Research article Special Issues

Some new results on the face index of certain polycyclic chemical networks


  • Silicate minerals make up the majority of the earth's crust and account for almost 92 percent of the total. Silicate sheets, often known as silicate networks, are characterised as definite connectivity parallel designs. A key idea in studying different generalised classes of graphs in terms of planarity is the face of the graph. It plays a significant role in the embedding of graphs as well. Face index is a recently created parameter that is based on the data from a graph's faces. The current draft is utilizing a newly established face index, to study different silicate networks. It consists of a generalized chain of silicate, silicate sheet, silicate network, carbon sheet, polyhedron generalized sheet, and also triangular honeycomb network. This study will help to understand the structural properties of chemical networks because the face index is more generalized than vertex degree based topological descriptors.

    Citation: Ricai Luo, Khadija Dawood, Muhammad Kamran Jamil, Muhammad Azeem. Some new results on the face index of certain polycyclic chemical networks[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8031-8048. doi: 10.3934/mbe.2023348

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  • Silicate minerals make up the majority of the earth's crust and account for almost 92 percent of the total. Silicate sheets, often known as silicate networks, are characterised as definite connectivity parallel designs. A key idea in studying different generalised classes of graphs in terms of planarity is the face of the graph. It plays a significant role in the embedding of graphs as well. Face index is a recently created parameter that is based on the data from a graph's faces. The current draft is utilizing a newly established face index, to study different silicate networks. It consists of a generalized chain of silicate, silicate sheet, silicate network, carbon sheet, polyhedron generalized sheet, and also triangular honeycomb network. This study will help to understand the structural properties of chemical networks because the face index is more generalized than vertex degree based topological descriptors.



    To exemplify the phenomena of compounds scientifically, researchers utilize the contraption of the diagrammatic hypothesis, it is a well-known branch of geometrical science named graph theory. This division of numerical science provides its services in different fields of sciences. The particular example in networking [1], from electronics [2], and for the polymer industry, we refer to see [3]. Particularly in chemical graph theory, this division has extra ordinary assistance to study giant and microscope-able chemical compounds. For such a study, researchers made some transformation rules to transfer a chemical compound to a discrete pattern of shapes (graph). Like, an atom represents as a vertex and the covalent bonding between atoms symbolized as edges. Such transformation is known as molecular graph theory. A major importance of this alteration is that the hydrogen atoms are omitted. Some chemical structures and compounds conversion are presented in [4,5,6].

    In cheminformatics, the topological index gains attraction due to its implementations. Various topological indices help to estimate a bio-activity and physicochemical characteristics of a chemical compound. Some interesting and useful topological indices for various chemical compounds are studied in [3,7,8]. A topological index modeled a molecular graph or a chemical compound into a numerical value. Since 1947, topological index implemented in chemistry [9], biology [10], and information science [11,12]. Sombor index and degree-related properties of simplicial networks [13], Nordhaus–Gaddum-type results for the Steiner Gutman index of graphs [14], Lower bounds for Gaussian Estrada index of graphs [15], On the sum and spread of reciprocal distance Laplacian eigenvalues of graphs in terms of Harary index [16], the expected values for the Gutman index, Schultz index, and some Sombor indices of a random cyclooctane chain [17,18,19], bounds on the partition dimension of convex polytopes [20,21], computing and analyzing the normalized Laplacian spectrum and spanning tree of the strong prism of the dicyclobutadieno derivative of linear phenylenes [22], on the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks [23,24], Zagreb indices and multiplicative Zagreb indices of Eulerian graphs [25], Minimizing Kirchhoff index among graphs with a given vertex bipartiteness, [26], asymptotic Laplacian energy like invariant of lattices [27]. Few interesting studies regarding the chemical graph theory can be found in [28,29,30,31,32].

    Recently, the researchers of [33] introduced a topological descriptor and called the face index. Moreover, the idea of computing structure-boiling point and energy of a structure, motivated them to introduced this parameter without heavy computation. They computed these parameters for different models compare the results with previous literature and found approximate solutions with comparatively less computations. This is all the blessings of face index of a graph. The major concepts of this research work are elaborated in the given below definitions.

    Definition 1.1. [33] Let a graph G=(V(G),E(G),F(G)) having face, edge and vertex sets notation with F(G),E(G),V(G), respectively. It is mandatory that the graph is connected, simple and planar. If e from the edge set E(G), is one of those edges which surrounds a face, then the face f from the face set F(G), is incident to the edge e. Likewise, if a vertex α from the vertex set V(G) is at the end of those incident edges, then a face f is incident to that vertex. This face-vertex incident relation is symbolized here by the notation αf. The face degree of f in G is described as d(f)=αfd(α), which are elaborated in the Figure 1.

    Figure 1.  An example of face degree.

    Definition 1.2. [33] The face index FI(G), for a graph G, is formulated as

    FI(G)=fF(G)d(f)=αf,fF(G)d(α).

    In the Figure 1, we can see that there are two faces with degree 4, exactly two with five count and four with count of 6. Moreover, there is an external face with count of face degree 28, which is the count of vertices.

    As the information given above that the face index is quite new and introduced in the year 2020, so there is not so much literature is available. A few recent studies on this topic are summarized here. A chemical compound of silicon carbides is elaborated with such novel definition in [34]. Some carbon nanotubes are discussed in [35]. Except for the face index, there are distance and degree-based graphical descriptors available in the literature. For example, distance-based descriptors of phenylene nanotube are studied in [36], and in [37] titania nanotubes are discussed with the same concept. Star networks are studied in [38], with the concept of degree-based descriptors. Bounds on the descriptors of some generalized graphs are discussed in [39]. General Sierpinski graph is discussed in [40], in terms of different topological descriptor aspects. The study of hyaluronic acid-doxorubicin ar found in [41], with the same concept of the index. The curvilinear regression model of the topological index for the COVID-19 treatment is discussed in [42]. For further reading and interesting advancements of topological indices, polynomials of zero-divisor structures are found in [43], zero divisor graph of commutative rings [44], swapped networks modeled by optical transpose interconnection system [45], metal trihalides network [46], some novel drugs used in the cancer treatment [47], para-line graph of Remdesivir used in the prevention of corona virus [48], tightest nonadjacently configured stable pentagonal structure of carbon nanocones [49]. In order to address a novel preventive category (P) in the HIV system known as the HIPV mathematical model, the goal of this study is to offer a design of a Morlet wavelet neural network (MWNN) [50].

    In the next section, we discussed the newly developed face index or face-based index for different chemical compounds. Silicate network, triangular honeycomb network, carbon sheet, polyhedron generalized sheet, and generalized chain of silicate network are studied with the concept of the face-based index. Given that the face index is more versatile than vertex degree-based topological descriptors, this study will aid in understanding the structural characteristics of chemical networks. Only the difficulty authors will face to compute the face degree of a generalized network or structure, because it is more generalized version and taking degree based partition of edges into this umbrella of face index.

    Silicates are formed when metal carbonates or metal oxides react with sand. The SiO4, which has a tetrahedron structure, is the fundamental chemical unit of silicates. The central vertex of the SiO4 tetrahedron is occupied by silicon ions, while the end vertices are occupied by oxygen ions [51,52,53]. A silicate sheet is made up of rings of tetrahedrons that are joined together in a two-dimensional plane by oxygen ions from one ring to the other to form a sheet-like structure. The silicate network SLn symbol, where n represents the total number of hexagons occurring between the borderline and center of the silicate network SLn. The silicate network of dimension one is depicted in Figure 2. It contain total 3n(5n+1) vertices are 36n2 edges. Moreover, the result required is detailed are available in Table 1.

    Figure 2.  A silicate network SL1.
    Table 1.  The number of f12, f15 and f36 in each dimension.
    Dimension |f12| |f15| |f36|
    1 24 48 7
    2 32 94 14
    3 40 152 23
    4 48 222 34
    5 56 304 47
    6 64 398 62
    7 72 504 79
    8 80 622 98
    . . . .
    . . . .
    . . . .
    n 8n+16 6n2+28n+14 n2+4n+2

     | Show Table
    DownLoad: CSV

    Theorem 2.1. Let SLn be the silicate network of dimension n1. Then the face index of SLn is

    FI(SLn)=126n2+720n+558.

    Proof. Consider SLn the graph of silicate network with dimension n. Suppose fi denotes the faces of graph SLn having degree i. that is, d(fi)=αfid(α)=i and |fi| denotes the number of faces with degree i. The graph SLn contains three types of internal faces f12, f15, f36, and single external face which is usually denoted by f.

    If SLn has one dimension then sum of degree of vertices incident to the external face is 144 and when SLn has two dimension then sum of degree of incident vertices to the external face is 204 whenever SLn has three dimension then sum of degree of incident vertices to the external face is 264. Similarly, SLn has ndimension then sum of degree of incident vertices to the external face is 60n+84.

    The number of internal faces with degree in each dimension is mentioned in Table 1.

    By using the definition of face index FI we have

    FI(SLn)=αfF(SLn)d(α)=αf12F(SLn)d(α)+αf15F(SLn)d(α)+αf36F(SLn)d(α)+αfF(SLn)d(α)=|f12|(12)+|f15|(15)+|f36|(36)+(60n+84)=(8n+16)(12)+(6n2+28n+14)(15)+(n2+4n+2)(36)+60n+84=126n2+72n+558.

    Hence, this is our required result.

    A chain silicate network of dimension (m,n) is symbolized as CSL(m,n) which is made by arranging (m,n) tetrahedron molecules linearly. A chain silicate network of dimension (m,n) with m,n1 where m denotes the number of rows and each row has n number of tetrahedrons. The following theorem formulates the face index FI for chain silicate network.

    Theorem 2.2. Let CSL(m,n) be the chain of silicate network of dimension m,n1. Then the face index FI of the graph CSL(m,n) is

    FI(CSL(m,n))={48n12if m=1, n1;96m12if n=1, m2;168m60if n=2,m2;45m9n+36mn42if both m,n are even45m9n+36mn21otherwise.

    Proof. Let CSL(m,n) be the graph of chain silicate network of dimension (m,n) with m,n1 where m represents the number of rows and n is the number of tetrahedrons in each row. A graph CSL(m,n) for m=1 contains three type of internal faces f9, f12 and f15 with one external face f. While for m2, it has four type of internal faces f9, f12, f15 and f36 with one external face f. We want to evaluate the algorithm of face index FI for chain silicate network. We will discuss it in two different cases.

    Case 1: When CSL(m,n) has one row (m=1) with n number of tetrahedrons as shown in the Figure 3.

    Figure 3.  Chain silicate network CSL(m,n) with particular value of m=1.

    The graph has three type of internal faces f9, f12 and f15 with one external face f. The sum of degree of incident vertices to the external face is 9n and number of faces are |f9|=2, |f12|=2n and |f15|=n2. Now the face index FI of the graph CSL(m,n) is given by

    FI(CSL(m,n))=αfF(CSL(m,n))d(α)=αf9F(CSL(m,n))d(α)+αf12F(CSL(m,n))d(α)+αf15F(CSL(m,n))d(α)+αfF(CSL(m,n))d(α)=|f9|(9)+|f12|(12)+|f15|(15)+(9n)=(2)(9)+(2n)(12)+(n2)(15)+9n=48n12.

    Case 2: When CSL(m,n) has more than one rows (m1) with n number of tetrahedrons in each row as shown in the Figure 4.

    Figure 4.  Chain silicate network CSL(m,n).

    The graph has four type of internal faces f9, f12, f15 and f36 with one external face f. The sum of degree of incident vertices to the external face is

    αfF(CSL(m,n))d(α)={18mif n=1, m1;27mif n=2, m1;30m+15n30if both m,n are even30m+15n33otherwise.

    The number of faces are |f9|, |f12|, f15 and |f36| are given by

    |f9|={2if  m is odd3+(1)nif  m is even.|f12|={2(2m+n1)if m is odd4(n+12+2m1)if m is even|f15|=(3m2)nm|f36|={(m12)(n1)if m is odd(2n+(1)n14)(m22)nif m is even.

    Now the face index FI of the graph CSL(m,n) is given by

    FI(CSL(m,n))=αfF(CSL(m,n))d(α)=αf9F(CSL(m,n))d(α)+αf12F(CSL(m,n))d(α)+αf15F(CSL(m,n))d(α)+αf36F(CSL(m,n))d(α)+αfF(CSL(m,n))d(α)=|f9|(9)+|f12|(12)+|f15|(15)+|f36|(36)+αfF(CSL(m,n))d(α).

    After some mathematical simplifications, we can get

    FI(CSL(m,n))={48n12if m=196m12if n=1,m168m60if n=2,m45m9n+36mn42if both m,n are even45m9n+36mn21otherwise.

    There are three regular plane tessellations known to exist, each constituted from the same type of regular polygon: triangular, square, and hexagonal. The triangular tessellation is used to define the hexagonal network, which is extensively studied in [54]. A dimensioned hexagonal network THk has 3k23k+1 vertices and 9k215k+6 edges, where k is the number of vertices on one side of the hexagon. It has 2k2 diameter. There are six vertices of degree three that are referred to as corner vertices. Moreover, the result required detailed are available in the Table 2.

    Figure 5.  Triangular honeycomb network with dimension k=3.
    Table 2.  The number of f12, f14, f17 and f18 in each dimension.
    Dimension |f12| |f14| |f17| |f18|
    1 6 0 0 0
    2 6 12 12 12
    3 6 24 24 60
    4 6 36 36 144
    5 6 48 48 264
    6 6 60 60 420
    7 6 72 72 612
    8 6 84 84 840
    . . . . .
    . . . . .
    . . . . .
    k 6 12(k1) 12(k1) 18k242k+24

     | Show Table
    DownLoad: CSV

    Theorem 2.3. Let THk be the triangular honeycomb network of dimension k1. Then the face index of graph THk is

    FI(THk)=324k2336k+102.

    Proof. Consider THk be a graph of triangular honeycomb network. The graph TH1 has one internal and only one external face while graph THk with k2, contains four types of internal faces f12, f14, f17, and f18 with one external face f.

    For TH1 the sum of degree of incident vertices to the external face is 18 and in TH2 the sum of degree of incident vertices to the external face is 66. Whenever the graph TH3, the sum of degree of incident vertices to the external face is 114. Similarly, for THk has ndimension then sum of degree of incident vertices to the external face is 48k30.

    The number of internal faces with degree in each dimension is given in Table 2.

    By using the definition of face index FI we have

    FI(THk)=αfF(THk)d(α)=αf12F(THk)d(α)+αf14F(THk)d(α)+αf17F(THk)d(α)+αf18F(THk)d(α)+αfF(THk)d(α)=|f12|(12)+|f14|(14)+|f17|(17)+|f18|(18)+(48k30)=(6)(12)+(12(k1))(14)+(12(k1))(17)+(18k242k+24)(18)+48k30=324k2336k+102.

    Hence, this is our required result.

    Given carbon sheet in the Figure 6, is made by grid of hexagons. There are few types of carbon sheets are given in [55,56]. The carbon sheet is symbolize as HCSm,n, where n represents the total number of vertical hexagons and m denotes the horizontal hexagons. It contain total 4mn+2(n+m)1 vertices and 6nm+2m+n2 edges. Moreover, the result required detailed are available in Tables 3 and 4.

    Figure 6.  Carbon Sheet HCSm,n.
    Table 3.  The number of f15, f16, and f18 in each dimension.
    Dimension m |f15| |f16| |f18| |f|
    2 3 2(n1) n1 20n+7

     | Show Table
    DownLoad: CSV
    Table 4.  The number of f15, f16, f17, f18, and f in each dimension.
    Dimension m |f15| |f16| |f17| |f18| |f|
    2 3 2(n1) 0 n1 20n+7
    3 2 2n 1 3(n1) 20n+17
    4 2 2n 3 5(n1) 20n+27
    5 2 2n 5 7(n1) 20n+37
    6 2 2n 7 9(n1) 20n+47
    . . . . . .
    . . . . . .
    . . . . . .
    m 2 2n 2m5 2mn2m3n+3 20n+10m13

     | Show Table
    DownLoad: CSV

    Theorem 2.4. Let HCSm,n be the carbon sheet of dimension (m,n) and m,n2. Then the face index of HCSm,n is

    FI(HCSm,n)={70n+2ifm=236mn142(n4m)ifm3.

    Proof. Consider HCSm,n be the carbon sheet of dimension (m,n) and m,n2. Let fi denotes the faces of graph HCSm,n having degree i, which is d(fi)=αfid(α)=i, and |fi| denotes the number of faces with degree i. A graph HCSm,n for a particular value of m=2 contains three types of internal faces f15, f16, f17 and f18 with one external face f. While for the generalize values of m3, it contain four types of internal faces f15, f16 and f17 with one external face f in usual manner. For the face index of generalize nanotube, we will divide into two cases on the values of m.

    Case 1: When HCSm,n has one row or HCS2,n.

    A graph HCSm,n for a this particular value of m=2 contains three types of internal faces |f15|=3, |f16|=2(n1) and |f18|=n1 with one external face f. For the face index of carbon sheet, details are given in the Table 3. Now the face index FI of the graph NT2,n is given by

    FI(HCS2,n)=αfF(HCS2,n)d(α)=αf15F(HCS2,n)d(α)+αf16F(HCS2,n)d(α)+αf18F(HCS2,n)d(α)+αfF(HCS2,n)d(α)=|f15|(15)+|f16|(16)+|f18|(18)+20n+7.=3(15)+2(n1)(16)+(n1)(18)+20n+7.=70n+2.

    Case 2: When HCSm,n has m3 rows.

    A graph HCSm,n for generalize values of m3 contains four types of internal faces |f15|=2, |f16|=2n, |f17|=2m5 and |f18|=2mn2m3n+3 with one external face f. For the face index of carbon sheet, details are given in the Table 4. Now the face index FI of the graph NTm,n is given by

    FI(HCSm,n)=αfF(HCSm,n)d(α)=αf15F(HCSm,n)d(α)+αf16F(HCSm,n)d(α)+αf17F(HCSm,n)d(α)+αf18F(HCSm,n)d(α)+αfF(HCSm,n)d(α)=|f15|(15)+|f16|(16)+|f17|(17)+|f18|(18)+20n+10m13.=36mn2n+8m14.

    Given structure of polyhedron generalized sheet of C28 in the Figure 7, is made by generalizing a C28 polyhedron structure which is shown in the Figure 8. This particular structure of C28 polyhedron are given in [57]. The polyhedron generalized sheet of C28 is as symbolize PHSm,n, where n represents the total number of vertical C28 polyhedrons and m denotes the horizontal C28 polyhedrons. It contain total 23nm+3n+2m vertices and 33nm+n+m edges. Moreover, the result required detailed are available in Tables 3 and 5.

    Figure 7.  Polyhedron generalized sheet of C28 for m=n=1, or PHS1,1.
    Figure 8.  Polyhedron generalized sheet of C28 or PHSm,n.
    Table 5.  The number of f14,f15,f16,f17,f18,f20, and f35 in each dimension.
    m |f14| |f15| |f16| |f17| |f18| |f20| |f35|
    1 2n+1 2 4n2 0 0 2n1 0
    2 2n+2 2 8n2 2 2n2 4n2 2n1
    3 2n+3 2 12n2 4 4n4 6n3 4n2
    . . . . . . . .
    . . . . . . . .
    . . . . . . . .
    m 2n+m 2 4mn2 2m2 2mn2(m+n)+2 2mnm 2mn(m+2n)+1

     | Show Table
    DownLoad: CSV

    Theorem 2.5. Let PHSm,n be the polyhedron generalized sheet of C28 of dimension (m,n) and m,n1. Then the face index of PHSm,n is

    FI(PHSm,n)=210mn2(3m+5n).

    Proof. Consider PHSm,n be the polyhedron generalized sheet of C28 of dimension (m,n) and m,n1. Let fi denotes the faces of graph PHSm,n having degree i, which is d(fi)=αfid(α)=i, and |fi| denotes the number of faces with degree i. A graph PHSm,n for the generalize values of m,n1, it contain seven types of internal faces f14,f15,f16,f17,f18,f20 and f35 with one external face f in usual manner. For the face index of polyhedron generalized sheet, details are given in the Table 5.

    A graph PHSm,n for generalize values of m,n1 contains seven types of internal faces |f14|=2n+m, |f15|=2, |f16|=4nm2, |f17|=2(m1), |f18|=2nm2(m+n)+2, |f20|=2nm2mnm, and |f35|=2mnm2n+1 with one external face f. Now the face index FI of the graph PHSm,n is given by

    FI(PHSm,n)=αfF(PHSm,n)d(α)=αf14F(PHSm,n)d(α)+αf15F(PHSm,n)d(α)+αf16F(PHSm,n)d(α)+αf17F(PHSm,n)d(α)+αf18F(PHSm,n)d(α)+αf20F(PHSm,n)d(α)+αf35F(PHSm,n)d(α)+αfF(PHSm,n)d(α)=|f14|(14)+|f15|(15)+|f16|(16)+|f17|(17)+|f18|(18)+|f20|(20)+|f35|(35)+37m+68n35.=210mn6m10n.

    With the advancement of technology, types of equipment and apparatuses of studying different chemical compounds are evolved. But topological descriptors or indices are still preferable and useful tools to develop numerical science of compounds. Therefore, from time to time new topological indices are introduced to study different chemical compounds deeply. In this study, we discussed a newly developed tool of some silicate type networks and generalized sheets, carbon sheet, polyhedron generalized sheet, with the face index concept. It provides numerical values of these networks based on the information of faces. It also helps to study physicochemical characteristics based on the faces of silicate networks.

    M. K. Jamil conceived of the presented idea. K. Dawood developed the theory and performed the computations. M. Azeem verified the analytical methods, R. Luo investigated and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.

    This work was supported by the National Science Foundation of China (11961021 and 11561019), Guangxi Natural Science Foundation (2020GXNSFAA159084), and Hechi University Research Fund for Advanced Talents (2019GCC005).

    The authors declare that they have no conflicts of interest.



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