma 2.4. ψ(Jm)=4, odd integers m6.

    Proof. To demonstrate that ψ(Jm)=4, where m6 is an odd integer, it is enough to find a DRS of order 4. Now, by using the sets Sτ(f0) from Table 1, the metric coordinate vectors are demonstrated in Table 5 for each vertex of Jm with respect to the set DJm={c0,cm32,fm12,g0}.

    Table 5.  Metric coordinate vectors for Jm if m6 is odd.
    τ Sτ(f0) DJm={c0,cm32,fm12,g0}
    0 f0 (1,m12,m+32,1)
    1 c0 (0,m32,m+12,2)
    g0 (2,m+12,m+12,0)
    h0 (2,m+12,m+12,2)
    2 c1 (1,m52,m12,3)
    cm1 (1,m12,m+12,3)
    g1 (3,m12,m12,1)
    gm1 (3,m+32,m+12,3)
    h1 (3,m12,m12,3)
    hm1 (3,m+32,m+12,1)
    3τm12 cτ1 (τ1,m2τ12,m2τ+32,τ+1)
    cmτ+1 (τ1,m2τ+52,m2τ+52,τ+1)
    fτ2 (τ1,m2τ+32,m2τ+72,τ1)
    fmτ+2 ={(2,m+12,m+32,2), if τ=3;(τ1,m2τ+92,m2τ+92,τ1), if τ>3.
    gτ1 (τ+1,m2τ+32,m2τ+32,τ1)
    gmτ+1 (τ+1,m2τ+92,m2τ+52,τ+1)
    hτ1 (τ+1,m2τ+32,m2τ+32,τ+1)
    hmτ+1 (τ+1,m2τ+92,m2τ+52,τ1)
    m+12 cm12 (m12,1,1,m+32)
    cm+12 (m12,2,2,m+32)
    fm32 (m12,1,3,m12)
    fm+32 (m12,4,4,m12)
    gm12 (m+32,3,1,m12)
    gm+12 (m+32,4,2,m+12)
    hm12 (m+32,3,1,m+12)
    hm+12 (m+32,4,2,m12)
    m+32 fm12 (m+12,2,0,m+12)
    fm+12 (m+12,3,3,m+12)

     | Show Table
    DownLoad: CSV

    Now, consider Table 5, where the vector of f0Sτ(f0) has its first metric coordinate equal to 1. We may also check that no two vertices a1,a2Sτ(f0) exist such as r(a1,DJm)r(a2,DJm)=0I, where I is the unit vector and τ{1,2,,m+32}. Furthermore, no two vertices a1Sτ(f0) and a2Sυ(f0) exist such as r(a1,DJm)r(a2,DJm)=τυ, for some τ,υ{1,2,,m+32} and τυ. Finally, the set DJm={c0,cm32,fm12,g0} is the MDRS. As a result, Lemma 2.4 holds true.

    The combination of Lemmas 2.3 and 2.4 yields the following main theorem:

    Theorem 2.5. Let Jm be the flower snark, then for m6,

    ψ(Jm)={4,if m is odd ;5,if m is even .

    In this section, we will concentrate on determining the DMD of quasi-flower snarks Gm.

    Figure 2 demonstrates the network of quasi-flower snark Gm, which has 4m vertices and 6m edges. The vertex set for the quasi-flower snarks is VGm = {cτ,fτ,gτ,hτ: 0τm1}, and the edge set is EGm = {cτcτ+1,cτfτ,fτgτ,fτhτ,gτgτ+1,hτhτ+1: 0τm2} {cm1c0,cm1fm1,fm1gm1,fm1hm1,gm1g0,hm1h0}.

    Figure 2.  Quasi-flower snark G9.

    As shown in Figure 2, the inner cycle vertices are labeled as {cτ : 0τm1}, the set of central vertices are labeled as {fτ : 0τm1}, and the outer cycle vertices are labeled as {gτ : 0τm1} {hτ : 0τm1}.

    MDRSs for Gm are to be found in this section. It follows from Theorem 1.4 that

    ψ(Gm){3,if m is odd;4,  otherwise,

    for all m6. We shall also show that ψ(Gm)=4, for all m6.

    Define the vertex subset Sτ(f0)={gVGm:d(f0,g)=τ}VGm having distance τ from f0. The Table 6 is a simple formulation for Sτ(f0) and will be used to compute the distances between each pair of vertices in VGm.

    Table 6.  Sτ(f0) for Gm.
    m τ Sτ(f0)
    1 {c0, g0, h0}
    2 {c1, cm1, g1, gm1, h1, hm1}
    3τm2 {cτ1, cmτ+1, fτ2, fmτ+2, gτ1, gmτ+1, hτ1, hmτ+1}
    even m+22 {cm2, fm22, fm+22, gm2, hm2}
    m+42 {fm2}
    odd m+12 {cm12, cm+12, fm32, fm+32, gm12, gm+12, hm12, hm+12}
    m+32 {fm12, fm+12}

     | Show Table
    DownLoad: CSV

    The following facts can be demonstrated as a result of the symmetry of Gm, where m6:

    d(fτ,gυ)=d(fτ,hυ)=d(f0,g|τυ|) if 0|τυ|m1;

    d(cτ,cυ)=d(f0,c|τυ|)1 if 0|τυ|m1;

    d(cτ,fυ)=d(f0,c|τυ|) if 0|τυ|m1;

    d(cτ,gυ)=d(cτ,hυ)=d(f0,c|τυ|)+1 if 0|τυ|m1;

    d(gτ,hυ)=d(f0,c|τυ|)+1 if 0|τυ|m1;

    d(fτ,fυ)=d(f0,f|τυ|) if 0|τυ|m1;

    d(gτ,gυ)=d(hτ,hυ)=d(f0,g|τυ|)1 if 0|τυ|m1.

    Therefore, we can recalculate the distances between each pair of vertices in VGm if d(f0,g) is known for each gVGm.

    Lemma 3.1. For any positive integer m6, ψ(Gm)>3.

    Proof. To prove the DMD ψ(Gm)>3, we must show that every subset DGmVGm of cardinality 3 is not a DRS for Gm. Table 7 demonstrates the non-doubly resolved pairs of vertices from the set VGm as well as all possible forms of subset DGm. Let us compute that any two vertices of the set {cτ,cυ,f0;0τ<υm1} do not doubly resolve the vertices f0,g0. For example, it is clear that d(f0,f0)=0,d(f0,g0)=1. Now, for 0τm2,1υm2, we have

    Table 7.  Non-doubly resolving pairs of vertices for Gm, where m6.
    DGm Non-doubly resolved pairs
    {cτ,cυ,f0}, 0τ<υm1 {f0,g0}
    {cτ,f0,gυ}, 0τ,υm1 {f0,h0}
    {cτ,f0,fυ}, 0τm1, 1υm1 {g0,h0}
    {cτ,f0,hυ}, 0τ,υm1 {f0,g0}
    {f0,fτ,gυ}, 1τm1, 0υm1 {c0,h0}
    {f0,fτ,fυ}, 1τ<υm1 {c0,g0}
    {f0,gτ,hυ}, 0τ,υm1 {c0,f0}
    {f0,fτ,hυ}, 1τm1, 0υm1 {c0,g0}
    {f0,hτ,hυ}, 0τ<υm1 {f0,g0}
    {f0,gτ,gυ}, 0τ<υm1 ={{c0,h0}, if m is even {c0,f0}, if m is odd 

     | Show Table
    DownLoad: CSV

    d(cτ,f0)=d(f0,cτ)=τ+1;

    d(cτ,g0)=d(f0,cτ)+1=τ+2;

    d(cυ,f0)=d(f0,cυ)=υ+1;

    d(cυ,g0)=d(f0,cυ)+1=υ+2.

    Also, for m+22τm2,m+22υm1, we have

    d(cτ,f0)=d(f0,cτ)=mτ+1;

    d(cτ,g0)=d(f0,cτ)+1=mτ+2;

    d(cυ,f0)=d(f0,cυ)=mυ+1;

    d(cυ,g0)=d(f0,cυ)+1=mυ+2.

    So, d(cτ,f0)d(cτ,g0)=d(cυ,f0)d(cυ,g0)=d(f0,f0)d(f0,g0)=1, that is, the set {cτ,cυ,f0;0τ<υm1} is not a DRS of Gm. Accordingly, we can investigate all the possible types of subset DGm shown in Table 7 and verify their associated non-doubly resolved pairs of vertices.

    Lemma 3.2. ψ(Gm)=4, even integers m6.

    Proof. To demonstrate that ψ(Gm)=4, where m6 is an even integer, it is enough to find a DRS of order 4. Now, by using the sets Sτ(f0) from Table 6, the metric coordinate vectors are demonstrated in Table 8 for every vertex of Gm with respect to the set DGm={c0,cm2,f1,gm+22}.

    Table 8.  Metric coordinate vectors for Gm if m6 is even.
    τ Sτ(f0) DGm={c0,cm2,f1,gm+22}
    0 f0 (1,m+22,3,m2)
    1 c0 (0,m2,2,m+22)
    g0 (2,m+42,2,m22)
    h0 (2,m+42,2,m+22)
    2 c1 (1,m22,1,m+42)
    cm1 (1,m22,3,m2)
    g1 (3,m+22,1,m2)
    gm1 (3,m+22,3,m42)
    h1 (3,m+22,1,m+42)
    hm1 (3,m+22,3,m2)
    3τm2 cτ1 (τ1,m2τ+22,τ1,m2τ+82)
    cmτ+1 (τ1,m2τ+22,τ+1,m2τ+42)
    fτ2 ={(2,m2,0,m+22), if τ=3;(τ1,m2τ+62,τ1,m2τ+82), if τ>3.
    fmτ+2 (τ1,m2τ+62,τ+1,m2τ+42)
    gτ1 (τ+1,m2τ+62,τ1,m2τ+42)
    gmτ+1 (τ+1,m2τ+62,τ+1,m2τ2)
    hτ1 (τ+1,m2τ+62,τ1,m2τ+82)
    hmτ+1 (τ+1,m2τ+62,τ+1,m2τ+42)
    m+22 cm2 (m2,0,m2,3)
    fm22 (m2,2,m2,3)
    fm+22 (m2,2,m+42,1)
    gm2 (m+42,2,m2,1)
    hm2 (m+42,2,m2,3)
    m+42 fm2 (m+22,1,m+22,2)

     | Show Table
    DownLoad: CSV

    Now, consider Table 8, where the vector of f0Sτ(f0) has its first metric coordinate equal to 1. We may also check that no two vertices b1,b2Sτ(f0) exist such as r(b1,DGm)r(b2,DGm)=0I, where I is the unit vector and τ{1,2,,m+42}. Furthermore, no two vertices b1Sτ(f0) and b2Sυ(f0) exist such as r(b1,DGm)r(b2,DGm)=τυ, for some τ,υ{1,2,,m+42} and τυ. Finally, the set DGm={c0,cm2,f1,gm+22} is the MDRS. As a result, Lemma 3.2 holds true.

    Lemma 3.3. ψ(Gm)=4, odd integers m6.

    Proof. To demonstrate that ψ(Gm)=4, where m6 is an odd integer, it is enough to find a DRS of order 4. Now, by using the sets Sτ(f0) from Table 6, the metric coordinate vectors are demonstrated in Table 9 for every vertex of Gm with respect to the set DGm={c0,cm32,fm12,g0}.

    Table 9.  Metric coordinate vectors for Gm if m6 is odd.
    τ Sτ(f0) DGm={c0,cm32,fm12,g0}
    0 f0 (1,m12,m+32,1)
    1 c0 (0,m32,m+12,2)
    g0 (2,m+12,m+12,0)
    h0 (2,m+12,m+12,2)
    2 c1 (1,m52,m12,3)
    cm1 (1,m12,m+12,3)
    g1 (3,m12,m12,1)
    gm1 (3,m+32,m+12,1)
    h1 (3,m12,m12,3)
    hm1 (3,m+32,m+12,3)
    3τm12 cτ1 (τ1,m2τ12,m2τ+32,τ+1)
    cmτ+1 (τ1,m2τ+52,m2τ+52,τ+1)
    fτ2 (τ1,m2τ+32,m2τ+72,τ1)
    fmτ+2 ={(2,m+12,m+32,2), if τ=3;(τ1,m2τ+92,m2τ+92,τ1), if τ>3.
    gτ1 (τ+1,m2τ+32,m2τ+32,τ1)
    gmτ+1 (τ+1,m2τ+92,m2τ+52,τ1)
    hτ1 (τ+1,m2τ+32,m2τ+32,τ+1)
    hmτ+1 (τ+1,m2τ+92,m2τ+52,τ+1)
    m+12 cm12 (m12,1,1,m+32)
    cm+12 (m12,2,2,m+32)
    fm32 (m12,1,3,m12)
    fm+32 (m12,4,4,m12)
    gm12 (m+32,3,1,m12)
    gm+12 (m+32,4,2,m12)
    hm12 (m+32,3,1,m+32)
    hm+12 (m+32,4,2,m+32)
    m+32 fm12 (m+12,2,0,m+12)
    fm+12 (m+12,3,3,m+12)

     | Show Table
    DownLoad: CSV

    Now, consider Table 9, where the vector of f0Sτ(f0) has its first metric coordinate equal to 1. We may also check that no two vertices b1,b2Sτ(f0) exist such as r(b1,DGm)r(b2,DGm)=0I, where I denotes the unit vector and τ{1,2,,m+32}. Furthermore, no two vertices b1Sτ(f0) and b2Sυ(f0) exist such as r(b1,DGm)r(b2,DGm)=τυ, for some τ,υ{1,2,,m+32} and τυ. Finally, the set DGm={c0,cm32,fm12,g0} is the MDRS. As a result, Lemma 3.3 holds true.

    The combination of Lemmas 3.2 and 3.3 yields the following main theorem:

    Theorem 3.4. Let Gm be the quasi-flower snark for m6. Then, ψ(Gm)=4.

    DRSs have recently been employed to find the origin of epidemics in several complex networks, including social networks, human epidemics, the cause of a disease outbreak, etc. (see [51,52]). To get the best possible detection of an epidemic source, we specifically take into account a network that uses DRSs to lower the number of observers.

    Assume a social network set up as a flower snark network J5, where the node set VJ5 = {cτ,fτ,gτ,hτ: 0τ4} stands in for the individuals and the edge set EJ5 = {cτcτ+1,cτfτ,fτgτ,fτhτ,gτgτ+1,hτhτ+1,g0hn1,gn1h0: 0τ4} between the nodes stands in for the connections between the individuals. An immediate solution can be found when observers are placed throughout the network, and the inter-node distances are reliable and well-known. However, the entire process would be very costly and time-consuming.

    Then, what number of observers is necessary to account for epidemic sources with an unknown starting time and node-to-node transmission delays? Each node has a unique representation depending upon the minimum distance from the observer nodes. We employed the MDRS DJ5={c0,c1,f2,g0} of the flower snark network J5 to reduce the number of observers.

    We only placed the observers on the nodes c0,c1,f2,g0 by applying Lemma 2.4 and Theorem 2.5. It is clear from Table 5 and Figure 3 that each node has a unique representation and the epidemic propagation will be optimal by placing observers at the nodes c0,c1,f2,g0. This hypothetical situation clarifies how MDRSs can be used to solve source localization issues.

    Figure 3.  Source detection in social network.

    Moreover, the DRSs are valuable for identifying faults in communication networks, assisting administrators in promptly locating and repairing issues for reliable operations. Additionally, they are beneficial in cybersecurity to detect and localize cyberattacks, allowing for quick responses to breaches. In environmental monitoring, DRSs aid in identifying sources of pollution, facilitating swift actions to address environmental hazards. The findings of this study provide a fundamental understanding of DRSs and their potential to enhance source detection in various real-world scenarios.

    In this article, the DMD for the flower snarks Jm and quasi-flower snarks Gm is computed by describing their MDRSs. We conclude that the DMD for the flower snarks Jm is finite and depends upon the order of the network. Also, the DMD for the quasi-flower snarks Gm is finite and free from the order of the network. An algorithm is created to calculate the MDRSs and DMD of a network with the help of Matlab or any other simulation tool. Further, an application of the work done in the context of source localization is also provided. It is also clearly observed that computing MDRSs for a network is an NP-hard problem (see [35]), which explains the limitations of computing this parameter and the importance of the obtained results. Exploring future directions for DRSs in various families of networks can help in the development of combinatorial designs, error-correcting codes, network analysis, and optimization problems, among other fields. Following are a few points that address the limitations of the study:

    ● Constructing DRSs with desirable properties can be a challenging task.

    ● The problem of finding MDRSs is known to be NP-hard, which means that there is no known efficient algorithm to solve it for all cases.

    ● As a result, finding optimal DRSs can require significant computational effort.

    In conclusion, DRSs have limitations in terms of size, optimality, ambiguity, construction difficulty, practical implementation, and generalization to various problem domains. These restrictions emphasize the importance of carefully taking into account the particular problem requirements and limits and illustrate the difficulties and complexities involved with using DRSs in different scenarios.

    Open Problem 5.1. Investigate the DRSs for the power paths Prn for all integers r, and n, where r<n2.

    Muhammad Ahmad was an engaged participant in initiating the problem, and ensuring the validation of results. Muhammad Faheem contributed to the methodology and conducting the final review. Sanaa A. Bajri played a crucial role in analyzing and computing the results, as well as initiating the drafting process of the study. Zohaib Zahid played a pivotal role in problem discussion, gathering essential data, and providing continuous supervision. Muhammad Javaid participated in characterizing and calculating the results and assisted with software in drafting the study. Hamiden Abd El-Wahed Khalifa contributed to the discussing the problem, and meticulously proofread the final version. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This research is funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R527), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

    The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.



    Authors contributions



    David Raymond: Conceptualization, methodology, writing original draft, visualization. Induni Weerarathna: Conceptualization, writing-review and editing, visualization. John Jallah: Conceptualization, writing-review and editing, visualization, Praveen Kumar: Conceptualization, methodology, writing-review and editing, visualization, supervision.

    Conflict of interest



    There is neither potential conflict of interest was reported by the author(s), nor funding associated with the work featured in this article.

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