Research article

Investigation of a nutrient-plankton model with stochastic fluctuation and impulsive control


  • In this paper, we investigate a stochastic nutrient-plankton model with impulsive control of the nutrient concentration and zooplankton population. Analytically, we find that the population size is nonnegative for a sufficiently long time. We derive some sufficient conditions for the existence of stable periodic oscillations, which indicate that the plankton populations will behave periodically. The numerical results show that the plankton system experiences a transition from extinction to the coexistence of species due to the emergence of impulsive control. Additionally, we observe that the nutrient pulse has a stronger relationship with phytoplankton growth than the zooplankton pulse. Although the frequency of impulsive control and appropriate environmental fluctuations can promote the coexistence of plankton populations, an excessive intensity of noise can result in the collapse of the entire ecosystem. Our findings may provide some insights into the relationships among nutrients, phytoplankton and zooplankton in a stochastic environment.

    Citation: Xin Zhao, Lijun Wang, Pankaj Kumar Tiwari, He Liu, Yi Wang, Jianbing Li, Min Zhao, Chuanjun Dai, Qing Guo. Investigation of a nutrient-plankton model with stochastic fluctuation and impulsive control[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 15496-15523. doi: 10.3934/mbe.2023692

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  • In this paper, we investigate a stochastic nutrient-plankton model with impulsive control of the nutrient concentration and zooplankton population. Analytically, we find that the population size is nonnegative for a sufficiently long time. We derive some sufficient conditions for the existence of stable periodic oscillations, which indicate that the plankton populations will behave periodically. The numerical results show that the plankton system experiences a transition from extinction to the coexistence of species due to the emergence of impulsive control. Additionally, we observe that the nutrient pulse has a stronger relationship with phytoplankton growth than the zooplankton pulse. Although the frequency of impulsive control and appropriate environmental fluctuations can promote the coexistence of plankton populations, an excessive intensity of noise can result in the collapse of the entire ecosystem. Our findings may provide some insights into the relationships among nutrients, phytoplankton and zooplankton in a stochastic environment.



    In mathematics chemistry and biology, a chemical compound can be represented by a molecular graph by converting atoms to vertices and bonds to edges. One of the primary mission of QSAR/QSPR research is to accurately convert molecular graphs into numerical values. Graph theoretic invariants of molecular graphs are called molecular descriptors which can be utilized to simulate the structural information of molecules, in order to make worthwhile physical and chemical properties of these molecules can be acquired by single numerical values. Such kinds of molecular descriptors are also referred to as topological indices.

    In the chemical literature, various topological indices relying only on vertex degrees of the molecular graphs can be utilized in QSPR/QSAR investigation on account of them can be obtained directly from the molecular architecture, and can be rapidly calculated for generous molecules (see [1,2]), and we call them VDB (vertex–degree–based) topological indices. To be more precise, for designated nonnegative real numbers {ψij} (1ijn1), a VDB topological index of a an n-order (molecular) graph G is expressed as

    TI(G)=1ijn1mijψij, (1.1)

    where mij is the amount of edges connecting an i-vertex and a j-vertex of G. A great deal of well–known VDB topological indices can be obtained by different ψij in expression (1.1). We list some VDB topological indices in Table 1.

    Table 1.  Some well-known VDB topological indices.
    ψij name
    i+j First Zagreb index
    1ij Randić index
    2iji+j GA index
    i+j2ij ABC index
    1i+j Sum–connectivity index
    (ij)3(i+j2)3 AZI index
    2i+j Harmonic index
    |ij| Albertson index
    i2+j2 Sombor index
    iji+j ISI index

     | Show Table
    DownLoad: CSV

    The first Zagreb index [3] is the very first VDB topological index, as powerful molecular structure-descriptors [2], Zagreb indices can describe the peculiarities of the degree of branching in molecular carbon-atom skeleton. Thereafter, many VDB topological indices have been put forward to simulate physical, chemical, biological, and other attributes of molecules [4,5,6,7]. In 2021, Gutman [8] introduced a new VDB topological index named as the Sombor index which has a linear correlation with the entropy and the enthalpy of vaporization of octanes [9]. Das et al., give sharp bounds for Sombor index of graphs by means of some useful graph parameters and they reveal the relationships between the Sombor index and Zagreb indices of graphs [10]. Recently, Steiner Gutman index was introduced by Mao and Das [11] which incorporate Steiner distance of a connected graph G. Nordhaus-Gaddum-type results for the Steiner Gutman index of graphs were given in [12]. In 2022, Shang study the Sombor index and degree-related properties of simplicial networks [13]. For more details of VDB topological indices, one can see [3,14,15,16,17,18,19,20,21,22,23,24,25,26] and the books [27,28,29].

    Fluoranthene is a eminent conjugated hydrocarbon which abound in coal tar [30]. A fluoranthene–type benzenoid system (f-benzenoid for short) is formed from two benzenoid units joined by a pentagon [31,32]. The ordinary structure modality of a f-benzenoid F is shown in Figure 1, where segments X and Y are two benzenoid systems. Each f-benzenoid possesses exactly one pentagon [32]. More and more attention is paid to f-benzenoids after the flash vacuum pyrolysis experiments of these nonalternant polycyclic aromatic hydrocarbons [33].

    Figure 1.  The ordinary structure modality of a f-benzenoid (F) and its construction from two benzenoid systems X and Y.

    In the whole article, the terminology and notation are chiefly derived from [34,35,36,37,38,39,40,41]. A vertex of degree k is called a k-vertex, and an edge linking a k-vertex and a j-vertex is designated as a (k,j)-edge. Let nk be the number of k-vertices and let mkj be the number of (k,j)-edges in the molecular graph G. A benzenoid system without internal vertices is said to be catacondensed. Analogously, a f-benzenoid F containing a unique internal vertex is referred to as catacatacondensed. We use h-hexagon benzenoid system (or h-hexagon f-benzenoid) to represent a benzenoid system (or f-benzenoid) containing h hexagons.

    Let Lh represent the h-hexagon linear chain (as shown in Figure 2(a)). An f-benzenoid FLh (h3) obtaining from pieces X=L2 and Y=Lh2 is named as f-linear chain (as shown in Figure 2(b)).

    Figure 2.  Linear chain and f-linear chain.

    A fissure (resp. bay, cove, fjord and lagoon) of a f-benzenoid F is a path of degree sequences (2,3,2) (resp. (2,3,3,2), (2,3,3,3,2), (2,3,3,3,3,2) and (2,3,3,3,3,3,2)) on the perimeter of F (see Figure 3). Fissures, bays, coves, fjords and lagoons are said to be different kinds of inlets and their number are signified by f, B, C, Fj and L, respectively [32,37]. Inlets determine many electronic and topological properties of f-benzenoids. Then, it can be found that f+2B+3C+4FJ+5L is the number of 3-vertices on the perimeter of F. It is noted that lagoons cannot occur in the theory of benzenoid systems. For convenience, let r=f+B+C+Fj+L to represent the total number of inlets and b=B+2C+3Fj+4L is referred to as the quantity of bay regions, In addition, b is exactly the quantity of (3,3)-edges on the perimeter of F. It is obvious that b2 for any f-benzenoid F.

    Figure 3.  Structural features occurring on the perimeter of f-benzenoids.

    It is noted that any f-benzenoid F contains merely either 2-vertex or 3-vertex. The vertices not on the perimeter are said to be internal, and we use ni to represent their number.

    Lemma 1.1. [32] Let F be an n-order, h-hexagon f-benzenoid with m edges and ni internal vertices. Then

    (i) n=4h+5ni;

    (ii) m=5h+5ni.

    Lemma 1.2. [32] Let F be an n-order and h-hexagon f-benzenoid with r inlets, Then

    (i) m22=n2hr;

    (ii) m23=2r;

    (iii) m33=3hr.

    From the perspective of mathematics and chemistry, finding the extremal values of some useful TI for significant classes of graphs is very interesting [14,19,23,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56].

    As a matter of convenience, we use Γm to represent the collection of f-benzenoids containing exactly m edges. In [45], we derived extremal values for TI among all f-benzenoids with given order. It is noted that structure of f-benzenoids with given order is different from that of f-benzenoids with given number of edges. And we found that the technique for studying TI among all f-benzenoids with given order can not be used directly to investigate TI for all f-benzenoids with fixed number of edges. For this reason, we concentrate on the research of extremal values for TI among all f-benzenoids with given size.

    The main idea of this work is to construct f-benzenoids owning maximal r and minimal h at the same time in Γm depending on the number m is congruent to 0,1,2,3 or 4 modulo 5. By making use of this technique, we obtain the extremum of TI over Γm and characterize their corresponding graphs on the basis of m is congruent to 0,1,2,3 or 4 modulo 5. Afterwards the extremums of some well-known TI over Γm can be got by use of the previous results.

    The structure of this paper is as below. We first determine the maximal r in the set Γm in Section 2. By utilizing these results, we find the extremum of several famed TI over Γm in Section 3.

    We will find the f-benzenoids with maximal r in Γm in this section. Figure 4 illustrates three f-benzenoids pertaining to Γ42.

    Figure 4.  Some f-benzenoids in Γ42.

    At first, we try to obtain the maximum and minimum number of hexagons in any FΓm.

    The spiral benzenoid system [57] Th is a benzenoid system whose structure is in a "spiral" manner as illustrated in Figure 5. Th has maximal ni in all h-hexagon benzenoid systems.

    Figure 5.  The spiral benzenoid system Th with maximal number of internal vertices.

    As a matter of convenience, let SHh (h3) represent the collection of f-benzenoids formed by two spiral benzenoids X and Y. Particularly, a f-spiral benzenoid is a f-benzenoid FSHh in which X=Th1 and Y=T1 (as shown in Figure 6). It is easy to see that that

    ni(F)=2h12(h1)3.
    Figure 6.  f-benzenoid FSHh whose two pieces X and Y are both spiral benzenoid systems, and f-spiral benzenoid FSHh with two pieces X=Th1 and Y=T1.

    In [40], we proved that for every FSHh (h3), the inequality

    ni(F)ni(F) (2.1)

    holds, and the following graph operations were introduced.

    Operation 1. For any h-hexagon f-benzenoid F having two segments X and Y, let h1=h(X) and h2=h(Y). By substituting spiral benzenoid systems Th1 and Th2 for X and Y, severally, another f-benzenoid FSHh can be obtained (as shown in Figure 7).

    Figure 7.  f-benzenoid FSHh is obtained from F by applying Operation 1 to it.

    For any h-hexagon f-benzenoid F, when h=3, it is easily checked that

    ni(F)=1=2×312(31)3. (2.2)

    When h4, let h1=h(X) and h2=h(Y). Another FSHh (as shown in Figure 7) in which X=Th1 and Y=Th2 can be acquired by applying Operation 1 to F. It is apparently that ni(X)ni(Th1), ni(Y)ni(Th2), therefore

    ni(F)=ni(X)+ni(Y)+1ni(Th1)+ni(Th2)+1=ni(F). (2.3)

    So, the following Lemma can be deduced by Eqs (2.1) and (2.3).

    Lemma 2.1. [41] Let F be an h(h3)-hexagon f-benzenoid. Then

    ni(F)2h12(h1)3, (2.4)

    and the equality is established when F is F.

    For any FΓm, h(F) over Γm is variable. Sharp bounds for h(F) in Γm is given below.

    Theorem 2.1. For any f-benzenoid FΓm,

    15(m4)h(F)m113(2m+4m31), (2.5)

    where x is the smallest integer larger or equal to x.

    Proof. On one hand, from Lemma 1.1 (ii) we know that m=5h(F)+5ni(F). Combining the fact that ni(F)1 for any FΓm, we get

    h(F)15(m4).

    On the other hand, by Lemma 2.1 we know that ni(F)ni(F). Consequently, from m=5h(F)+5ni(F) we have

    m3h(F)512(h(F)1)312(h(F)1)3.

    Hence,

    (3h(F)+(3m))24m31.

    Due to the fact that 3h(F)+(3m)<0, we deduce

    3h(F)+(3m)4m31,

    i.e., h(F)m113(2m+4m31).

    Remark 1. Theorem 2.1 implies that f-spiral benzenoid F has the maximal number of hexagons over Γm.

    For the sake of obtaining the extremum TI among all f-benzenoids in Γm, we need to find the f-benzenoids FΓm possessing maximal r.

    Recall that convex benzenoid systems (CBS for brevity) are a particular sort of benzenoid systems lack of bay regions [14]. Let HSh be the collection of benzenoid systems containing h hexagons.

    Lemma 2.2. [42] Let HHSh. Under the below cases, H is definitely not a CBS:

    (i) If h4 and ni=1;

    (ii) If h5 and ni=2;

    (iii) If h6 and ni=3.

    Lemma 2.3. [52] Let HHSh such that ni(H)=4. Then H is bound to embody a subbenzenoid system given in Figure 8, there does not exist hexagons which are adjacent to fissures.

    Figure 8.  Benzenoid systems with 1, 2, 3 and 4 internal vertices, respectively.

    Lemma 2.4. Let SHSh. If h7 and ni(S)=4, then S is not a CBS.

    Proof. Let S be an h (h7)-hexagon benzenoid system, ni(S)=4, then by Lemma 2.3 S must contain one of the benzenoid systems of the form given in Figure 7. The proof is carried out in two cases.

    Case 1. If these four internal vertices form a path P4 or a K1,3, then S contains one of benzenoid systems (d)(f) in Figure 7 as its subbenzenoid systems. It is noted that h7, by Lemma 2.2, it must not exist hexagons contiguous to the fissures, so, S has at least one hexagon contiguous to a (2,2)-edge, by means of such hexagons, it is succeeded in converting one of the fissures into a cove, bay or fjord. Hence, b(S)1.

    Case 2. If these four internal vertices are not adjacent then S has possibility subbenzenoid systems as follows.

    1) There exist one type (a) and one type (c) benzenoid systems in S;

    2) There exist two type (b) benzenoid systems in S;

    3) There exist two type (a) and one type (b) benzenoid systems in S.

    4) There exist four type (a) benzenoid systems in S

    By Lemma 2.2, neither hexagons may be adjacent to the fissures in any of the cases indicated above. Since h7, S has at least one hexagon contiguous to a (2,2)-edge, by means of such hexagons, it is succeeded in making one of the fissures become a cove, bay or fjord. Therefore, b(S)1.

    The proof is completed.

    Lemma 2.5. [45] Let F be an h-hexagon f-benzenoid. Then

    1) If ni=1, then r(F)r(FLh)=2h3 (h3);

    2) If ni=2, then r(F)r(Gh)=2h4 (h4);

    3) If ni=3, then r(F)r(Rh)=2h5 (h5);

    4) If ni=4, then r(F)r(Zh)=2h6 (h6).

    Next we find the f-benzenoids with maximal r in Γm with a fixed ni. Recall that Mh, Nh and Qh (see Figure 9) are benzenoid systems, and Gh (see Figure 10), Rh (see Figure 11), Zh (see Figure 12) are f-benzenoids.

    Figure 9.  Three types of benzenoid systems.
    Figure 10.  f-benzenoids G4, and Gh (h5).
    Figure 11.  f-benzenoids R5, and Rh (h6).
    Figure 12.  f-benzenoids Z6, and Zh (h7).
    Figure 13.  f-benzenoids U7, and Uh (h8).

    Lemma 2.6. [41] Let F be an h-hexagon f-benzenoid. Then

    r(F)r(FLh)=2h3.

    Lemma 2.7. [32] For any h-hexagon f-benzenoid including ni internal vertices and b bay regions, the number of (2,2)-edge and (2,3)-edge are m22=b+5,m23=4h2ni2b, respectively.

    From Lemmas 1.2 (ii) and 8, we get

    r=2hnib (2.6)

    Furthermore, by Lemma 1.1 (ii) and Eq (2.6), we deduce

    r=m3h5b (2.7)

    Theorem 2.2. Let F be an h-hexagon f-benzenoid. If ni=5, then r(F)r(Uh)=2h7 (h7).

    Proof. Let h1=h(X) and h2=h(Y), X and Y are two segments of F. If ni=5, by the structure of f-benzenoid, equality ni(X)+ni(Y)=4 holds, so, we have the following five cases.

    Case 1. ni(X)=1, ni(Y)=3, i.e., there exist one internal vertex and three internal vertices in X and Y, respectively.

    Subcase 1.1. If h1=3, then X=M3.

    Subcase 1.1.1. If h2=5, i.e., Y=Q5, then F is the f-benzenoid D1, D2 or D3 (see Figure 14). It is clear that r(F)=r(D1)=82h7, r(F)=r(D2)=72h7 or r(F)=r(D3)=82h7.

    Figure 14.  f-benzenoids D1, D2, D3, D4 and D5.

    Subcase 1.1.2. If h26, by Lemma 2.2 and the hypothesis that ni(Y)=3, Y is not a CBS, so b(Y)1. Furthermore, b(F)3, combining Eq (2.6) we obtain r=2hnib2h8<2h7.

    Subcase 1.2. If h14, according to Lemma 2.2, X is definitely not a CBS, i.e., b(X)1.

    Subcase 1.2.1. If h2=5, i.e., Y=Q5. It is clear that b(F)4, then Eq (2.6) deduces r2h9<2h7.

    Subcase 1.2.2. If h26, Y is definitely not not a CBS according to Lemma 2.2, so, b(Y)1. It is clear that b(F)5, consequently from Eq (2.6) we obtain r2h10<2h7.

    Case 2. ni(X)=3 and ni(Y)=1.

    Subcase 2.1. If h1=5, then X=Q5.

    Subcase 2.1.1. If h2=3, i.e., Y=M3, then F is the f-benzenoid D4, D5, D6 (see Figure 14), or D7 (as shown in Figure 15). r(F)=r(D4)=82h7, r(F)=r(D5)=72h7, r(F)=r(D6)=82h7, r(F)=r(D7)=72h7.

    Figure 15.  f-benzenoids D7, D8 and D9.

    Subcase 2.1.2. If h24, Y is surely not a CBS in light of Lemma 2.2, i.e., b(X)1. Hence, we have b(F)4, it follows from Eq (2.6) that r2h9<2h7.

    Subcase 2.2. If h16, by Lemma 2.2, X is definitely not a CBS, hence b(X)1.

    Subcase 2.2.1. If h2=3, i.e., Y=M3. We have b(F)4, and Eq (2.6) infers that r2h9<2h7.

    Subcase 2.2.2. f h24, by Lemma 2.2, Y is certainly not a CBS, i.e., b(X)1. Hence we have b(F)5, by Eq (2.6), r2h10<2h7.

    Case 3. ni(X)=2, ni(Y)=2, i.e., X and Y both have two internal vertices.

    Subcase 3.1. If h1=4, we note that ni(X)=2, so X must be the benzenoid system (b) in Figure 9.

    Subcase 3.1.1. If h2=4, Y is surely the benzenoid system (b) in Figure 9 according to the hypothesis ni(Y)=2, therefore, F is D8 or D9 (as shown in Figure 15). We get r(F)=r(D8)=8<2h7 or r(F)=r(D9)=7<2h7.

    Subcase 3.1.2. If h25, by Lemma 2.2 and that ni(Y)=2, Y is not a CBS, so we know that b(X)1. Then b(F)4, by Eq (2.6) and the fact that ni=5, r2h9<2h7.

    Subcase 3.2. If h2=4, we note that ni(Y)=2, so Y must be the benzenoid system (b) in Figure 8.

    Subcase 3.2.1. If h1=4, X must also be the benzenoid system (b) in Figure 9. Hence, F is D8 or D9 (as shown in Figure 15). r(F)=r(D8)=82h7 or r(F)=r(D9)=72h7.

    Subcase 3.2.2. If h15, by Lemma 2.2 and ni(X)=2, X is definitely not a CBS, i.e., b(X)1. Hence, b(F)4, by Eq (2.6) and the fact that ni=5, we have r2h9<2h7.

    Subcase 3.3. If h15, h25, it is noted that ni(X)=ni(Y)=2, neither X nor Y are definitely CBS according to Lemma 2.2. So, both b(X) and b(Y) are greater than 1. Hence, b(F)5, on the basis of Eq (2.6) we get r2h10<2h7.

    Case 4. ni(X)=4 and ni(Y)=0, i.e., X contains four internal vertices, Y is a catacondensed benzenoid system.

    Subcase 4.1. If h1=6, then X is the benzenoid system (d), (e) or (f) in Figure 9.

    Subcase 4.1.1. If h2=1, F is the f-benzenoid D10, D11, D12 (see Figure 16), D13 (see Figure 17) or U7 (see Figure 12). r(F)=r(D10)=62h7, r(F)=r(D11)=62h7, r(F)=r(D12)=62h7, r(F)=r(D13)=62h7 or r(F)=r(U7)=7=2h7.

    Figure 16.  f-benzenoids D10, D11 and D12.
    Figure 17.  f-benzenoids D13, D14, D15, D16, D17, D18, D19, D20 and D21.

    Subcase 4.1.2. If h22, we have b(F)2, by Eq (2.6), r2h7.

    Subcase 4.2. If h17, in the light of Lemma 2.4, X is definitely not a CBS, hence b(Y)1. In this situation b(F)3, we get the inequality r2h8<2h7 according to Eq (2.6).

    Case 5. ni(X)=0 and ni(Y)=4, i.e., X is a catacondensed benzenoid system, Y has four internal vertices.

    Subcase 5.1. If h2=6, then Y is the benzenoid system (d), (e) or (f) in Figure 8.

    Subcase 5.1.1. If h1=2, X must be the linear chain L2. In this event, F is D14, D15, D16, D17, D18, D19, D20 or D21 (see Figure 17). By further checking, we gain that r(F)=r(D14)=72h7, r(F)=r(D15)=82h7, r(F)=r(D16)=82h7, r(F)=r(D17)=72h7, r(F)=r(D18)=72h7, r(F)=r(D19)=82h7, r(F)=r(D20)=62h7 or r(F)=r(D21)=62h7.

    Subcase 5.1.2. If h13, bearing in mind that X is a catacondensed benzenoid system and Y is the benzenoid system (d), (e) or (f) in Figure 8, then F must have f-benzenoid D14, D15, D16, D17, D18, D19, D20 or D21 (see Figure 17) as its subgraph.

    Subcase 5.1.2.1. If D14 is a subgraph in F, it is obvious that D14 has two coves. Since X is a catacondensed benzenoid system and h13, F has at least one hexagon contiguous to a (2,2)-edge of X, and such hexagons can convert one fissure into a bay, or convert one cove into a fjord, or convert one fjord into a lagoon. In this instance b(F)4. Consequently, r2h9<2h7 can be got according to Eq (2.6).

    Subcase 5.1.2.2. If D15, D16 or D19 is a subpart f-benzenoid in F, it is obvious each one of D15, D16 and D19 has a bay and a cove. Since X is a catacondensed benzenoid system and h13, F contains at least one hexagon adjoining a (2,2)-edge of X, and such hexagons will make one fissure become a bay, or make one cove become a fjord, or make one fjord become a lagoon. Consequently, b(F)4, by Eq (2.6) it follows that r2h9<2h7.

    Subcase 5.1.2.3. If D17 is a subpart f-benzenoid in F, it is obvious that D17 has a fjord and a bay. Since X is a catacondensed benzenoid system and h13, F has at least one hexagon adjoining a (2,2)-edge of X, and such hexagons will convert one fissure into a bay, or convert one cove into a fjord, or convert one fjord into a lagoon. Consequently, b(F)4, by Eq (2.6) it follows that r2h9<2h7.

    Subcase 5.1.2.4. If D18 is a subpart f-benzenoid in F, it is obvious that D18 has a fjord and two bays. Since X is a catacondensed benzenoid system and h13, there exists has at least one hexagon adjoining a (2,2)-edge of X in F, and these hexagons will convert one of the fissures into a bay, or convert one cove into a fjord, or convert one fjord into a lagoon. Consequently, b(F)4, in light of Eq (2.6), r2h9<2h7.

    Subcase 5.1.2.5. If D20 or D21 is a subpart f-benzenoid in F, it is obvious that both D20 and D21 have a bay and two fjords. Since X is a catacondensed benzenoid system and h13, F contains at least one hexagon adjoining a (2,2)-edge of X, and such hexagons will make one fissure become a bay, or make one cove become a fjord, or make one fjord become a lagoon. Consequently, b(F)4, according to Eq (2.6), r2h9<2h7.

    Subcase 5.2. If h27, by Lemma 2.4 and the fact that ni(Y)=4, Y is certainly not a CBS, i.e., b(Y)1.

    Subcase 5.2.1. If h1=2, i.e., X=L2. From the structure of f-benzenoid, F is formed from X and Y joined by a pentagon, it is easily seen that there are at least one bay or one cove arisen in the process of construction of F. It is clear that b(F)2, by Eq (2.6) we have r2h7.

    Subcase 5.2.2. If h13, we know that F is formed by joining from X and Y through a pentagon, in this construction process of F, it is easily seen that there are at least one bay or one cove arisen. Then b(F)2, by Eq (2.6), r2h7.

    The proof is completed.

    We recall that FLh is the f-linear chain with h hexagons [40]. Extremal f-benzenoids with maximal r in Γm were determined in the following theorem.

    Theorem 2.3. Let FΓm. Then

    1) If m0(mod5), then r(F)2m355=r(Um5);

    2) If m1(mod5), then r(F)2m325=r(Zm15);

    3) If m2(mod5), then r(F)2m295=r(Rm25);

    4) If m3(mod5), then r(F)2m265=r(Gm35);

    5) If m4(mod5), then r(F)2m235=r(FLm45).

    Proof. We know by Eq (2.5) that

    15(m4)h(F)m113(2m+4m31).

    1) If m0(mod5), then 15(m4)=m5. If h=m5, then by Lemma 1.1 (ii)

    m=5h(F)+5ni(F)=m+5ni(F),

    it means that ni(F)=5. Furthermore, Theorem 2.2 infers that r(F)r(Um5) and we are done. So assume now that h(F)m5+1, then by equality (2.7) and the fact that b(F)2

    r(F)=m53h(F)b(F)m53(m5+1)b(F)
    2m510=2m5052m355=r(Um5).

    2) If m1(mod5), then 15(m4)=m15. If h(F)=m15, then by Lemma 1.1 (ii)

    m=5h(F)+5ni(F)=m+4ni(F),

    thus ni(F)=4. Then r(F)r(Zm15) by part 4 of Lemma 2.5. Otherwise h(F)m15+1, then by equality (2.7) and the obvious fact that b(F)2

    r(F)=m53h(F)b(F)m53(m15+1)b(F)
    2m+3510=2m4752m325=r(Zm15).

    3) If m2(mod5), then 15(m4)=m25. If h(F)=m25, then by Lemma 1.1 (ii)

    m=5h(F)+5ni(F)=m+3ni(F),

    and so ni(F)=3. Then r(F)r(Rm25) by part 3 of Lemma 2.5. So assume now that h(F)m25+1, then by Eq (2.7) and the fact that b(F)2

    r(F)=m53h(F)b(F)m53(m25+1)b(F)
    2m+6510=2m4452m295=r(Rm25).

    4) If m3(mod5), then 15(m4)=m35. If h(F)=m35, then by Lemma 1.1 (ii)

    m=5h(F)+5ni(F)=m+2ni(F),

    thus ni(F)=2. By Lemma 2.5, r(F)r(Gm35) and we are done. If h(F)m35+1, then by equality (2.7) and the fact that b(F)2

    r(F)=m53h(F)b(F)m53(m35+1)b(F)
    2m+9510=2m4152m265=r(Gm35).

    5) If m4(mod5), then 15(m4)=m45. Since hm45 and b(F)2, then by Eq (2.7), we have

    r(F)=m53h(F)b(F)m53m125b(F)
    2m+1257=2m235=r(FLm45).

    In this part, we attempt to find the extremal values of TI over Γm.

    It is noted that a f-benzenoid F contains only 2-vertex and 3-vertex. Hence, equation (1.1) reduces to

    TI(F)=m22ψ22+m23ψ23+m33ψ33, (3.1)

    In the light of Lemmas 1.1 and 1.2,

    TI(F)=ψ22m+3(ψ33ψ22)h+(2ψ23ψ22ψ33)r, (3.2)

    If U,VΓm then clearly

    TI(U)TI(V)=3(ψ33ψ22)(h(U)h(V))          +(2ψ23ψ22ψ33)(r(U)r(V)). (3.3)

    For convenience, we set s=ψ33ψ22, q=2ψ23ψ22ψ33.

    Theorem 3.1. For any FΓm, we have the following results.

    a. If s0 and q0,

    TI(F){TI(Um5),if m0(mod 5)TI(Zm15),if m1(mod 5)TI(Rm25),if m2(mod 5)TI(Gm35),if m3(mod 5)TI(FLm45),if m4(mod 5)

    b. If s0 and q0,

    TI(F){TI(Um5),if m0(mod 5)TI(Zm15),if m1(mod 5)TI(Rm25),if m2(mod 5)TI(Gm35),if m3(mod 5)TI(FLm45),if m4(mod 5)

    Proof. Let FΓm. By Eq (2.5)

    h(F)15(m4)={h(Um5),if m0(mod 5)h(Zm15),if m1(mod 5)h(Rm25),if m2(mod 5)h(Gm35),if m3(mod 5)h(FLm45),if m4(mod 5)

    i.e., f-benzenoids Um5, Zm15, Rm25, Gm35 and FLm45 have minimal h over the set Γm. Meanwhile, by Theorem 2.3, we have

    r(F){r(Um5),if m0(mod 5)r(Zm15),if m1(mod 5)r(Rm25),if m2(mod 5)r(Gm35),if m3(mod 5)r(FLm45),if m4(mod 5)

    i.e., these five f-benzenoids have maximal number of inlets over Γm. Hence, for any f-benzenoids FΓm and V{Um5,Zm15,Rm25,Gm35,FLm45}, h(F)h(V)0 and r(F)r(V)0 hold simultaneously, from Eq (2.7), we have

    TI(F)TI(V)=3s(h(F)h(V))+q(r(F)r(V)).

    If s0 and q0, then TI(F)TI(V)0, i.e., V reaches the maximum value of TI over Γm. If s0 and q0, then TI(F)TI(V)0, i.e., V reaches the minimum value of TI over Γm. Furthermore, which V{Um5,Zm15,Rm25,Gm35,FLm45} is the extremal graph depending on m is congruent to 0,1,2,3 or 4 modulo 5.

    Example 1. Values of s and q for several famous TI are listed in Table 2:

    Table 2.  Values of s and q for six famous TI.
    ij 1ij 2iji+j 1i+j (ij)3(i+j2)3 i+j2ij
    q -1 -0.0168 -0.0404 -0.0138 -3.390 0.040
    s 5 -0.1667 0 -0.091 3.390 -0.040

     | Show Table
    DownLoad: CSV

    Therefore, the minimum extreme value of TI for the second Zagreb index, GA index and the AZI index can be determined in the light of Theorems 2.3 and 3.1, and we can obtain the maximum extreme value of TI for the ABC index.

    If f-benzenoid FΓm, then from the Eqs (2.3) and (2.6) and Lemma 1.1(ii) we have

    TI(F)=(2ψ23ψ33)m+6(ψ33ψ23)h(2ψ23ψ22ψ33)b               5(2ψ23ψ22ψ33). (3.4)

    Consequently, for f-benzenoids U,VΓm

    TI(U)TI(V)=6(ψ33ψ23)(h(U)h(V))          +(2ψ23+ψ22+ψ33)(b(U)b(V)). (3.5)

    Set u=6(ψ33ψ23) and keep in mind that q=2ψ23ψ22ψ33. Then

    TI(U)TI(V)=u(h(U)h(V))q(b(U)b(V)). (3.6)

    It is noted that Eq (3.6) can be decided only by h, b and the signs of u and q. For any FΓm, We know that

    h(F)m113(2m+4m31),

    and the equality can be achieved precisely when F is the f-spiral benzenoid F [41].

    In [41], we proved that ni(F)=2h12(h1)3. But, b(F)2 may occur. It is noticeable if X in F is a CBS, F is a f-benzenoid satisfying that b(F)=2 or 3. For the sake of simplicity, Let N be the set of positive integers.

    The CBS, W=H(l1,l2,l3,l4,l5,l6) (as shown in Figure 18), can be completely determined by the positive integers l1,l2,l3,l4 [14].

    Figure 18.  The general form of a CBS. The parameters li1,i=1,2,,6, count the number of hexagons on the respective side of CBS.

    The following lemma gave requirements that there exists CBS with maximal ni [53].

    Lemma 3.1. [53] Let hN. The conditions below are isovalent:

    (a) There is a CBS W containing h hexagons and 2h+112h3  number of internal vertices.

    (b) There exist l1,l2,l3,l4N satisfying the following equation

    h=l1l3+l1l4+l2l3+l2l4l2l312l1(l1+1)12l4(l4+1)+112h3 =l1+2l2+2l3+l43} (3.7)

    If for hN, Eq (3.7) has a solution l1,l2,l3,l4N, then there is a CBS W meeting the conditions that ni(W)=ni(Th).

    Now, we concentrate on the research for TI of f-benzenoids. For a h1N, supposing that the system below

    h1=l1l3+l1l4+l2l3+l2l4l2l312l1(l1+1)12l4(l4+1)+112(h1)3 =l1+2l2+2l3+l43 li{l1,l2,l3,l4,l5,l6}, li=2} (3.8)

    has a solution {l1,l2,l3,l4}, then a CBS Wh1 containing ni(Wh1)=2(h1)+112(h1)3 number of internal vertices exists. Note that li=2 in system (3.8), i.e., there exists one fissure on the side of li of Wh1, let u,w,v in Figure 1 represent the three vertices of this fissure. Now, we obtain an f-spiral benzenoid F1 in which X=Wh1 and Y=L1. It is obvious that

    ni(F1)=2h12(h1)3 (3.9)

    and b(F1)=2. (as shown in Figure 19)

    Figure 19.  A f-spiral benzenoid F1 whose fragment X is a convex spiral benzenoid system Wh1.

    Theorem 3.2. Let h1N such that the Eq (3.8) has a solution, and m=3h+5+12(h1)3. Then for any FΓm

    1) TI(F1)TI(F), when u0 and q0;

    2) TI(F1)TI(F), when u0 and q0.

    Proof. From Lemma 1.1 (ii) and Eq (3.9), we have

    m(F1)=5h+5(2h12(h1)3)=3h+5+12(h1)3

    and so

    h=m113(2m+4m31).

    It is obvious that b(F1)=2 and b(F)2 for any FΓm. Hence by Eq (3.6), we have

    TI(F)TI(F1)=u(h(F)h(F1))q(b(F)b(F1))
    =u[h(F)(m113(2m+4m31))]q[b(F)2].

    And by Eq (2.5)

    h(F)m113(2m+4m31).

    If u0 and q0 then TI(F)TI(F1)0, i.e., F1 achieves maximal TI in Γm. Similarly, if u0 and q0 then TI(F)TI(F1)0, i.e., F1 obtains minimal TI in Γm.

    Example 2. The values of u and q for some famous TI are listed in the following Table 3:

    Table 3.  Values of u and q for six famous TI.
    ij 1ij 2iji+j 1i+j (ij)3(i+j2)3 i+j2ij
    q -1 -0.0168 -0.0404 -0.0138 -3.390 0.040
    u 18 -0.449 0.121 -0.233 20.344 -0.242

     | Show Table
    DownLoad: CSV

    Hence, by Theorem 3.1 we can deduce the minimal values of the Randć index and the the sum–connectivity index in f-spiral benzenoid F1 for those h such that Eq (3.8) holds.

    Example 3. Take consideration of the generalized Randć index

    Rα(G)=1ijn1mij(ij)α,

    where αR. Note that

    q=2(6α)4α9α=4α((32)α1)20

    for all αR. Moreover, s=9α4α0 if and only if α0 if and only if u=6(9α6α)0. Hence, by Theorem 3.1, the minimal value of Rα(G) is obtained for all α0, and for any α0, the minimal value of Rα(G) can be attained by the f-spiral benzenoid F1 for those h such that Eq (3.8) holds.

    This work investigates extremum TI over the collection of f-benzenoids having same number of edges. In practical terms, there are many other types of very useful topological indices for instance graph energy [58,59,60,61,62], Wiener index [63], Randić energy [64], Wiener polarity index [65], incidence energy [66], Harary index [67], entropy measures [68,69] and HOMO-LUMO index [70]. So, determining these topological indices for f-benzenoids is going to be extraordinary fascinating.

    It is noted that the current framework is for studying topological indices of deterministic networks. But random networks would be a very promising direction. In [71,72], the distance Estrada index of random graphs was discussed, and the author went deeply into (Laplacian) Estrada index for random interdependent graphs. So, studying VDB topological indices of random and random interdependent graphs is another interesting problem.

    This work was supported by Ningbo Natural Science Foundation (No. 2021J234). The authors are very grateful to anonymous referees and editor for their constructive suggestions and insightful comments, which have considerably improved the presentation of this paper.

    The authors declare there is no conflict of interest.



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