Research article Special Issues

On the variable inverse sum deg index


  • Several important topological indices studied in mathematical chemistry are expressed in the following way uvE(G)F(du,dv), where F is a two variable function that satisfies the condition F(x,y)=F(y,x), uv denotes an edge of the graph G and du is the degree of the vertex u. Among them, the variable inverse sum deg index ISDa, with F(du,dv)=1/(dau+dav), was found to have several applications. In this paper, we solve some problems posed by Vukičević [1], and we characterize graphs with maximum and minimum values of the ISDa index, for a<0, in the following sets of graphs with n vertices: graphs with fixed minimum degree, connected graphs with fixed minimum degree, graphs with fixed maximum degree, and connected graphs with fixed maximum degree. Also, we performed a QSPR analysis to test the predictive power of this index for some physicochemical properties of polyaromatic hydrocarbons.

    Citation: Edil D. Molina, Paul Bosch, José M. Sigarreta, Eva Tourís. On the variable inverse sum deg index[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8800-8813. doi: 10.3934/mbe.2023387

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  • Several important topological indices studied in mathematical chemistry are expressed in the following way uvE(G)F(du,dv), where F is a two variable function that satisfies the condition F(x,y)=F(y,x), uv denotes an edge of the graph G and du is the degree of the vertex u. Among them, the variable inverse sum deg index ISDa, with F(du,dv)=1/(dau+dav), was found to have several applications. In this paper, we solve some problems posed by Vukičević [1], and we characterize graphs with maximum and minimum values of the ISDa index, for a<0, in the following sets of graphs with n vertices: graphs with fixed minimum degree, connected graphs with fixed minimum degree, graphs with fixed maximum degree, and connected graphs with fixed maximum degree. Also, we performed a QSPR analysis to test the predictive power of this index for some physicochemical properties of polyaromatic hydrocarbons.



    Topological indices have become an important research topic associated with the study of their mathematical and computational properties and, fundamentally, for their multiple applications to various areas of knowledge (see, e.g., [2,3,4,5,6,7,8,9]). Within the study of mathematical properties, we will contribute to the study of the optimization problems involved with topological indices (see, e.g., [10,11,12,13,14,15,16,17,18]).

    In [19,20] several degree-based topological indices, called adriatic indices, were presented; one of them is the inverse sum indeg index ISI. It is important to note that this index was selected as one of the most predictive, in particular associated with the total surface area of the isomers of octane.

    Let G be a graph and E(G) the set of all edges in G, denote by uv the edge of the graph G with vertices u,v and dz is the degree of the vertex z. the ISI index is defined by

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

    Nowadays, this index has become one of the most studied from the mathematical point of view (see, e.g., [21,22,23,24,25,26,27]). We study, here, the mathematical properties of 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.

    This research is motivated, in general, by the theoretical-mathematical importance of the topological indices and by their applicability in different areas of knowledge (see [28,29,30]). Additionally, in particular, by the work developed by Vukičević entitled "Bond Additive Modeling 5. Mathematical Properties of the Variable Sum Exdeg Index" (see [1]), where several open problems on the topological index ISDa were proposed. The novelty of this work is given in two main directions. The first one is associated with the solution of some of the problems posed in [1]. The second one is associated with the development of new optimization techniques and procedures related to the monotony and differentiation of symmetric functions, which allowed us to solve extremal problems and to present bounds for ISDa. Although, these techniques can be extended or applied in a natural way to obtain new relations and properties of other topological indices, it should be noted that their applicability requires the monotony of the function that determines the index to be studied.

    In Section 2, we find optimal bounds and solve extremal problems associated with the topological index ISDa, with a<0, for several families of graphs. In Proposition 4, we solve the extremal problems for connected graphs with a given number of vertices. Theorem 6 and Remark 1 solve these problems for graphs with a given number of vertices and minimum degree; similarly, Theorems 8 and 9 present solutions to extremal problems in connected graphs with a given number of vertices and maximum degree. In this direction, in Theorem 5, Proposition 7 and Theorem 10, we present optimal bounds for the studied index.

    In Section 3 of this research, a QSPR study related to the ISDa index in polyaromatic hydrocarbons is performed using experimental data. First, we determine the value of a that maximizes the Pearson's correlation coefficient between this index, and each of the studied physico-chemical properties. Finally, models for these properties are constructed using the simple linear regression method. A discussion of the results obtained is presented in Section 4, and some open problems for future research on this topic are raised.

    In this research, G=(V(G),E(G)) denotes an undirected finite simple graph without isolated vertices. By n, m, Δ and δ, we denote the cardinality of the set of vertices of G, the cardinality of the set of edges of G, its maximum degree and its minimum degree, respectively. Thus, we have 1δΔ<n. We denote by N(u) the set of neighbors of the vertex uV(G).

    Suppose δ<Δ, we say that a graph G is (δ,Δ)-quasi-regular if it contains a vertex w, such that δ=dw and Δ=dz for every zV(G){w}; G is (δ,Δ)-pseudo-regular if it contains a vertex w, such that Δ=dw and δ=dz for every zV(G){w}.

    In [31] appears the following result.

    Lemma 1. Let k be an integer, such that 2k<n.

    (1) If nk is even, then there exists a k-regular graph that is connected and has n vertices.

    (2) If nk is odd, then there exist a (k,k1)-quasi-regular and a connected (k+1,k)-pseudo-regular graphs, which are connected and have n vertices.

    The following result is basic to the development of this work.

    Lemma 2. For each a<0, the function f:R+×R+R+ given by

    f(x,y)=1xa+ya

    is strictly increasing in each variable.

    Proof. Since a<0, we have

    fx(x,y)=axa1(xa+ya)2>0.

    Then, f is a strictly increasing function in x, and since f is symmetric, it is also strictly decreasing in y.

    Using Lemma 2, we obtain the following result.

    Proposition 3. If G is a graph, u,vV(G) with uvE(G), and a<0, then ISDa(G{uv})>ISDa(G).

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

    Next, given integer numbers 1δΔ<n, we are going to define the following classes of graphs: let H(n,δ) (respectively, Hc(n,δ)) be the graphs (respectively, connected graphs) with n vertices and minimum degree δ, and let I(n,Δ) (respectively, Ic(n,Δ)) be the graphs (respectively, connected graphs) with maximum degree Δ and n vertices.

    First, let us state an optimization result for the ISDa index on Gc(n) and G(n) (see [32]).

    Proposition 4. Consider a<0 and an integer n2.

    (1) The graph that maximizes the ISDa index on Gc(n) or G(n) is unique and given by the complete graph Kn.

    (2) If a graph minimizes the ISDa index on Gc(n), then it is a tree.

    (3) If n is even, then the graph that minimizes the ISDa index on G(n) is unique and given by the union of n/2 paths P2. If n is odd, then the graph that minimizes the ISDa index on G(n) is unique and given by the union of (n3)/2 paths P2 with a path P3.

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

    Items (1) and (2) follow directly from Proposition 3.

    For the proof of item (3), we first assume that n is even. For any graph GG(n) Lemma 2 gives

    ISDα(G)=uvE(G)1dau+davuvE(G)11a+1a=m2,

    and the equality is attained if, and only if, {du,dv}={1} for each uvE(G), i.e., G is the union of n/2 path graphs P2.

    Now, we assume that n is odd. If du=1 for each uV(G), handshaking lemma gives 2m=n, a contradiction. So, there exists wV(G), such that dw2. Let N(w) be the set of neighbors of the vertex w, from Lemma 2, we obtain

    ISDα(G)=uvE(G),u,vw1dau+dav+uN(w)1dau+dawuvE(G),u,vw11a+1a+uN(w)11a+2am22+21+2a,

    and the equality is attained if, and only if, du=1 for each uV(G)w and dw=2. Hence, G is the union of (n3)/2 path graphs P2 and a path graph P3.

    Proposition 4 allows to obtain the following inequalities.

    Theorem 5. Consider a graph G with n vertices and a negative constant a.

    (1) Then,

    ISDa(G)14n(n1)1a,

    and equality holds if, and only if, G is the complete graph Kn.

    (2) If n is even, then

    ISDa(G)14n,

    and equality holds if, and only if, G is the union of n/2 path graphs P2.

    (3) If n is odd, then

    ISDa(G)14(n3)+21+2a,

    and equality holds if, and only if, G is the union of a path graph P3 and (n3)/2 path graphs P2.

    Proof. Proposition 4 gives

    ISDa(G)ISDa(Kn)=uvE(Kn)1dau+dav=n(n1)212(n1)a=14n(n1)1a.

    This argument gives that the bound is attained if, and only if, G is the complete graph Kn. Hence, item (1) holds.

    Suppose G has minimum degree δ. If n is even, handshaking lemma gives 2mnδn, using this and the proof of Proposition 4, we have

    ISDa(G)m2n4,

    and equality holds if, and only if, G is the union of n/2 path graphs P2. This gives item (2).

    If n is odd, handshaking lemma gives 2m(n1)δ+2n+1, using this and the proof of Proposition 4, we have

    ISDa(G)m22+21+2an+1222+21+2a=n34+21+2a,

    and equality holds if, and only if, G is the union of a path graph P3 and (n3)/2 path graphs P2. This gives item (3).

    Fix positive integers 1δ<n. Let Kδn be the n-vertex graph with minimum and maximum degrees δ and n1, respectively, obtained from Kn1 (the complete graph with n1 vertices) and an additional vertex w, as follows: If we fix δ vertices v1,,vδV(Kn1), then the vertices of Kδn are w and the vertices of Kn1, and the edges of Kδn are {v1w,,vδw} and the edges of Kn1.

    We consider now the optimization problem for the ISDa index on Hc(n,δ) and H(n,δ).

    Theorem 6. Consider a<0 and integers 1δ<n.

    (1) Then, the graph in Hc(n,δ) that maximizes the ISDa index is unique and given by Kδn.

    (2) If δ2 and nδ is even, then all the graphs in Hc(n,δ) that minimize ISDa are the connected δ-regular graphs.

    (3) If δ2 and nδ is odd, then all the graphs in Hc(n,δ) that minimize ISDa are the connected (δ+1,δ)-pseudo-regular graphs.

    Proof. Given a graph GHc(n,δ){Kδn}, fix any vertex uV(G) with du=δ. Since

    GG{vw:v,wV(G){u} and vwE(G)}=Kδn,

    Proposition 3 gives ISDa(Kδn)>ISDa(G). This proves item (1).

    Handshaking lemma gives 2mnδ.

    Since duδ for every uV(G), Lemma 2 gives

    ISDa(G)=uvE(G)1dau+davuvE(G)12δa=m2δanδ/22δa=14nδ1a,

    and the bound is attained if, and only if, δ=du for all uV(G).

    If δn is even, then Lemma 1 gives that there is a connected δ-regular graph with n vertices. Hence, the unique graphs in Hc(n,δ) that minimize the ISDa index are the connected δ-regular graphs.

    If δn is odd, then handshaking lemma gives that there is no regular graph. Hence, there exists a vertex w with dwδ+1. Since duδ for every uV(G), handshaking lemma gives 2m(n1)δ+δ+1=nδ+1. Lemma 2 gives

    ISDa(G)=uN(w)1dau+daw+uvE(G),u,vw1dau+davuN(w)1δa+(δ+1)a+uvE(G),u,vw12δaδ+1δa+(δ+1)a+mδ12δaδ+1δa+(δ+1)a+(nδ+1)/2δ12δa,

    and the bound is attained if, and only if, du=δ for all uV(G){w}, and dw=δ+1. Lemma 1 gives that there is a connected (δ+1,δ)-pseudo-regular graph with n vertices. Therefore, the unique graphs in Hc(n,δ) that minimize the ISDa index are the connected (δ+1,δ)-pseudo-regular graphs.

    Remark 1. If we replace Hc(n,δ) with H(n,δ) everywhere in the statement of Theorem 6, then the argument in its proof gives that the same conclusions hold if we remove everywhere the word "connected".

    Theorem 6 and Remark 1 have the following consequence.

    Proposition 7. Consider a graph G with minimum degree δ and n vertices, and a negative constant a.

    (1) Then,

    ISDa(G)(nδ1)(nδ2)4(n2)a+δδa+(n1)a+δ(δ1)4(n1)a+δ(nδ1)(n2)a+(n1)a,

    and the bound is attained if, and only if, G is isomorphic to Kδn.

    (2) If δ2 and δn is even, then

    ISDa(G)14nδ1a,

    and the bound is attained if, and only if, G is δ-regular.

    (3) If δ2 and nδ is odd, then

    ISDa(G)δ(n2)14δa+δ+1δa+(δ+1)a,

    and the bound is attained if, and only if, G is (δ+1,δ)-pseudo-regular.

    Let us deal with the optimization problem for the ISDa index on Ic(n,Δ).

    Theorem 8. Consider a<0 and integers 2Δ<n.

    (1) If nΔ is even, then all the graphs that maximize ISDa on Ic(n,Δ) are the connected Δ-regular graphs.

    (2) If nΔ is odd, then all the graphs that maximize ISDa on Ic(n,Δ) are the connected (Δ,Δ1)-quasi-regular graphs.

    (3) If a graph minimizes ISDa on Ic(n,Δ), then it is a tree.

    Proof. Handshaking lemma gives 2mnΔ. Since duΔ for every uV(G), Lemma 2 gives

    ISDa(G)=uvE(G)1dau+davuvE(G)12Δa=m2ΔanΔ/22Δa=14nΔ1a,

    and the bound is attained if, and only if, Δ=du for all uV(G).

    If nΔ is even, then Lemma 1 gives that there is a connected Δ-regular graph with n vertices. Hence, the unique graphs in Ic(n,Δ) that maximize the ISDa index are the connected Δ-regular graphs.

    If nΔ is odd, then handshaking lemma gives that there is no regular graph in Ic(n,Δ). Let GIc(n,Δ). Hence, there exists a vertex w with dwΔ1. Then, 2mΔ(n1)+Δ1=Δn1. Lemma 2 gives

    ISDa(G)=uN(w)1dau+daw+uvE(G),u,vw1dau+davuN(w)1Δa+(Δ1)a+uvE(G),u,vw12ΔaΔ1Δa+(Δ1)a+mΔ+12ΔaΔ1Δa+(Δ1)a+(Δn1)/2Δ+12Δa,

    and the bound is attained if, and only if, du=Δ for all uV(G){w}, and dw=Δ1. Lemma 1 gives that there is a connected (Δ,Δ1)-quasi-regular graph with n vertices. Therefore, the unique graphs in Ic(n,δ) that maximize the ISDa index are the connected (Δ,Δ1)-quasi-regular graphs.

    Given any graph GIc(n,Δ) which is not a tree, fix any vertex uV(G) with du=Δ. Since G is not a tree, there exists a cycle C in G. Since C has at least three edges, there exists vwE(G)C, such that u{v,w}. Since vw is contained in a cycle of G, then G{vw} is a connected graph. Thus, G{vw}Ic(n,Δ) and Proposition 3 gives ISDa(G)>ISDa(G{vw}). By iterating this argument, we obtain that if a graph minimizes the ISDa index on Ic(n,Δ), then it is a tree.

    The following result deals with the optimization problem for the ISDa index on I(n,Δ).

    Theorem 9. Consider a<0 and integers 2Δ<n.

    (1) If nΔ is even, then all the graphs that maximize the ISDa index on I(n,Δ) are the Δ-regular graphs.

    (2) If nΔ is odd, then all the graphs that maximize the ISDa index on I(n,Δ) are the (Δ,Δ1)-quasi-regular graphs.

    (3) If nΔ is odd, then the graph that minimizes the ISDa index on I(n,Δ) is unique and given by the union of the star graph SΔ+1 and (nΔ1)/2 path graphs P2.

    (4) If n=Δ+2, then the graph that minimizes the ISDa index on I(n,Δ) is unique and given by the star graph SΔ+1 with an additional edge attached to a vertex of degree 1 in SΔ+1.

    (5) If nΔ+4 and nΔ is even, then the graph that minimizes the ISDa index on I(n,Δ) is unique and given by the union of the star graph SΔ+1, (nΔ4)/2 path graphs P2 and a path graph P3.

    Proof. The argument in Theorem 8 gives directly items (1) and (2).

    Let GI(n,Δ) and wV(G) a vertex with dw=Δ.

    Assume first that nΔ is odd. Handshaking lemma gives 2mn1+Δ. Note that n1+Δ=nΔ+2Δ1 is even. Lemma 2 gives

    ISDa(G)=uN(w)1dau+daw+uvE(G),u,vw1dau+davuN(w)11a+Δa+uvE(G),u,vw11a+1a=Δ1+Δa+mΔ2Δ1+Δa+(n1+Δ)/2Δ2=Δ1+Δa+nΔ14,

    and the bound is attained if, and only if, 1=du for all uV(G){w}, i.e., G is the union of the star graph SΔ+1 and (nΔ1)/2 path graphs P2.

    Assume now that n=Δ+2. Let zV(G)N(w) be the vertex with V(G)={w,z}N(w). Choose pN(z); since zN(w), we have pN(w) and so, dp2. Handshaking lemma gives 2m(n2)+Δ+2=n+Δ. Lemma 2 gives

    ISDa(G)=uN(w)1dau+daw+uvE(G),u,vw1dau+davΔ11+Δa+12a+Δa+11+2a,

    and the bound is attained if, and only if, 1=du for all uV(G){w,p} and dp=2, i.e., G is the star graph SΔ+1 with an additional edge attached to a vertex of degree 1 in SΔ+1.

    Assume that nΔ+4 and nΔ is even. If du=1 for every uV(G){w}, then handshaking lemma gives 2m=n1+Δ, a contradiction since n1+Δ=nΔ+2Δ1 is odd. Thus, there exists a vertex pV(G){w} with dp2. Handshaking lemma gives 2m(n2)+2+Δ=n+Δ.

    If pN(w), then Lemma 2 gives

    ISDa(G)=uN(w)1dau+daw+uN(p)1dau+dap+uvE(G),u,v{w,p}1dau+davuN(w)11a+Δa+uN(p)11a+2a+uvE(G),u,v{w,p}11a+1aΔ1+Δa+21+2a+mΔ22Δ1+Δa+21+2a+(n+Δ)/2Δ22=Δ1+Δa+21+2a+nΔ44,

    and the bound is attained if, and only if, du=1 for all uV(G){w,p}, and dp=2, i.e., G is the union of the star graph SΔ+1, (nΔ4)/2 path graphs P2 and a path graph P3.

    If pN(w), then

    ISDa(G)=uN(w){p}1dau+daw+uN(p){w}1dau+dap+1dap+daw+uvE(G),u,v{w,p}1dau+davuN(w){p}11a+Δa+uN(p){w}11a+dap+1dap+Δa+uvE(G),u,v{w,p}11a+1aΔ11+Δa+11+2a+12a+Δa+mΔ12Δ11+Δa+11+2a+12a+Δa+(n+Δ)/2Δ12=Δ11+Δa+11+2a+12a+Δa+nΔ24.

    Hence, in order to finish the proof of item (5), it suffices to show that

    Δ11+Δa+11+2a+12a+Δa+nΔ24>Δ1+Δa+21+2a+nΔ44.

    We have

    (12a)(1Δa)>0,1+2aΔa>2a+Δa,2aΔa+Δa+2a+1>2(2a+Δa),(Δa+1)(1+2a)(2+2a+Δa)>2(2a+Δa)(2a+Δa+2),12a+Δa+12>11+Δa+11+2a,Δ11+Δa+11+2a+12a+Δa+nΔ24>Δ1+Δa+21+2a+nΔ44,

    and so, (5) holds.

    Remark 2. Note that the case Δ=1 in Theorem 9 is trivial: if Δ=1, then G is a union of isolated edges.

    Also, we can state the following inequalities.

    Theorem 10. Consider a graph G with maximum degree Δ and n vertices, and a negative constant a.

    (1) If nΔ is even, then

    ISDa(G)14nΔ1a,

    and the bound is attained if, and only if, G is a regular graph.

    (2) If nΔ is odd, then

    ISDa(G)Δ1Δa+(Δ1)a+Δ(n2)+14Δa,

    and the bound is attained if, and only if, G is a (Δ,Δ1)-quasi-regular graph.

    (3) If nΔ is odd, then

    ISDa(G)Δ1+Δa+nΔ14,

    and the bound is attained if, and only if, G is the union of the star graph SΔ+1 and (nΔ1)/2 path graphs P2.

    (4) If n=Δ+2, then

    ISDa(G)Δ11+Δa+12a+Δa+11+2a,

    and the bound is attained if, and only if, G is the star graph SΔ+1 with an additional edge attached to a vertex of degree 1 in SΔ+1.

    (5) If nΔ+4 and nΔ is even, then

    ISDa(G)Δ1+Δa+21+2a+nΔ44,

    and the bound is attained if, and only if, G is the union of the star graph SΔ+1, (nΔ4)/2 path graphs P2 and a path graph P3.

    Proof. The argument in the proof of Theorem 8 gives items (1) and (2), since the variable inverse sum deg index of a regular graph is

    14nΔ1a,

    and the ISDa index of a (Δ,Δ1)-quasi-regular graph is

    Δ1Δa+(Δ1)a+Δ(n2)+14Δa.

    The argument in the proof of Theorem 9 gives directly items (3)(5).

    The variable inverse sum deg index ISD1.950 was selected in [33] as a significant predictor of standard enthalpy of formation for octane isomers. In this section, we will test the predictive power of the ISDa index using experimental data on three physicochemical properties of 82 polyaromatic hydrocarbons (PAH). The properties studied are the melting point (MP), boiling point (BP) and octanol-water partition coefficient (LogP) (the experimental data were obtained from [34]). In order to obtain the values of the ISDa index, we constructed the hydrogen-suppressed graph of each molecule, then we use a program of our own elaboration to compute the index for each value of a analyzed.

    We calculated the Pearson's correlation coefficient r between the three analyzed properties and the ISDa index, for values of a in the interval [5,5] with a spacing of 0.01; the results are shown in Figure 1. The dashed red line indicates the value of a that maximizes r.

    Figure 1.  Pearson's correlation coefficient r between the ISDa index and the following properties of polyaromatic hydrocarbons (a) melting point (MP), (b) boiling point (BP), and (c) octanol-water partition coefficient (LogP). Red dashed vertical line indicates the value of a for which r is maximized.

    Figure 2 shows the ISDa index (for values of a that maximize r) vs. the studied properties of PAH. In addition, in Figure 2, we test the following linear regression models (red lines)

    MP=31.54ISD0.15121.15BP=58.94ISD0.3913.35LogP=0.68ISD0.63+1.55.
    Figure 2.  Properties of polyaromatic hydrocarbons vs. ISDa index for the values of a that maximize the correlation coefficient r: (a) a=0.15, (b) a=0.39, and (c) a=0.63. Red lines are the regression models obtained.

    Table 1 summarizes the statistical and regression parameters of these models.

    Table 1.  Parameters of the linear QSPR models. Here, r, c, m, SE, F, and SF are the correlation coefficient, intercept, slope, standard error, F-test, and statistical significance, respectively.
    Property a r c m SE F SF
    MP 0.15 0.856 122.15 31.54 54.61 214.47 4.31×1024
    BP 0.39 0.989 13.35 58.94 12.42 2272.86 5.69×1044
    LogP 0.63 0.943 1.55 0.68 0.34 282.04 2.53×1018

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    Motivated by a paper of Vukičević [1], and based on the practical applications found for the variable inverse sum deg index ISDa, we focus our research on the study of optimal graphs associated with ISDa, when a<0. In this direction, it is wise to study the extremal properties of ISDa, when a<0 in general graphs. Specifically, in this paper, we characterize the graphs with extremal values in the following significant classes of graphs with a fixed number of vertices:

    ● graphs with a fixed minimum degree;

    ● connected graphs with a fixed minimum degree;

    ● graphs with a fixed maximum degree;

    ● connected graphs with a fixed maximum degree.

    From the QSPR study performed on polyaromatic hydrocarbons, it can be concluded that the ISDa index presents a strong correlation with the boiling point and octanol-water partition coefficient properties, with maximum values of r higher than 0.98 and 0.94, respectively. Further, the melting point property presents some correlation with the ISDa index with maximum value of r close to 0.85.

    For future research, we suggests:

    ● To study the extreme problems for the ISDa index for values of a>0.

    ● To consider the problem of finding which tree/trees with n vertices (with a fixed maximum degree or not) minimize the index ISDa (a<0).

    ● To analyze the behavior of the ISDa index in other important families of graphs, such as graph products and graph operators.

    ● To explore the mathematical properties and possible applications of the exponential extension of the ISDa index.

    ● To study the predictive power of this index on other physicochemical properties of PAH, and on other classes of molecules.

    The authors were supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain.

    The authors declare there are no conflicts of interest.



    [1] D. Vukičević, Bond additive modeling 5. Mathematical properties of the variable sum exdeg index, Croat. Chem. Acta, 84 (2011), 93–101.
    [2] E. Estrada, Quantifying network heterogeneity, Phys. Rev. E, 82 (2010), 066102. https://doi.org/10.1103/PhysRevE.82.066102 doi: 10.1103/PhysRevE.82.066102
    [3] I. Gutman, B. Furtula, V. Katanić, Randić index and information, AKCE Int. J. Graphs Comb., 15 (2018), 307–312. https://doi.org/10.1016/j.akcej.2017.09.006 doi: 10.1016/j.akcej.2017.09.006
    [4] V. R. Kulli, F-Revan index and F-Revan polynomial of some families of benzenoid systems, J. Global Res. Math. Arch., 5 (2018), 1–6.
    [5] V. R. Kulli, Revan indices of oxide and honeycomb networks, Int. J. Math. Appl., 55 (2017), 7.
    [6] A. Miličević, S. Nikolić, On variable Zagreb indices, Croat. Chem. Acta, 77 (2004), 97–101.
    [7] E. D. Molina, J. M. Rodríguez, J. L. Sánchez, J. M. Sigarreta, Some properties of the arithmetic–geometric index, Symmetry, 13 (2021), 857. https://doi.org/10.3390/sym13050857 doi: 10.3390/sym13050857
    [8] J. Pineda, C. Martínez, J. A. Méndez, J. Muños, J. M. Sigarreta, Application of bipartite networks to the study of water quality, Sustainability, 12 (2020), 5143. https://doi.org/10.3390/su12125143 doi: 10.3390/su12125143
    [9] N. Zahra, M. Ibrahim, M. K. Siddiqui, On topological indices for swapped networks modeled by optical transpose interconnection system, IEEE Access, 8 (2020), 200091–200099. https://doi.org10.1109/ACCESS.2020.3034439 doi: 10.1109/ACCESS.2020.3034439
    [10] A. Ali, L. Zhong, I. Gutman, Harmonic index and its generalizations: extremal results and bounds, MATCH Commun. Math. Comput. Chem., 81 (2020), 249–311.
    [11] K. C. Das, On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem., 63 (2010), 433–440.
    [12] Z. Du, B. Zhou, N. Trinajstić, Minimum sum-connectivity indices of trees and unicyclic graphs of a given matching number, J. Math. Chem., 47 (2010), 842–855. https://doi.org/10.1007/s10910-009-9604-7 doi: 10.1007/s10910-009-9604-7
    [13] R. Cruz, J. Monsalve, J. Rada, On chemical trees that maximize atombond connectivity index, its exponential version, and minimize exponential geometric-arithmetic index, MATCH Commun. Math. Comput. Chem., 84 (2020), 691–718.
    [14] R. Cruz, J. Monsalve, J. Rada, Trees with maximum exponential Randic index, Discrete Appl. Math., 283 (2020), 634–643. https://doi.org/10.1016/j.dam.2020.03.009 doi: 10.1016/j.dam.2020.03.009
    [15] R. Cruz, J. Rada, Extremal values of exponential vertex-degree-based topological indices over graphs, Kragujevac J. Math., 46 (2022), 105–113.
    [16] K. C. Das, Y. Shang, Some extremal graphs with respect to sombor index, Mathematics, 9 (2021), 1202. https://doi.org/10.3390/math9111202 doi: 10.3390/math9111202
    [17] M. A. Iranmanesh, M. Saheli, On the harmonic index and harmonic polynomial of Caterpillars with diameter four, Iran. J. Math. Chem., 5 (2014), 35–43. https://doi.org/10.22052/IJMC.2015.9044 doi: 10.22052/IJMC.2015.9044
    [18] X. Li, I. Gutman, Mathematical aspects of Randić-type molecular structure descriptors, Croat. Chem. Acta, 79 (2006).
    [19] D. Vukičević, M. Gašperov, Bond additive modeling 1. Adriatic indices, Croat. Chem. Acta, 83 (2010), 243–260.
    [20] D. Vukičević, Bond additive modeling 2. Mathematical properties of max-min rodeg index, Croat. Chem. Acta, 83 (2010), 261–273.
    [21] W. Carballosa, J. A. Méndez-Bermúdez, J. M. Rodríguez, J. M. Sigarreta, Inequalities for the variable inverse sum deg index, Submitted.
    [22] H. Chen, H. Deng, The inverse sum indeg index of graphs with some given parameters, Discr. Math. Algor. Appl., 10 (2018), 1850006. https://doi.org/10.1142/S1793830918500064 doi: 10.1142/S1793830918500064
    [23] F. Falahati-Nezhad, M. Azari, T. Došlić, Sharp bounds on the inverse sum indeg index, Discrete Appl. Math., 217 (2017), 185–195. https://doi.org/10.1016/j.dam.2016.09.014 doi: 10.1016/j.dam.2016.09.014
    [24] I. Gutman, M. Matejić, E. Milovanović, I. Milovanović, Lower bounds for inverse sum indeg index of graphs, Kragujevac J. Math., 44 (2020), 551–562.
    [25] I. Gutman, J. M. Rodríguez, J. M. Sigarreta, Linear and non-linear inequalities on the inverse sum indeg index, Discrete Appl. Math., 258 (2019), 123–134. https://doi.org/10.1016/j.dam.2018.10.041 doi: 10.1016/j.dam.2018.10.041
    [26] M. An, L. Xiong, Some results on the inverse sum indeg index of a graph, Inf. Process. Lett., 134 (2018), 42–46. https://doi.org/10.1016/j.ipl.2018.02.006 doi: 10.1016/j.ipl.2018.02.006
    [27] J. Sedlar, D. Stevanović, A. Vasilyev, On the inverse sum indeg index, Discrete Appl. Math., 184 (2015), 202–212. https://doi.org/10.1016/j.dam.2014.11.013 doi: 10.1016/j.dam.2014.11.013
    [28] M. A. Rashid, S. Ahmad, M. K. Siddiqui, M. K. A. Kaabar, On computation and analysis of topological index-based invariants for complex coronoid systems, Complexity, 2021 (2021), 4646501. https://doi.org/10.1155/2021/4646501 doi: 10.1155/2021/4646501
    [29] M. K. Siddiqui, S. Manzoor, S. Ahmad, M. K. A. Kaabar, On computation and analysis of entropy measures for crystal structures, Math. Probl. Eng., 2021 (2021), 9936949. https://doi.org/10.1155/2021/9936949 doi: 10.1155/2021/9936949
    [30] D. A. Xavier, E. S. Varghese, A. Baby, D. Mathew, M. K. A. Kaabar, Distance based topological descriptors of zinc porphyrin dendrimer, J. Mol. Struct., 1268 (2022), 133614. https://doi.org/10.1016/j.molstruc.2022.133614 doi: 10.1016/j.molstruc.2022.133614
    [31] W. Carballosa, J. M. Rodríguez, J. M. Sigarreta, Extremal problems on the variable sum exdeg index, MATCH Commun. Math. Comput. Chem., 84 (2020), 753–772.
    [32] J. C. Hernández, J. M. Rodríguez, O. Rosario, J. M. Sigarreta, Extremal problems on the general Sombor index of graph, AIMS Math., 7 (2022), 8330–8334. https://doi.org/10.3934/math.2022464 doi: 10.3934/math.2022464
    [33] D. Vukičević, Bond additive modeling 4. QSPR and QSAR studies of the variable Adriatic indices, Croat. Chem. Acta, 84 (2011), 87–91.
    [34] R. Todeschini, P. Gramatica, E. Marengo, R. Provenzani, Weighted holistic invariant molecular descriptors. Part 2. Theory development and applications on modeling physicochemical properties of polyaromatic hydrocarbons, Chemom. Intell. Lab. Syst., 27 (1995), 221–229. https://doi.org/10.1016/0169-7439(95)80026-6 doi: 10.1016/0169-7439(95)80026-6
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