1 Introduction

A topological descriptor is a single number that represents a chemical structure in graph-theoretical terms via the molecular graph, they play a significant role in mathematical chemistry especially in the QSPR/QSAR investigations. Those topological descriptors which correlate with some molecular property are called topological indices. It is a well known fact that the main application of topological indices focuses on the understanding of physicochemical properties of chemical compounds. Hundreds of topological indices have been introduced and its mathematical properties and chemical applications have been intensively studied, starting with the seminal work by H. Wiener [1], and more recently, we can mention the work [2] which includes some chemical applications in a similar way to the present work.

Although only about 1000 benzenoid hydrocarbons are known, the number of possible benzenoid hydrocarbons is huge. For instance, the number of possible benzenoid hydrocarbons with 35 benzene rings is 5, 851, 000, 265, 625, 801, 806, 530 (cf., [3]). Therefore, modeling their physicochemical properties is crucial for predicting properties of currently unknown species. The main reason for using topological indices is to predict properties of molecular graphs. Therefore, given certain fixed parameters, a natural problem is to find, when possible, upper and lower bounds for such topological indices (see, e.g., [2] and the references therein).

Topological indices based on end-vertex degrees of edges have been used over 40 years. Probably, among such descriptors, the best known is the Randić connectivity index (R) [4]. There are more than one thousand papers and a couple of books dealing with this molecular descriptor (see, e.g., [5,6,7,8,9] and the references therein). For many years, scientists have been trying to improve the predictive power of the Randić index. These efforts led to the introduction of a large number of new topological descriptors resembling the original Randić index. Two of the main successors of the latter are the first and second Zagreb indices, denoted by \(M_1\) and \(M_2\), respectively, and defined as

$$\begin{aligned} M_1(G) = \sum _{uv\in E(G)} (d_u + d_v) = \sum _{u\in V(G)} d_u^2, \qquad M_2(G) = \sum _{uv\in E(G)} d_u d_v, \qquad \end{aligned}$$

where uv denotes the edge of the graph G connecting the vertices u and v, and \(d_u\) is the degree of the vertex u. These indices have attracted increasing interest, see e.g., [10,11,12,13]. In particular, they are included in a number of programs used for the routine computation of topological indices.

The inverse degree index ID(G) of a graph G is defined by

$$\begin{aligned} ID(G) = \sum _{u\in V(G)} \frac{1}{d_u} = \sum _{uv\in E(G)} \left( \frac{1}{d_u^2} + \frac{1}{d_v^2}\right) = \sum _{uv\in E(G)}\frac{d_u^2 + d_v^2}{d_u^2 d_v^2}. \end{aligned}$$

The inverse degree index first attracted attention through numerous conjectures generated by the computer programme Graffiti [14]. Since then, its relationship with other graph invariants, such as diameter, edge-connectivity, matching number, and Wiener index have been studied by several authors (see, e.g., [15,16,17,18,19]).

Miličević and Nikolić defined in [20] the first and second variable Zagreb indices as

$$\begin{aligned} {}^{\alpha }M_1(G) = \sum _{u\in V(G)} d_u^{2\alpha }, \qquad {}^{\alpha }M_2(G) = \sum _{uv\in E(G)} (d_u d_v)^\alpha , \end{aligned}$$

with \(\alpha \in \mathbb {R}\). In [21] and [22] the first and second general Zagreb indices are introduced as

$$\begin{aligned} M_1^{\alpha }(G) = \sum _{u\in V(G)} d_u^{\alpha }, \qquad M_2^{\alpha }(G) = \sum _{uv\in E(G)} (d_u d_v)^\alpha , \end{aligned}$$

respectively. It is clear that these indices are equivalent to the previous ones, since \(^{\alpha }M_1(G)=M_1^{2\alpha }(G)\) and \(^{\alpha }M_2(G)=M_2^{\alpha }(G)\). Furthermore, the first general Zagreb, \(M_1^{\alpha }(G)\) also has the following representation

$$\begin{aligned} M_1^{\alpha }(G)=\sum _{uv\in E(G)} \left( d_u^{\alpha -1}+d_v^{\alpha -1}\right) . \end{aligned}$$
(1)

In what follows, \(M_j^{\alpha }(G)\) will be used instead of \(^{\alpha }M_j(G)\), for \(j=1,2,\) since the inequalities obtained in this paper become simpler with them.

Note that \(M_1^0=n\), \(M_1^{1}=2m\), \(M_1^{2}\) is the first Zagreb index \(M_1\), \(M_1^{-1}\) is the inverse index ID, \(M_1^{3}\) is the forgotten index F, etc.; also, \(M_2^{0}=m\), \(M_2^{-1/2}\) is the usual Randić index R, \(M_2^{1}\) is the second Zagreb index \(M_2\), \(M_2^{-1}\) is the modified Zagreb index, etc.

The concept of the variable molecular descriptors was proposed as a new way of characterizing heteroatoms in molecules (see [23, 24]), but also to assess structural differences (e.g., the relative role of carbon atoms of acyclic and cyclic parts in alkylcycloalkanes [25]). The idea behind the variable molecular descriptors is that the variables are determined during the regression so that the standard error of an estimate for a studied property to be as small as possible.

In the paper of I. Gutman and J. Tosovic [26], the correlation abilities of 20 vertex-degree-based topological indices occurring in the chemical literature were tested for the case of standard heats of formation and normal boiling points of octane isomers. It is remarkable that the second general Zagreb index \(M_2^\alpha \) with exponent \(\alpha = -1\) (and to a lesser extent with exponent \(\alpha = -2\)) performs significantly better than the Randić index (\(R=M_2^{-1/2}\)).

The second variable Zagreb index is used in the structure-boiling point modeling of benzenoid hydrocarbons [27]. Various properties and relations of these indices are discussed in several papers (see, e.g., [28,29,30,31,32,33]). The interested reader can find recent and interesting results involving several topological indices and their applications in [34,35,36].

The aim of this work is to provide new inequalities relating several classes of variable topological indices including the first and second general Zagreb indices, the general sum-connectivity index and the variable inverse sum deg index. Also, upper and lower bounds on the inverse degree in terms of the first general Zagreb are shown. Moreover, the characterization of extremal graphs with respect to many of such inequalities is obtained. Finally, some applications are given to the study of the physico-chemical properties of the octane isomers, in particular to the study of Entropy, Motor octane number, Standard enthalpy of vaporization and Acentric factor.

Throughout this paper, \(G=(V (G),E (G))\) denotes a (non-oriented) finite simple (without multiple edges and loops) non-trivial (each vertex belongs to some edge) graph. Also, m and n will denote, respectively, the cardinality of the sets E(G) and V(G).

2 Main inequalities

The sum-connectivity index was proposed in [37]. It has been shown that this index correlates well with the \(\pi \)-electronic energy of benzenoid hydrocarbons [38]. More applications of the sum-connectivity index can be found in [39]. Recently, this concept was extended to the general sum-connectivity index in [40], which is defined by

$$\begin{aligned} \chi _{a}(G) = \sum _{uv \in E(G)} (d_u + d_v)^a. \end{aligned}$$

Note that \(\chi _{-1/2}\) is the sum-connectivity index, \(\chi _{1}\) is the first Zagreb index and \(\chi _{-1}\) is half the harmonic index.

Let us start with the following elementary fact (see, for instance [41]).

Lemma 1

If \(f \in C^1[a,b]\) and \(f'=g_1g_2\) with \(g_1,g_2\in C[a,b]\), \(g_1\) positive and \(g_2\) non-increasing (resp. non-decreasing) on [a, b], then f attains its minimum (resp. maximum) value on [a, b] on the set \(\{a,b\}\).

The following result relates the general sum-connectivity and the first general Zagreb indices.

Theorem 2

Let G be a graph with maximum degree \(\Delta \) and minimum degree \(\delta \), and \(a,b\in \mathbb {R}\).

If \(b\ge a\) and \(b\ge 1\), then

$$\begin{aligned} 2^{1-a}\delta ^{b-a}\, \chi _{a}(G) \le M_{1}^{b+1}(G)\le \max \left\{ (\Delta +\delta )^{-a}\left( \Delta ^{b}+\delta ^{b}\right) ,\, 2^{1-a} \Delta ^{b-a} \right\} \, \chi _a(G). \end{aligned}$$

If \(b\le a\) and \(b\le 0\), then

$$\begin{aligned} 2^{1-a}\Delta ^{b-a}\, \chi _{a}(G)\le M_{1}^{b+1}(G)\le \max \left\{ 2^{1-a} \delta ^{b-a},\, (\Delta +\delta )^{-a}\left( \Delta ^{b}+\delta ^{b}\right) \right\} \, \chi _a(G). \end{aligned}$$

Proof

For each \(\delta \le x,y \le \Delta \), define the function

$$\begin{aligned} \Gamma (x,y) = \frac{x^{b}+y^{b}}{(x+y)^{a}} = (x+y)^{-a}\left( x^{b}+y^{b}\right) . \end{aligned}$$

A computation gives

$$\begin{aligned} \frac{\partial \Gamma }{\partial x}(x,y)= & {} (x+y)^{-a-1}\left( -ax^{b}-ay^{b} +b (x+y) x^{b-1}\right) \nonumber \\= & {} (x+y)^{-a-1}\left( -ax^{b}+b x^{b}-ay^{b}+byx^{b-1}\right) . \end{aligned}$$
(2)

Assume that \(b\ge a\) and \(b\ge 1\). By symmetry, we also can assume that \(x \ge y\), then

$$\begin{aligned} \begin{aligned} \frac{\partial \Gamma }{\partial x}(x,y)&= (x+y)^{-a-1}\left( -ax^{b}+b x^{b}-ay^{b}+byx^{b-1}\right) \\&\ge a(x+y)^{-a-1}\left( -x^{b} + x^{b}-y^{b}+yx^{b-1}\right) \\&=a(x+y)^{-a-1}y\left( x^{b-1}-y^{b-1}\right) \ge 0. \end{aligned} \end{aligned}$$

Hence, \(\Gamma (y,y) \le \Gamma (x,y) \le \Gamma (\Delta ,y)\).

Set

$$\begin{aligned} \Theta (y) = \Gamma (y,y) = (y+y)^{-a}\big (y^{b}+y^{b}\big ) = 2^{1-a}y^{b-a}. \end{aligned}$$

Since \(b\ge a\), \(\Theta \) is a non-decreasing function and

$$\begin{aligned} \Gamma (x,y) \ge \Gamma (y,y) =\Theta (y) \ge \Theta (\delta ) = 2^{1-a}\delta ^{b-a}, \end{aligned}$$

we get

$$\begin{aligned} d_u^{b}+d_v^{b}\ge 2^{1-a}\delta ^{b-a}(d_u+d_v)^{a}, \end{aligned}$$

for every \(uv \in E(G)\). Hence, using the representation (1) for \(M_{1}^{b+1}(G)\), we obtain

$$\begin{aligned} M_{1}^{b+1}(G) \ge 2^{1-a}\delta ^{b-a}\, \chi _{a}(G). \end{aligned}$$

Now, let

$$\begin{aligned} \Lambda (y) = \Gamma (\Delta ,y) = (\Delta +y)^{-a}\left( \Delta ^{b}+y^{b}\right) \end{aligned}$$

on \([\delta ,\Delta ]\).

We have

$$\begin{aligned} \begin{aligned} \Lambda '(y)&= \left( \Delta +y\right) ^{-a-1}\left( -a\Delta ^b +(b-a)y^b+\Delta by^{b-1}\right) . \end{aligned} \end{aligned}$$

Let us consider the function

$$\begin{aligned} \begin{aligned} \Psi (y)&= -a\Delta ^b +(b-a)y^b+\Delta by^{b-1} \end{aligned} \end{aligned}$$

on \([\delta ,\Delta ]\). Since \(b\ge a\) and \(b\ge 1\), we have

$$\begin{aligned} \begin{aligned} \Psi '(y)&= b(b-a)y^{b-1}+\Delta b (b-1)y^{b-2}\ge 0. \end{aligned} \end{aligned}$$

Consequently, \(\Psi \) is a non-decreasing function. Since \(\Lambda '(y) = \left( \Delta +y\right) ^{-a-1} \Psi (y)\), Lemma 1 gives

$$\begin{aligned} \begin{aligned} \Gamma (x,y)&\le \Lambda (y) \le \max \left\{ \Lambda (\delta ),\, \Lambda (\Delta ) \right\} = \max \left\{ \Gamma (\Delta ,\delta ),\, \Gamma (\Delta ,\Delta ) \right\} \\&= \max \left\{ (\Delta +\delta )^{-a}\left( \Delta ^{b}+\delta ^{b}\right) ,\, 2^{1-a} \Delta ^{b-a} \right\} \end{aligned} \end{aligned}$$

for every \(y \in [\delta ,\Delta ]\).

Therefore,

$$\begin{aligned} \begin{aligned} \frac{x^{b}+y^{b}}{(x+y)^a}&=\Gamma (x, y) \le \max \left\{ (\Delta +\delta )^{-a}\left( \Delta ^{b}+\delta ^{b}\right) ,\, 2^{1-a} \Delta ^{b-a} \right\} , \end{aligned} \end{aligned}$$

and this last inequality implies that

$$\begin{aligned} d_u^{b}+d_v^{b}\le \max \left\{ (\Delta +\delta )^{-a}\left( \Delta ^{b}+\delta ^{b}\right) ,\, 2^{1-a} \Delta ^{b-a} \right\} (d_u+d_v)^{a}, \end{aligned}$$

for every \(uv \in E(G)\). Hence, it follows from (1) that

$$\begin{aligned} M_{1}^{b+1}(G) \le \max \left\{ (\Delta +\delta )^{-a}\left( \Delta ^{b}+\delta ^{b}\right) ,\, 2^{1-a} \Delta ^{b-a} \right\} \, \chi _a(G). \end{aligned}$$

Now, assume that \(b\le a\) and \(b\le 0\). By symmetry, we can assume also that \(x\le y\), then

$$\begin{aligned} \begin{aligned} \frac{\partial \Gamma }{\partial x}(x,y)&= (x+y)^{-a-1}\left( -ax^{b}+b x^{b}-ay^{b}+byx^{b-1}\right) \\&\le a(x+y)^{-a-1}\left( -x^{b} + x^{b}-y^{b}+yx^{b-1}\right) \\&=a(x+y)^{-a-1}y\left( x^{b-1}-y^{b-1}\right) \le 0. \end{aligned} \end{aligned}$$

Hence, \(\Gamma (y,y) \le \Gamma (x,y) \le \Gamma (\delta ,y)\).

Consider the function

$$\begin{aligned} \Theta (y) = \Gamma (y,y) = (y+y)^{-a}\big (y^{b}+y^{b}\big ) = 2^{1-a}y^{b-a}. \end{aligned}$$

Since \(b\le a\), \(\Theta \) is a non-increasing function and

$$\begin{aligned} \Gamma (x,y) \ge \Gamma (y,y) =\Theta (y) \ge \Theta (\Delta ) = 2^{1-a}\Delta ^{b-a}, \end{aligned}$$

we get

$$\begin{aligned} d_u^{b}+d_v^{b}\ge 2^{1-a}\Delta ^{b-a}(d_u+d_v)^{a}, \end{aligned}$$

for every \(uv \in E(G)\). Hence, using the representation (1) for \(M_{1}^{b+1}(G)\), we obtain

$$\begin{aligned} M_{1}^{b+1}(G) \ge 2^{1-a}\Delta ^{b-a}\, \chi _{a}(G). \end{aligned}$$

Now, consider the function

$$\begin{aligned} \Lambda _{1}(y) = \Gamma (\delta ,y) = (\delta +y)^{-a}\left( \delta ^{b}+y^{b}\right) \end{aligned}$$

on \([\delta ,\Delta ]\).

We have

$$\begin{aligned} \begin{aligned} \Lambda '_{1}(y)&= \left( \delta +y\right) ^{-a-1}\left( -a\delta ^b +(b-a)y^b+\delta by^{b-1}\right) . \end{aligned} \end{aligned}$$

Let us consider the function

$$\begin{aligned} \begin{aligned} \Psi _{1}(y)&= -a\delta ^b +(b-a)y^b+\delta by^{b-1} \end{aligned} \end{aligned}$$

on \([\delta ,\Delta ]\). Since \(b\le a\) and \(b\le 0\), we have

$$\begin{aligned} \begin{aligned} \Psi '_{1}(y)&= b(b-a)y^{b-1}+\delta b (b-1)y^{b-2}\ge 0. \end{aligned} \end{aligned}$$

Consequently, \(\Psi _{1}\) is a non-decreasing function. Since \(\Lambda '_{1}(y) = \left( \delta +y\right) ^{-a-1} \Psi _{1}(y)\), Lemma 1 gives

$$\begin{aligned} \begin{aligned} \Gamma (x,y)&\le \Lambda _{1}(y) \le \max \left\{ \Lambda _{1}(\delta ),\, \Lambda _{1}(\Delta ) \right\} = \max \left\{ \Gamma (\delta ,\delta ),\, \Gamma (\delta ,\Delta ) \right\} \\&= \max \left\{ 2^{1-a} \delta ^{b-a},\, (\Delta +\delta )^{-a}\left( \Delta ^{b}+\delta ^{b}\right) \right\} \end{aligned} \end{aligned}$$

for every \(y \in [\delta ,\Delta ]\).

Consequently,

$$\begin{aligned} \begin{aligned} \frac{x^{b}+y^{b}}{(x+y)^a}&=\Gamma (x, y) \le \max \left\{ 2^{1-a} \delta ^{b-a},\, (\Delta +\delta )^{-a}\left( \Delta ^{b}+\delta ^{b}\right) \right\} , \end{aligned} \end{aligned}$$

and this last inequality implies that

$$\begin{aligned} d_u^{b}+d_v^{b}\le \max \left\{ 2^{1-a} \delta ^{b-a},\, (\Delta +\delta )^{-a}\left( \Delta ^{b}+\delta ^{b}\right) \right\} (d_u+d_v)^{a}, \end{aligned}$$

for every \(uv \in E(G)\). Hence, it follows from (1) that

$$\begin{aligned} M_{1}^{b+1}(G) \le \max \left\{ 2^{1-a} \delta ^{b-a},\, (\Delta +\delta )^{-a}\left( \Delta ^{b}+\delta ^{b}\right) \right\} \, \chi _a(G). \end{aligned}$$

This completes the proof of the theorem. \(\square \)

The following proposition relates the first Zagreb and the inverse degree indices.

Proposition 3

If G is a graph with m edges, then

$$\begin{aligned} 2\, ID(G) +M_2(G)\ge 4m. \end{aligned}$$

Proof

Since \(x+1/x \ge 2\) for every \(x>0\) and \(2xy \le x^2+y^2\) for every \(x,y \in \mathbb {R}\), we have

$$\begin{aligned} \begin{aligned} \frac{d_u^2 + d_v^2}{d_u^2d_v^2}+\frac{d_u^2d_v^2}{d_u^2 + d_v^2}&\ge 2, \\ \sum _{uv\in E(G)}\frac{d_u^2 + d_v^2}{d_u^2d_v^2}+\sum _{uv\in E(G)}\frac{d_u^2d_v^2}{d_u^2 + d_v^2}&\ge \sum _{uv\in E(G)}2, \\ ID(G)+\sum _{uv\in E(G)}\frac{d_u^2d_v^2}{2d_ud_v}&\ge 2m, \\ 2ID(G)+ M_2(G)&\ge 4m. \end{aligned} \end{aligned}$$

\(\square \)

We need the following converse Hölder inequality in [42, Theorem 3], which is interesting on its own. This result improves the inequality in [43, Theorem 2].

Theorem 4

Let \((X,\mu )\) be a measure space, \(f,g: X \rightarrow \mathbb {R}\) measurable functions, and \(1<p,q<\infty \) with \(1/p+1/q=1\). If there exist positive constants a, b with \(a |g|^q \le |f|^p \le b |g|^q\) \(\mu \)-a.e., then:

$$\begin{aligned} \Vert f\Vert _p \Vert g\Vert _q \le K_p(a,b ) \Vert fg\Vert _1, \end{aligned}$$
(3)

with:

$$\begin{aligned} K_p(a,b ) = {\left\{ \begin{array}{ll} \,\displaystyle \frac{1}{p} \Big ( \frac{a }{b } \Big )^{1/(2q)} + \frac{1}{q} \Big ( \frac{b }{a } \Big )^{1/(2p)}, &{} \quad \text {if } 1<p<2, \\ \, &{} \, \\ \,\displaystyle \frac{1}{p} \Big ( \frac{b }{a } \Big )^{1/(2q)} + \frac{1}{q} \Big ( \frac{a }{b } \Big )^{1/(2p)}, &{} \quad \text {if } p \ge 2. \end{array}\right. } \end{aligned}$$

If these norms are finite, the equality in the bound is attained if and only if \(a = b\) and \(|f|^p = a |g|^q\) \(\mu \)-a.e. or \(f = g = 0\) \(\mu \)-a.e.

Theorem 4 has the following consequence.

Corollary 5

If \(1<p,q<\infty \) with \(1/p+1/q=1\), \(x_j,y_j\ge 0\) and \(a y_j^q \le x_j^p \le b y_j^q\) for \(1\le j \le k\) and some positive constants a, b,  then:

$$\begin{aligned} \left( \sum _{j=1}^k x_j^p \right) ^{1/p} \left( \sum _{j=1}^k y_j^q \right) ^{1/q} \le K_p(a,b ) \sum _{j=1}^k x_jy_j, \end{aligned}$$

where \(K_p(a,b)\) is the constant in Theorem 4. If \(x_j>0\) for some \(1\le j \le k\), then the equality in the bound is attained if and only if \(a =b\) and \(x_j^p=a y_j^q\) for every \(1\le j \le k\).

The next result relates several first general Zagreb indices. It generalizes [43, Theorem 2.12].

Theorem 6

Let G be a nontrivial graph with n vertices, maximum degree \(\Delta \) and minimum degree \(\delta \), and \(\alpha ,p,q \in \mathbb {R}\) with \(1/p+1/q=1\). Then

$$\begin{aligned} C_p(\delta ,\Delta ) \, n^{1/q} \left( M_1^{\alpha p}(G)\right) ^{1/p} \le M_1^{\alpha }(G)\le n^{1/q} \left( M_1^{\alpha p}(G)\right) ^{1/p}. \end{aligned}$$

with:

$$\begin{aligned} C_p(\delta ,\Delta ) = {\left\{ \begin{array}{ll} \,\displaystyle \frac{\left( \delta \Delta ^{p/q}\right) ^{\alpha /2}}{\frac{1}{p}\left( \delta ^{\alpha }\right) ^{p/2} + \frac{1}{q}\left( \Delta ^{\alpha }\right) ^{p/2}}, &{} \quad \text {if } 1<p<2, \\ \, &{} \, \\ \,\displaystyle \frac{\left( \Delta \delta ^{p/q}\right) ^{\alpha /2}}{\frac{1}{p}\left( \Delta ^{\alpha }\right) ^{p/2} + \frac{1}{q}\left( \delta ^{\alpha }\right) ^{p/2}}, &{} \quad \text {if } p \ge 2. \end{array}\right. } \end{aligned}$$

The lower bound is attained for every value of \(\alpha \) if G is regular. The upper bound is attained for some \(\alpha \ne 0\) if and only if G is regular.

Proof

Applying Hölder inequality

$$\begin{aligned} M_1^\alpha (G) = \sum _{u\in V(G)} d_u^{\alpha } \le \left( \sum _{u\in V(G)} d_u^{\alpha p} \right) ^{1/p} \left( \sum _{u\in V(G)} 1 \right) ^{1/q} = n^{1/q} \left( M_1^{\alpha p}(G)\right) ^{1/p}. \end{aligned}$$

Note that

$$\begin{aligned} \begin{aligned} \delta ^{\alpha } \le d_u^{\alpha } \le \Delta ^{\alpha }&\qquad \text { if }\, \alpha \ge 0, \\ \Delta ^{\alpha } \le d_u^{\alpha } \le \delta ^{\alpha }&\qquad \text { if }\, \alpha \le 0, \end{aligned} \end{aligned}$$

In order to prove the other inequality we are going to use Corollary 5 with \(a=\delta ^{\alpha p}\) and \(b=\Delta ^{\alpha p}\).

$$\begin{aligned} \begin{aligned} M_1^\alpha (G)&= \sum _{u\in V(G)} d_u^{\alpha } \ge \frac{\Big (\sum _{u\in V(G)} d_u^{\alpha p} \Big )^{1/p} \Big (\sum _{u\in V(G)} 1 \Big )^{1/q}}{K_p(\delta ^{\alpha p}, \Delta ^{\alpha p})} \\&= C_p(\delta ,\Delta ) \, n^{1/q} \left( M_1^{\alpha p}(G)\right) ^{1/p} . \end{aligned} \end{aligned}$$

For \(\alpha \ne 0\), by Hölder inequality the upper bound is sharp if and only if the graph is regular. In this case \(M_1^{\alpha }(G)=n\delta ^{\alpha }=n\Delta ^{\alpha }\) and both bounds coincide. \(\square \)

The following result relates the inverse degree and the first general Zagreb indices.

Theorem 7

If \(\alpha \in \mathbb {R}\) and G is a non-trivial graph with n vertices, m edges, minimum degree \(\delta \) and maximum degree \(\Delta \), then the following inequalities hold:

(1) if \(\alpha < -1\), then

$$\begin{aligned} ID(G)^{-\alpha }n^{-(\alpha +1)} \le M_1^\alpha (G)\le K_{-\alpha }^{-\alpha }\left( \Delta ^{\alpha }, \delta ^{\alpha }\right) ID(G)^{-\alpha }n^{-(\alpha +1)}; \end{aligned}$$

(2) if \(-1<\alpha < 0\), then

$$\begin{aligned} K_{-\frac{1}{\alpha }}^{-1}\left( \Delta ^{-1}, \delta ^{-1}\right) ID(G)^{-\alpha } n^{1+\alpha } \le M_1^\alpha (G)\le ID(G)^{-\alpha } n^{1+\alpha }; \end{aligned}$$

(3) if \(0<\alpha <1\), then

$$\begin{aligned} K_{\frac{1}{\alpha }}^{-1}(\delta , \Delta )(2m)^{\alpha } n^{1-\alpha } \le M_1^\alpha (G)\le (2m)^{\alpha } n^{1-\alpha }; \end{aligned}$$

(4) if \(\alpha >1\), then

$$\begin{aligned} (2m)^{\alpha } n^{1-\alpha } \le M_1^\alpha (G)\le K_{\alpha }^{\alpha }(\delta ^{\alpha }, \Delta ^{\alpha })(2m)^{\alpha } n^{1-\alpha }, \end{aligned}$$

where \(K_p(a,b)\) is the constant in Theorem 4. Moreover, the equalities are attained if and only if G is regular. Also, for the three special cases, one gets

$$\begin{aligned} M_1^{-1}(G)=ID(G),\qquad M_1^{0}(G)=n,\qquad M_1^{1}(G)=2m. \end{aligned}$$

Proof

For any graph G, the last three special cases are obtained straightforwardly by applying the definition of the first general Zagreb index. So, if \(\alpha =-1\) one gets \(M_1^{-1}(G)=ID(G)\); if \(\alpha =0\), then \(M_1^{0}(G)=\sum _{u\in V(G)} d_u^0=n\); and finally, if \(\alpha =1\), then \(M_1^{1}(G)=\sum _{uv\in E(G)} \left( d_u^0+d_v^0\right) =2\,m\).

Now, for the case \(\alpha > 1\), take \(p=\alpha \) and \(q=\alpha /(\alpha -1)\). Then, Hölder’s inequality gives

$$\begin{aligned} \begin{aligned} 2m = M_1^1(G)&= \sum _{u\in V(G)} d_u \le \left( \sum _{u\in V(G)} d_u^\alpha \right) ^{\frac{1}{\alpha }} \left( \sum _{u\in V(G)} 1^{\frac{\alpha }{\alpha -1}}\right) ^{\frac{\alpha -1}{\alpha }} \\&= M_1^{\alpha }(G)^{\frac{1}{\alpha }} n^{\frac{\alpha -1}{\alpha }}. \end{aligned} \end{aligned}$$
(4)

Next, the lower bound can be obtained applying Corollary 5 with \(a=\delta ^{\alpha }\) and \(b=\Delta ^{\alpha }\)

$$\begin{aligned} \begin{aligned} 2m = M_1^1(G)&= \sum _{u\in V(G)} d_u \ge K_\alpha ^{-1}(\delta ^\alpha , \Delta ^\alpha )\left( \sum _{u\in V(G)} d_u^\alpha \right) ^{\frac{1}{\alpha }} \left( \sum _{u\in V(G)} 1^{\frac{\alpha }{\alpha -1}}\right) ^{\frac{\alpha -1}{\alpha }} \\&=K_\alpha ^{-1}(\delta ^\alpha , \Delta ^\alpha )M_1^{\alpha }(G)^{\frac{1}{\alpha }} n^{\frac{\alpha -1}{\alpha }}. \end{aligned} \end{aligned}$$
(5)

Thus, Eqs. (4) and (5) give

$$\begin{aligned} (2m)^{\alpha } n^{1-\alpha }\le M_1^\alpha (G)\le K_{\alpha }^{\alpha }(\delta ^{\alpha }, \Delta ^{\alpha })(2m)^{\alpha } n^{1-\alpha }. \end{aligned}$$

The proofs of the remaining cases are similar choosing the appropriate values for the constants. Namely,

  • if \(\alpha <-1\), take \(p=-\alpha \), \(q=\alpha /(1+\alpha )\), \(a=\Delta ^{\alpha }\) and \(b=\delta ^{\alpha }\);

  • if \(-1<\alpha <0\), take \(p=-1/\alpha \), \(q=1/(1+\alpha )\), \(a=\Delta ^{-1}\) and \(b=\delta ^{-1}\);

  • if \(0<\alpha <1\), take \(p=1/\alpha \), \(q=1/(1-\alpha )\), \(a=\delta \) and \(b=\Delta \).

Note that for \(\alpha \ne -1,0,1,\) the tools used (Hölder inequality and Corollary 5) give that all inequalities are sharp if and only if the graph G is regular. \(\square \)

The \(\sigma \)-index is defined in [44] as

$$\begin{aligned} \sigma (G) = \sum _{uv\in E(G)} (d_u-d_v)^{2}. \end{aligned}$$

Note that \(\sigma (G) = F(G) - 2 M_2(G)\).

Theorem 8

Let G be a nontrivial graph with n vertices, maximum degree \(\Delta \) and minimum degree \(\delta \). Then

$$\begin{aligned} 2 M_2^{-1}(G) + \frac{\sigma (G)}{\Delta ^4} \le ID(G) \le 2 M_2^{-1}(G) + \frac{\sigma (G)}{\delta ^4} \end{aligned}$$

and the lower (respectively, upper) bound is attained if and only if G is regular.

Proof

Note that

$$\begin{aligned} \begin{aligned} d_u^{2} + d_v^2&= (d_u - d_v)^2 + 2 d_u d_v, \\ \frac{1}{d_u^{2}} + \frac{1}{d_v^{2}}&= \frac{2}{d_u d_v} + \frac{(d_u - d_v)^2}{d_u^2 d_v^2} . \end{aligned} \end{aligned}$$

Since \(\delta ^{4} \le d_u^2 d_v^2 \le \Delta ^{4}\), we deduce

$$\begin{aligned} \begin{aligned} \frac{1}{d_u^{2}} + \frac{1}{d_v^{2}}&\le \frac{2}{d_u d_v} + \frac{(d_u - d_v)^2}{\delta ^4} , \\ \frac{1}{d_u^{2}} + \frac{1}{d_v^{2}}&\ge \frac{2}{d_u d_v} + \frac{(d_u - d_v)^2}{\Delta ^4} , \end{aligned} \end{aligned}$$

and the desired inequalities hold.

If the graph is regular, then the lower and upper bounds are the same, and both are equal to ID(G). If the lower (respectively, upper) bound is attained, then \(d_u = d_v = \Delta \) (respectively, \(d_u = d_v = \delta \)) for every \(uv\in E(G)\) and so, G is regular. \(\square \)

The following result relates the inverse degree index with the first Zagreb and the second general Zagreb indices.

Theorem 9

Let G be a nontrivial graph with n vertices, maximum degree \(\Delta \) and minimum degree \(\delta \). Then

$$\begin{aligned} ID(G) \le M_2^{-1}(G) - \delta ^2 M_2^{-2}(G) + \frac{\delta }{\Delta ^4} \, M_1(G) \end{aligned}$$

and the bound is attained if and only if G is regular.

Proof

Since \((d_u - d_v)^2 + (d_u - \delta )^2 + (d_v - \delta )^2 \ge 0\), we have

$$\begin{aligned} \begin{aligned} d_u^{2} + d_v^2 + \delta ^2&\ge \delta (d_u + d_v) + d_u d_v, \\ \frac{1}{d_u^{2}} + \frac{1}{d_v^{2}} + \frac{\delta ^2}{d_u^2 d_v^2}&\ge \frac{1}{d_u d_v} + \delta \,\frac{d_u + d_v}{d_u^2 d_v^2} . \end{aligned} \end{aligned}$$

Since \(d_u^2 d_v^2 \le \Delta ^{4}\), we deduce

$$\begin{aligned} \begin{aligned} \frac{1}{d_u^{2}} + \frac{1}{d_v^{2}} + \frac{\delta ^2}{d_u^2 d_v^2}&\ge \frac{1}{d_u d_v} + \delta \,\frac{d_u + d_v}{\Delta ^4} , \end{aligned} \end{aligned}$$

and the inequality holds.

The bound is attained if and only if \(d_u = d_v = \delta \) and \(d_u = d_v = \Delta \) for every \(uv\in E(G)\), i.e., if G is regular. \(\square \)

In [45, 46] several degree-based topological indices called adriatic indices are introduced. The inverse sum indeg index ISI, defined by

$$\begin{aligned} ISI(G) = \sum _{uv\in E(G)} \frac{d_u\,d_v}{d_u + d_v}, \end{aligned}$$

is one of them. This index is one of the most predictive adriatic indices, associated with the total surface area of the isomers of octanes.

Also, this index has become one of the most studied from the mathematical point of view. We present here several inequalities relating the first variable Zagreb index with the variable inverse sum deg index defined, for each \(a \in \mathbb {R}\), as

$$\begin{aligned} ISD_a(G) = \sum _{uv \in E(G)} \frac{1}{d_u^a + d_v^a} . \end{aligned}$$

Note that \(ISD_{-1}\) is the inverse sum indeg index ISI.

The variable inverse sum deg index \(ISD_{-1.950}\) is a good predictor of standard enthalpy of formation [47].

Theorem 10

If G is a graph with m edges, and \(a \in \mathbb {R}\), then

$$\begin{aligned} ISD_a(G) + M_1^{a+1}(G) \ge \frac{5}{2} \,m, \qquad&\text {if } \, a > 0, \end{aligned}$$
(6)
$$\begin{aligned} ISD_a(G) + M_1^{a+1}(G) \ge 2m, \qquad&\text {if } \, a < 0. \end{aligned}$$
(7)

The equality in the first bound is attained if and only if G is a union of path graphs \(P_2\).

Proof

Recall that, for any function h,

$$\begin{aligned} \sum _{uv \in E(G)} \big ( h(d_u)+ h(d_v) \big ) = \sum _{u \in V(G)} d_u h(d_u). \end{aligned}$$

In particular,

$$\begin{aligned} \sum _{uv \in E(G)} \left( d_u^{a}+d_v^{a} \right) = \sum _{u \in V(G)} d_u^{a+1} = M_1^{a+1}(G). \end{aligned}$$

The function \(f(x)=x+1/x\) is strictly decreasing on (0, 1] and strictly increasing on \([1,\infty )\), and so, \(f(x) \ge f(1) = 2\) for every \(x>0\). Hence,

$$\begin{aligned} \begin{aligned} \frac{1}{d_u^{a}+d_v^{a}} + d_u^{a}+d_v^{a}&\ge 2, \\ ISD_a(G) + M_1^{a+1}(G)&\ge 2m. \end{aligned} \end{aligned}$$

If \(a>0\), then \(d_u^{a}+d_v^{a} \ge 2\) and

$$\begin{aligned} \begin{aligned} \frac{1}{d_u^{a}+d_v^{a}} + d_u^{a}+d_v^{a}&\ge f(2) = \frac{5}{2} , \\ ISD_a(G) + M_1^{a+1}(G)&\ge \frac{5}{2} \,m. \end{aligned} \end{aligned}$$

The previous argument gives that the equality is attained if and only if \(d_u=d_v=1\) for every \(uv \in E(G)\), i.e., G is a union of path graphs \(P_2\). \(\square \)

Proposition 11

Let G be a graph with minimum degree \(\delta >1\) and m edges. If \(a \le -\log 2/\log \delta \), then

$$\begin{aligned} \begin{aligned} ISD_a(G) + M_1^{a+1}(G) \ge \Big ( 2\delta ^a + \frac{1}{2\delta ^a} \Big ) m. \end{aligned} \end{aligned}$$

and the equality is attained if and only if G is regular.

Proof

Since \(\delta >1\) and \(a \le -\log 2 / \log \delta < 0\), then \(2 \delta ^a \le 1\) and \(d_u^{a}+d_v^{a} \le 2 \delta ^a \le 1\). Thus,

$$\begin{aligned} \begin{aligned} \frac{1}{d_u^{a}+d_v^{a}} + d_u^{a}+d_v^{a}&\ge f(2 \delta ^a) = 2\delta ^a + \frac{1}{2\delta ^a} , \\ ISD_a(G) + M_1^{a+1}(G)&\ge \Big ( 2\delta ^a + \frac{1}{2\delta ^a} \Big ) m. \end{aligned} \end{aligned}$$

The equality holds if and only if \(d_u^{a}+d_v^{a} = 2 \delta ^a\) for every \(uv \in E(G)\), i.e., \(d_u=d_v=\delta \) for every \(uv \in E(G)\). That is, if and only if G is regular. \(\square \)

3 Some applications: QSPR/QSAR models

In this section, the predictive power of the first general Zagreb index \(M_1^{a}\) will be investigated. For this purpose, experimental data on some physico-chemical properties of octane isomers are used. Namely,

  • Entropy (S).

  • Motor octane number (MON).

  • Standard enthalpy of vaporization (DHVAP).

  • Acentric factor (AcenFac).

Such experimental data are obtained from https://webbook.nist.gov. Later, they are processed with a self-developed program to calculate the absolute value of the Pearson’s correlation coefficient (|r|) for values of \(a\in [-5,5]\) with a spacing of 0.01. The Fig. 1 show a plot of the results obtained for S, MON, DHVAP and AcenFac properties. The dashed vertical lines indicate the value of a that maximize |r|.

Fig. 1
figure 1

Plots for S, MON, DHVAP and AcentFact

Full size image
Fig. 2
figure 2

Testing of linear regression models of the form (8)

Full size image

In the Fig. 2 we test linear regression models of the form:

$$\begin{aligned} {\mathcal {P}}=c_1\, M_1^a(G) + c_2\,, \end{aligned}$$
(8)

where \({\mathcal {P}}\) is the S, MON, DHVAP or AcenFac property, and \(c_1\), \(c_2\) are constants. In the Table 1 we collect, respectively, the regression and statistical parameters of the linear QSPR models for the properties S, MON, DHVAP and AcenFac. See the dashed lines in the Fig. 1 given by Eq. (8).

Table 1 Regression and statistical parameters of linear QSPR models for S, MON, DHVAP and AcenFac
Full size table
$$\begin{aligned} \begin{aligned} S&=-0.815\, M_1^{2.270} + 137.675 \\ MON&=84.244\, M_1^{-1} - 406.982 \\ DHV\!AP&=3.014\, M_1^{0.62} - 24.037 \\ AcentFac&=-0.087\, M_1^{1.34} + 1.903 \end{aligned} \end{aligned}$$

A topological index is considered a good predictor for a property when the absolute value of the Pearson’s correlation coefficient is greater than 0.9. From this analysis we can conclude that indices \(M_1^{2.270}\), \(M_1^{-1}\), \(M_1^{0.62}\) and \(M_1^{1.34}\) are good predictors, respectively, of the S, MON, DHVAP and AcenFac properties of the octane isomers. In particular, the index \(M_1^{1.34}\) has \(|r|=0.975.\)

4 Conclusion

The aim of our research was to determine novel inequalities relating several classes of variable topological indices, like the first and second general Zagreb indices, the general sum-connectivity index and the variable inverse sum deg index. It is worth noting that the results and methodology shown in this work allowed us to characterize extremal graphs with respect to many of such inequalities.

In addition, our analysis about the predictive power of the first general Zagreb index shows its applicability to the study of the physico-chemical properties of the octane isomers; in particular to the study of Entropy, Motor octane number, Standard enthalpy of vaporization and Acentric factor.