Research article Special Issues

Analytical and computational properties of the variable symmetric division deg index


  • The aim of this work is to obtain new inequalities for the variable symmetric division deg index SDDα(G)=uvE(G)(dαu/dαv+dαv/dαu), and to characterize graphs extremal with respect to them. Here, by uv we mean the edge of a graph G joining the vertices u and v, and du denotes the degree of u, and αR. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the SDDα(G) index on random graphs and we demonstrate that the ratio SDDα(G)/n (n is the order of the graph) depends only on the average degree d.

    Citation: J. A. Méndez-Bermúdez, José M. Rodríguez, José L. Sánchez, José M. Sigarreta. Analytical and computational properties of the variable symmetric division deg index[J]. Mathematical Biosciences and Engineering, 2022, 19(9): 8908-8922. doi: 10.3934/mbe.2022413

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  • The aim of this work is to obtain new inequalities for the variable symmetric division deg index SDDα(G)=uvE(G)(dαu/dαv+dαv/dαu), and to characterize graphs extremal with respect to them. Here, by uv we mean the edge of a graph G joining the vertices u and v, and du denotes the degree of u, and αR. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the SDDα(G) index on random graphs and we demonstrate that the ratio SDDα(G)/n (n is the order of the graph) depends only on the average degree d.



    By a graph G=(V(G),E(G)), with V(G) and E(G) the set of vertices and edges of G respectively, we mean an undirected simple graph without isolated vertices (i.e., each vertex has at least a neighbor).

    Given a graph G, representing a chemical structure,

    X(G)=uvE(G)F(du,dv)

    is said a topological descriptor and, if also it correlates with a molecular property, it is called a topological index. Above, by uv we mean the edge of a graph G joining the vertices u and v, du denotes the degree of u, and F is an appropriate chosen function. Remarkably, topological indices capture physical properties of a chemical compound in a single number.

    A great number of topological indices have been defined and studied over more than four decades. Among them, probably the most popular topological indices are the Randić and the Zagreb indices. The first and second Zagreb indices, denoted by M1 and M2, respectively, were defined by Gutman and Trinajstić (see [1]) in 1972 by

    M1(G)=uV(G)d2u,M2(G)=uvE(G)dudv.

    For more details of the applications and mathematical properties of Zagreb indices see [2,3,4], and the references therein. Zagreb indices have many connections with other topological indices, see e.g., [5,6].

    The concept of variable molecular descriptors was proposed as a way of characterizing heteroatoms (see [7,8]), but also to assess structural differences in alkylcycloalkanes [9]. The idea behind the variable molecular descriptors is that the variables are determined during the regression in order to minimize the error of estimate for a particular chemical property (see, e.g., [10]).

    In this line of ideas, the variable versions of the first and second Zagreb indices were introduced as [10,11,12]

    Mα1(G)=uV(G)dαu,Mα2(G)=uvE(G)(dudv)α,

    with αR. Evidently, M21 and M12 are the first and second Zagreb indices, respectively. In addition, the first and second variable Zagreb indices include several known indices. As examples we note that M1/22 is the Randić index, M31 is the forgotten index F, M11 is the inverse index ID, and M12 is the modified Zagreb index.

    In 2011, Vukičević proposed the variable symmetric division deg index [13]

    SDDα(G)=uvE(G)(dαudαv+dαvdαu). (1.1)

    Note that SDDα(G)=SDDα(G) and so, it suffices to consider positive values of α. The symmetric division deg index is the best predictor of total surface area for polychlorobiphenyls [14].

    In this work we perform studies of the variable symmetric division deg index from analytical and computational viewpoints. We obtain new inequalities for the variable symmetric division deg index SDDα(G) and we characterize graphs extremal with respect to them. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the SDDα(G) index on random graphs and we demonstrate that the ratio SDDα(G)/n (n denotes the order of the graph) depends only on the average degree d.

    One of our main results is Theorem 8, which provides upper and lower bounds of SDDα(G) in terms of the number of edges, the maximum and the minimum degree of G.

    Let us start by proving a monotonicity property of these indices.

    Theorem 1. Let G be a graph and 0<α<β.Then

    SDDα(G)SDDβ(G),

    and the equality in the bound is attained if and only if each connected component of G is a regular graph.

    Proof. Let us consider x1. Thus, xαxβ and

    xβα10,xα(xβα1)xβ(xβα1),xβxαxαxβ,xβ+xβxα+xα,

    for every x1. Since u(x)=xα+xα satisfies u(1/x)=u(x) for every x>0, we have xβ+xβxα+xα for every x>0. Note that we obtain the equality if and only if x=1.

    Thus, we have

    SDDβ(G)=uvE(G)(dβudβv+dβvdβu)uvE(G)(dαudαv+dαvdαu)=SDDα(G).

    The previous argument gives that we have the equality in the bound if and only if du/dv=1 for every uvE(G), i.e., each connected component of G is a regular graph.

    Our next results in this section provide bounds of SDDα(G) involving the maximum and minimum degree of the graph G. Since scientists often estimate average degree of large networks, we present in the next section results involving the average degree.

    Our next theorem relates the SDDα and the variable Zagreb indices.

    Theorem 2. If G is a graph with minimum degree δ and maximum degree Δ, and α>0, then

    2δ2αMα2(G)SDDα(G)2Δ2αMα2(G),Δ2αM2α+11(G)SDDα(G)δ2αM2α+11(G),

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

    Proof. First of all recall that for every function f the following equality

    uvE(G)(f(du)+f(dv))=uV(G)duf(du)

    holds. In particular,

    uvE(G)(d2αu+d2αv)=uV(G)d2α+1u=M2a+11(G).

    Since

    SDDα(G)=uvE(G)(dαudαv+dαvdαu)=uvE(G)d2αu+d2αv(dudv)α.

    and α>0, we obtain

    SDDα(G)=uvE(G)d2αu+d2αv(dudv)α2Δ2αuvE(G)(dudv)α=2Δ2αMα2(G),

    and

    SDDα(G)=uvE(G)d2αu+d2αv(dudv)α2δ2αuvE(G)(dudv)α=2δ2αMα2(G).

    We also have

    SDDα(G)=uvE(G)d2αu+d2αv(dudv)αδ2αuvE(G)(d2αu+d2αv)=δ2αuV(G)d2α+1u=δ2αM2α+11(G),

    and

    SDDα(G)=uvE(G)d2αu+d2αv(dudv)αΔ2αuvE(G)(d2αu+d2αv)=Δ2αuV(G)d2α+1u=Δ2αM2α+11(G).

    If G is a regular graph, then each lower bound and its corresponding upper bound are the same, and both are equal to SDDα(G).

    Assume now that the equality in either the first or second bound holds. The previous argument gives that we have either d2αu+d2αv=2Δ2α for any uvE(G) or d2αu+d2αv=2δ2α for any uvE(G). Since α>0, we have δ2αd2αu,d2αvΔ2α, and we conclude that du=dv=Δ for any uvE(G), or du=dv=δ for any uvE(G). Hence, G is a regular graph.

    Finally, assume that the equality in either the third or fourth bound holds. The previous argument gives that we have either (dudv)α=δ2α for any uvE(G) or (dudv)α=Δ2α for any uvE(G). Since α>0, we have δαdαu,dαvΔα, and we conclude that du=dv=δ for any uvE(G), or du=dv=Δ for every uvE(G). Therefore, G is a regular graph.

    We will need the following technical result.

    Lemma 3. Let 0<a<A. Then

    ax2+y2x+yA

    for every ax,yA.The lower bound is attained if and only if x=y=a.The upper bound is attained if and only if x=y=A.

    Proof. If ax,yA, then ax+ayx2+y2Ax+Ay, and the statement holds.

    A large kind of topological indices, named Adriatic indices, was introduced in [14,15]. Twenty of them were selected as significant predictors of chemical properties. One of them, the inverse sum indeg index, defined by

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

    appears in [14,15] as a good predictor of total surface area of octane isomers.

    Next, we relate SDDα(G) with the variable inverse sum deg index defined, for each aR, as

    ISDa(G)=uvE(G)1dau+dav.

    Note that ISD1 is the inverse sum indeg index ISI.

    Theorem 4. If G is a graph with m edges and minimum degree δ, and α>0, then

    SDDα(G)δαm2ISDα(G),

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

    Proof. Lemma 3 gives

    δαx2α+y2αxα+yαΔα,1x2α+y2αδαxα+yα,1d2αu+d2αvδαdαu+dαv,

    for every δx,yΔ. This last inequality and Cauchy-Schwarz inequality give

    m2=(uvE(G)1)2=(uvE(G)(d2αu+d2αvdαudαv)1/2(dαudαvd2αu+d2αv)1/2)2uvE(G)d2αu+d2αvdαudαvuvE(G)dαudαvd2αu+d2αvδαSDDα(G)uvE(G)dαudαvdαu+dαv=δαSDDα(G)uvE(G)1dαu+dαv=δαSDDα(G)ISDα(G).

    If G is a regular graph, then SDDα(G)=2m, ISDα(G)=mδα/2 and the equality in the bound holds.

    Assume now that the equality in the bound holds. Thus, by the previous argument,

    1d2αu+d2αv=δαdαu+dαv

    for any uvE(G). Then Lemma 3 gives du=dv=δ for any uvE(G). Hence, G is a regular graph.

    The modified Narumi-Katayama index is defined by

    NK(G)=uV(G)dduu=uvE(G)dudv

    in [16], inspired in the Narumi-Katayama index [17]. Next, we present an inequality relating SDDα(G) and NK(G).

    Theorem 5. Let G be a graph with m edges and minimum degree δ, and α>0.Then

    SDDα(G)2δ2αmNK(G)α/m,

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

    Proof. Since the geometric mean is at most the arithmetic mean, we have

    1mSDDα(G)=1muvE(G)(dαudαv+dαvdαu)=1muvE(G)d2αu+d2αv(dudv)α2δ2α1muvE(G)1(dudv)α2δ2α(uvE(G)1(dudv)α)1/m=2δ2αNK(G)α/m.

    If G is a regular graph, then

    2δ2αmNK(G)α/m=2δ2αm(δ2m)α/m=2m=SDDα(G).

    Finally, assume that the equality in the bound holds. The previous argument gives that d2αu+d2αv=2δ2α for any uvE(G). Since α>0, we obtain δ2αd2αu,d2αv, and we have du=dv=δ for any uvE(G). Hence, G is regular.

    Next, we obtain additional bounds of SDDα.

    Theorem 6. Let G be a graph with m edges, minimum degree δ and maximum degree δ+1, α>0 and let A be the number of edges uvE(G) with dudv.Then A is an even integer and

    SDDα(G)=2m+A((δ+1)αδα+δα(δ+1)α2).

    Proof. Let F={uvE(G):dudv}, then A is the cardinality of the set F. Since the maximum degree of G is δ+1 and its minimum degree is δ, if uvF, then du=δ and dv=δ+1 or viceversa, and therefore

    dαudαv+dαvdαu=(δ+1)αδα+δα(δ+1)α.

    If uvFc=E(G)F, then du=dv=δ or du=dv=δ+1, and therefore

    dαudαv+dαvdαu=2.

    Since there are exactly A edges in F and mA edges in Fc, we have

    SDDα(G)=uvE(G)(dαudαv+dαvdαu)=uvFc(dαudαv+dαvdαu)+uvF(dαudαv+dαvdαu)=uvFc2+uvF((δ+1)αδα+δα(δ+1)α)=2m2A+A((δ+1)αδα+δα(δ+1)α).

    This gives the equality.

    Seeking for a contradiction assume that A is an odd integer.

    Let Γ1 be a subgraph of G obtained as follows: Γ1 is induced by the n1 vertices with degree δ in V(G); denote by m1 the number of edges of Γ1. Handshaking Lemma implies n1δA=2m1. Since A is an odd integer, δ is also an odd integer. Thus, δ+1 is even.

    Let Γ2 be the subgraph of G induced by the n2 vertices with degree δ+1 in V(G), and denote by m2 the number of edges of Γ2. Handshaking Lemma implies n2(δ+1)A=2m2, a contradiction, since A is odd and δ+1 is even.

    Thus, we conclude that A is an even integer.

    We will need the following result in the proof of Theorem 8 below.

    Lemma 7. Given α>0, consider the function u:(0,)(0,) defined as u(t)=tα+tα.Then u strictly decreases on (0,1], u strictly increases on [1,) and u(t)u(1)=2.

    Proof. We have

    u(t)=αtα1αtα1=αtα1(t2α1).

    Since α>0, we have u<0 on (0,1) and u>0 on (1,). This gives the result. The following figure shows the function u(t) for some values of α.

    Theorem 6 gives the precise value of SDDα when Δ=δ+1. Theorem 8 below provides a lower bound when Δ>δ+1.

    Theorem 8. Let G be a graph with m edges, minimum degree δ and maximum degree Δ>δ+1. Denote by A0,A1,A2, the cardinality of the subsets of edges F0={uvE(G):du=δ,dv=Δ}, F1={uvE(G):du=δ,δ<dv<Δ}, F2={uvE(G):du=Δ,δ<dv<Δ}, respectively. If α>0, then

    SDDα(G)(mA1A2)(Δαδα+δαΔα)+A1((Δ1)αδα+δα(Δ1)α)+A2(Δα(δ+1)α+(δ+1)αΔα),SDDα(G)2m+A0(Δαδα+δαΔα2)+A1((δ+1)αδα+δα(δ+1)α2)+A2(Δα(Δ1)α+(Δ1)αΔα2).

    Proof. Lemma 7 gives that the function

    dαvδα+δαdαv=u(dvδ)

    is increasing in dv[δ+1,Δ1] and so,

    (δ+1)αδα+δα(δ+1)αdαvδα+δαdαv(Δ1)αδα+δα(Δ1)α,

    for every uvF1.

    In a similar way, Lemma 7 gives that the function

    Δαdαv+dαvΔα=u(dvΔ)

    is decreasing in dv[δ+1,Δ1] and so,

    Δα(Δ1)α+(Δ1)αΔαΔαdαv+dαvΔαΔα(δ+1)α+(δ+1)αΔα,

    for every uvF2.

    Also,

    2dαudαv+dαvdαuΔαδα+δαΔα

    for any uvE(G).

    We obtain

    SDDα(G)=uvE(G)(F0F1F2)(dαudαv+dαvdαu)+uvF0(dαudαv+dαvdαu)+uvF1(dαudαv+dαvdαu)+uvF2(dαudαv+dαvdαu)uvE(G)(F0F1F2)2+uvF0(Δαδα+δαΔα)+uvF1(dαvδα+δαdαv)+uvF2(Δαdαv+dαvΔα).

    Hence,

    SDDα(G)2m2A02A12A2+A0(Δαδα+δαΔα)+A1((δ+1)αδα+δα(δ+1)α)+A2(Δα(Δ1)α+(Δ1)αΔα).

    We also have

    SDDα(G)=uvE(G)(F1F2)(dαudαv+dαvdαu)+uvF1(dαudαv+dαvdαu)+uvF2(dαudαv+dαvdαu)(mA1A2)(Δαδα+δαΔα)+A1((Δ1)αδα+δα(Δ1)α)+A2(Δα(δ+1)α+(δ+1)αΔα).

    Here we deal with two classes of random graphs G: Erdös-Rényi (ER) graphs G(n,p) and bipartite random (BR) graphs G(n1,n2,p). ER graphs are formed by n vertices connected independently with probability p[0,1]. While BR graphs are composed by two disjoint sets, sets 1 and 2, with n1 and n2 vertices each such that there are no adjacent vertices within the same set, being n=n1+n2 the total number of vertices in the bipartite graph. The vertices of the two sets are connected randomly with probability p[0,1]. Another work in this spirit is [18].

    We stress that the computational study of the variable symmetric division deg index we perform below is justified by the random nature of the graph models we want to explore. Since a given parameter set (n,p) [(n1,n2,p)] represents an infinite-size ensemble of ER graphs [BR graphs], the computation of SDDα(G) on a single graph is irrelevant. In contrast, the computation of SDDα(G) (where indicates ensemble average) over a large number of random graphs, all characterized by the same parameter set (n,p) [(n1,n2,p)], may provide useful average information about the full ensemble. This computational approach, well known in random matrix theory studies, is not widespread in studies involving topological indices, mainly because topological indices are not commonly applied to random graphs; for very recent exceptions see [19,20,21,22].

    From the definition of the variable symmetric division deg index, see Eq (1.1), we have that:

    (i) For α=0, SDD0(G) gives twice the average number of edges of the ER graph. That is,

    SDD0(G)=uvE(G)(d0ud0v+d0vd0u)=uvE(G)(1+1)=2|E(G)|=n(n1)p. (3.1)

    (ii) When np1, we can approximate dudvd, then

    SDDα(G)uvE(G)(1α+1α)=uvE(G)2=2|E(G)|=n(n1)p. (3.2)

    (iii) By recognizing that the average degree of the ER graph model reads as

    d=(n1)p, (3.3)

    we can rewrite Eq (3.2) as

    SDDα(G)nd. (3.4)

    We stress that Eq (3.4) is expected to be valid for np1.

    In Figure 2(a) we plot SDDα(G) as a function of the probability p of ER graphs of size n=500. All averages in Figure 2 are computed over ensembles of 107/n random graphs. In Figure 2(a) we show curves for α[0,4]. The dashed-magenta curve corresponds to the case α=0, which coincides with Eq (3.1). Moreover, we observe that

    SDDα0.5(G)SDD0(G)=n(n1)p.
    Figure 1.  Representations of the function u(t) of Lemma 7 for: α=1/4 (yellow), α=1/2 (green), α=1 (red) and α=2 (blue).
    Figure 2.  (a) Average variable symmetric division deg index SDDα(G) as a function of the probability p of Erdös-Rényi graphs of size n=500. Here we show curves for α[0,4] in steps of 0.5 (the arrow indicates increasing α). The dashed-magenta curve corresponds to the case α=0. (b) SDDα(G) as a function of the probability p of ER graphs of three different sizes: n=125, 250 and 500. (c) SDDα(G)/n as a function of the average degree d; same curves as in panel (b). The inset in (c) is the enlargement of the cyan rectangle. (d)–(f) Equivalent figures to (a)–(c), respectively, but for bipartite random graphs composed by sets of equal sizes: in (d) n1=n2=500 while in (e), (f) n1=n2={125,250,500}.

    However, once α>0.5, the curves SDDα(G) versus p deviate from Eq (3.1), at intermediate values of p, in the form of a bump which is enhanced the larger the value of α is. Also, in Figure 2(a) we can clearly see that Eq (3.2) is satisfied when np1, as expected.

    Now, in Figure 2(b) we present SDDα(G) as a function of the probability p of ER graphs of three different sizes. It is clear from this figure that the blocks of curves, characterized by the different graph sizes (and shown in different colors), display similar curves but displaced on both axes. Moreover, the fact that these blocks of curves, plotted in semi-log scale, are shifted the same amount on both x and yaxis when doubling n make us anticipate the scaling of SDDα(G). We stress that other average variable degree-based indices on ER random graphs (normalized to the graph size) have been shown to scale with the average degree [22]. Indeed, this statement is encoded in Eq (3.4), that we derived for np1 but should serve as the global scaling equation for SDDα(G).

    Therefore, in Figure 2(c) we show SDDα(G)/n as a function of the average degree d where the same curves of Figure 2(b) have been used. There we verify the global scaling of SDDα(G), as anticipated in Eq (3.4), by noticing that the blocks of curves (painted in different colors) for different graph sizes fall on top of each other.

    Also, from Figure 2(a)(c) we observe that the inequality of Theorem 1 is extended to the average variable symmetric division deg index on random graphs:

    SDDα(G)SDDβ(G),0<α<β; (3.5)

    see e.g., the blue arrow in Figure 2(a) which indicates increasing α. Here, the equality is attained if and only if p=1. However, we have observed that SDDα(G)SDDβ(G) already for d10.

    In Figure 2(d), (e) we present curves of the SDDα(G) as a function of the probability p of BR graphs. For simplicity we show results for BR graphs composed by sets of equal sizes n1=n2. In Figure 2(d) we consider the case of n1=n2=500 while in (e) we report n1=n2={125,250,500}. In both figures we show curves for α[0,4] in steps of 0.5.

    Since edges in a bipartite graph join vertices of different sets, and we are labeling here the sets as sets 1 and 2, we replace du by d1 and dv by d2 in the expression for the SDDα(G) index below. Thus,

    (i) For α=0, SDD0(G) gives twice the average number of edges of the BG graph. That is,

    SDD0(G)=E(G)(d01d02+d02d01)=E(G)(1+1)=2|E(G)|=2n1n2p. (3.6)

    (ii) When both n1p1 and n2p1, we can approximate d1d1 and d2d2, then

    SDDα(G)E(G)(d1αd2α+d2αd1α)=|E(G)|(d1αd2α+d2αd1α). (3.7)

    (iii) In the case we consider in Figure 2(d)(f), where n1=n2=n/2, so that d1=d2=d, Equation (3.7) reduces to

    SDDα(G)2|E(G)|=2n1n2p=n22p. (3.8)

    (iv) By recognizing that d=np/2 we can rewrite Eq (3.8) as

    SDDα(G)nd. (3.9)

    We stress that Eq (3.9) is expected to be valid for np1. We also note that Eq (3.9) has exactly the same form as Eq (3.4).

    From Figure 2(d), (e) we note that

    SDDα0.5(G)SDD0(G)=2n1n2p,

    see the dashed-magenta curve in Figure 2(d). But once α>0.5, the curves SDDα(G) versus p deviate from Eq (3.6), at intermediate values of p, in the form of bumps which are enhanced the larger the value of α is. These bumps make clear the validity of inequality (3.5) on BR graphs; see e.g., the blue arrow in Figure 2(d) which indicates increasing α.

    Finally, following the scaling analysis made in the previous subsection for ER graphs, in Figure 2(f) we plot the SDDα(G)/n as a function of the average degree d where the same data sets of Figure 2(e) have been used. Thus we verify that SDDα(G)/n scales with d, as anticipated in Eq (3.9); that is, the blocks of curves (painted in different colors) for different graph sizes coincide.

    In this paper we have performed analytical and computational studies of the variable symmetric division deg index SDDα(G). First, we provided a monotonicity property and obtained new inequalities connecting SDDα(G) with other well–known topological indices such as the variable inverse sum deg index, as well as the the modified Narumi-Katayama index. Then, we apply the index SDDα(G) on two ensembles of random graphs: Erd\H{o}s-Rényi graphs and bipartite random graphs. Thus, we computationally showed, for both random graph models, that the ratio SDDα(G)/n is a function of the average degree d only (n being the order of the graph). We note that this last result, also observed for other variable topological indices [22], is valid for random bipartite graphs only when they are formed by sets of the same size.

    Since many important topological indices can be written as

    X(G)=uvE(G)F(du,dv),

    an open problem is to extend the results of this paper to other indices of this kind.

    J. A. M.-B. has been supported by CONACyT (Grant No. CB-2016-286633-F). J. M. R. and J. M. S. have been supported by a grant from Agencia Estatal de Investigación (PID2019-106433GBI00/AEI/10.13039/501100011033), Spain. J. M. R. has been 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).

    The authors declare there is no conflict of interest.



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