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. A topological descriptor is called a topological index if it correlates with a molecular property. Topological indices are used to understand physicochemical properties of chemical compounds, since they capture some properties of a molecule in a single number. Hundreds of topological indices have been introduced and studied, starting with the seminal work by Wiener [42].

Topological indices based on end-vertex degrees of edges have been used over 40 years. Among them, several indices are recognized to be useful tools in chemical researches. Probably, the best know such descriptor is the Randić connectivity index (R) [33]. There are more than thousand papers and a couple of books dealing with this molecular descriptor (see, e.g., [20, 24, 25, 38, 39] and the references therein). During many years, scientists were trying to improve the predictive power of the Randić index. This led to the introduction of a large number of new topological descriptors resembling the original Randić index. Two of the main successors of the Randić index are the first and second Zagreb indices, denoted by \(M_1\) and \(M_2\), respectively, defined as

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

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 growing interest, see e.g. [2, 9, 18, 28] (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

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

The inverse degree index first attracted attention through numerous conjectures generated by the computer programme Graffiti [16]. Since then its relationship with other graph invariants, such as diameter, edge-connectivity, matching number, Wiener index has been studied by several authors (see, e.g., [8, 10, 13, 14, 45]).

Miličević and Nikolić defined in [30] the first and second variable Zagreb indices as

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

with \(\alpha \in \mathbb {R}\). In [27] and [43] the first and second general Zagreb indices are introduced as

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

respectively. It is clear that these indices are equivalent to the previous ones, since \(^{\alpha }M_1(G)=M_1^{2\alpha }(G)\) and \(^{\alpha }M_2(G)=M_2^{\alpha }(G)\). We prefer to use \(M_j^{\alpha }(G)\) instead of \(^{\alpha }M_j(G)\), for \(j=1,2,\) since the inequalities obtained in this paper become simpler with them.

Note that \(M_1^0\) is n, \(M_1^{1}\) is 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, \(M_2^{0}\) is 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 [34, 35]), but also to assess the structural differences (e.g., the relative role of carbon atoms of acyclic and cyclic parts in alkylcycloalkanes [36]). The idea behind the variable molecular descriptors is that the variables are determined during the regression so that the standard error of estimate for a studied property is as small as possible.

In the paper of Gutman and Tosovic [21], 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 to realize 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 [31]. Various properties and relations of these indices are discussed in several papers (see, e.g., [1, 26, 29, 41, 44, 46]).

The study of the exponential vertex-degree-based topological indices was initiated in [32] and has been successfully studied in [4,5,6,7]. Cruz et al. mentioned in 2020 some open problems on the exponential vertex-degree-based topological indices of trees [7].

In this paper we obtain new inequalities involving the inverse degree index and its exponential extension, and we characterize graphs which are extremal with respect to them. Also, through a QSPR study of this index and its exponential extension, we obtained linear models for some physicochemical properties of octane isomers.

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.

2 Inequalities involving the ID index

The Sombor index of G was defined in [19] as

$$\begin{aligned} SO( G) =\sum \limits _{uv\in E( G) }\sqrt{d_{u}^{2}+ d_{v}^{2}}. \end{aligned}$$

Many papers have continued the study of the Sombor index. In [37] it is shown that this index have good predictive potential.

Our next result relates the Sombor and the inverse degree indices.

Theorem 1

If G is a graph with maximum degree \(\Delta \) and minimum degree \(\delta \), then

$$\begin{aligned} \frac{\sqrt{2}}{\Delta ^3} \, SO(G) \le ID(G) \le \frac{\sqrt{2}}{\delta ^3} \, SO(G), \end{aligned}$$

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

Proof

If \(\delta \le x,y \le \Delta \), then the function

$$\begin{aligned} f(x,y) = \frac{\frac{x^2+y^2}{x^2y^2}}{\sqrt{x^2+y^2}} = \frac{\sqrt{x^2+y^2}}{x^2y^2} = \sqrt{\frac{x^2+y^2}{x^4y^4}} = \sqrt{\frac{1}{x^2y^4}+ \frac{1}{x^4y^2}} \end{aligned}$$

is strictly decreasing in each variable. Thus,

$$\begin{aligned} \begin{aligned}&\frac{\frac{x^2+y^2}{x^2y^2}}{\sqrt{x^2+y^2}} \le \frac{\sqrt{\delta ^2+\delta ^2}}{\delta ^2\delta ^2} = \frac{\sqrt{2}}{\delta ^3} , \\&\quad \frac{\frac{x^2+y^2}{x^2y^2}}{\sqrt{x^2+y^2}} \ge \frac{\sqrt{\Delta ^2+\Delta ^2}}{\Delta ^2\Delta ^2} = \frac{\sqrt{2}}{\Delta ^3} , \\&\quad \frac{\sqrt{2}}{\Delta ^3} \, \sqrt{d_u^2+d_v^2} \le \frac{d_u^2+d_v^2}{d_u^2 d_v^2} \le \frac{\sqrt{2}}{\delta ^3} \, \sqrt{d_u^2+d_v^2} , \end{aligned} \end{aligned}$$

for every \(uv \in E(G)\). Thus,

$$\begin{aligned} \frac{\sqrt{2}}{\Delta ^3} \, SO(G) \le ID(G) \le \frac{\sqrt{2}}{\delta ^3} \, SO(G), \end{aligned}$$

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

In 2015, Shegehall and Kanabur [40] introduced the arithmetic–geometric index as

$$\begin{aligned} AG(G) = \sum _{uv\in E(G)}\frac{d_u + d_v}{2\sqrt{d_u d_v}} , \end{aligned}$$

and it has been studied by many authors. The next result relates the arithmetic–geometric and the inverse degree indices.

Theorem 2

If G is a graph with maximum degree \(\Delta \) and minimum degree \(\delta \), then

$$\begin{aligned} \frac{2}{\Delta ^2} \, AG(G) \le ID(G) \le \frac{2}{\delta ^2} \, AG(G), \end{aligned}$$

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

Proof

If \(\delta \le x,y \le \Delta \), consider the function

$$\begin{aligned} g(x,y) = \frac{\frac{x^2+y^2}{x^2y^2}}{\frac{x+y}{2\sqrt{xy}}} = 2 \, \frac{x^2+y^2}{x+y} \, x^{-3/2}y^{-3/2}. \end{aligned}$$

We have

$$\begin{aligned} \begin{aligned} \frac{\partial g}{\partial x}(x,y)&= 2 \, \frac{2x(x+y)-x^2-y^2}{(x+y)^2} \, x^{-3/2}y^{-3/2} -3 \, \frac{x^2+y^2}{x+y} \, x^{-5/2}y^{-3/2} \\&= \frac{2x^2+4xy-2y^2}{(x+y)^2} \, x^{-3/2}y^{-3/2} -3 \, \frac{x^2+y^2}{x+y} \, x^{-5/2}y^{-3/2} \\&= \frac{x^{-5/2}y^{-3/2}}{(x+y)^2} \, \big ( (2x^2+4xy-2y^2)x - 3 (x^2+y^2)(x+y) \big ) \\&= \frac{x^{-5/2}y^{-3/2}}{(x+y)^2} \, \big ( -x^3+ x^2y-5xy^2 - 3 y^3 \big ). \\ \end{aligned} \end{aligned}$$

If \(x \ge y\), then \(-x^3+ x^2y=x^2(y-x) \le 0\). If \(x \le y\), then \(x^2y-5xy^2=xy(x-5y) \le 0\). Hence, \(\partial g/\partial x<0\) in any case.

In a similar way, one can check that

$$\begin{aligned} \frac{\partial g}{\partial y}(x,y) = \frac{y^{-5/2}x^{-3/2}}{(x+y)^2} \left( -y^3+ y^2x-5yx^2 - 3 x^3 \right) <0. \end{aligned}$$

Thus, g is a decreasing function in each variable and

$$\begin{aligned} \begin{aligned}&\frac{\frac{x^2+y^2}{x^2y^2}}{\frac{x+y}{2\sqrt{xy}}} \le 2 \, \frac{2\delta ^2}{2\delta } \, \delta ^{-3/2}\delta ^{-3/2} = \frac{2}{\delta ^2},\\&\frac{\frac{x^2+y^2}{x^2y^2}}{\frac{x+y}{2\sqrt{xy}}} \ge 2 \, \frac{2\Delta ^2}{2\Delta } \, \Delta ^{-3/2}\Delta ^{-3/2} = \frac{2}{\Delta ^2},\\&\frac{2}{\Delta ^2} \, \frac{d_u +d_v}{2\sqrt{d_u d_v}} \le \frac{d_u^2+d_v^2}{d_u^2 d_v^2} \le \frac{2}{\delta ^2} \, \frac{d_u +d_v}{2\sqrt{d_u d_v}}, \end{aligned} \end{aligned}$$

for every \(uv \in E(G)\). Thus,

$$\begin{aligned} \frac{2}{\Delta ^2} \, AG(G) \le ID(G) \le \frac{2}{\delta ^2} \, AG(G), \end{aligned}$$

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

As a natural extension of the Albertson irregularity index, Gutman et al. recently defined in [23] the \(\sigma \)-index of a graph as

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

The next result relates the inverse degree and the \(\sigma \) indices.

Theorem 3

If G is a graph with n vertices, maximum degree \(\Delta \) and minimum degree \(\delta \), then

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

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

Proof

Since

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

we obtain

$$\begin{aligned} ID(G) = \sum _{uv \in E(G)} \left( \frac{1}{d_u^2} + \frac{1}{d_v^2} \right) = \sum _{uv \in E(G)} \frac{(d_u-d_v)^2}{d_u^2 d_v^2} + 2 M_2^{-1}(G). \end{aligned}$$

Since

$$\begin{aligned} \begin{aligned} \sum _{uv \in E(G)} \frac{(d_u-d_v)^2}{d_u^2 d_v^2}&\le \sum _{uv \in E(G)} \frac{(d_u-d_v)^2}{\delta ^4} = \frac{\sigma (G)}{\delta ^4},\\ \sum _{uv \in E(G)} \frac{(d_u-d_v)^2}{d_u^2 d_v^2}&\ge \sum _{uv \in E(G)} \frac{(d_u-d_v)^2}{\Delta ^4} = \frac{\sigma (G)}{\Delta ^4},\end{aligned} \end{aligned}$$

we conclude

$$\begin{aligned} \begin{aligned} ID(G)&= \sum _{uv \in E(G)} \frac{(d_u-d_v)^2}{d_u^2 d_v^2} + 2 M_2^{-1}(G) \le \frac{\sigma (G)}{\delta ^4} + 2 M_2^{-1}(G), \\ ID(G)&= \sum _{uv \in E(G)} \frac{(d_u-d_v)^2}{d_u^2 d_v^2} + 2 M_2^{-1}(G) \ge \frac{\sigma (G)}{\Delta ^4} + 2 M_2^{-1}(G). \end{aligned} \end{aligned}$$

If G is a regular graph, then \(\Delta =\delta \) and \(\sigma (G)=0\), and so, both bounds are equal and they are equal to ID(G).

Assume now that the equality in the upper (respectively, lower) 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 a regular graph. \(\square \)

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

Theorem 4

If \(\alpha \in \mathbb {R}\) and G is a graph with minimum degree \(\delta \) and maximum degree \(\Delta \), then

$$\begin{aligned} \begin{aligned} \frac{M_1^{\alpha }(G)}{\delta ^{\alpha +1}}\le ID(G)\le \frac{M_1^{\alpha }(G)}{\Delta ^{\alpha +1}}, \quad&\text {if }\, \alpha \le -1,\\ \frac{M_1^{\alpha }(G)}{\Delta ^{\alpha +1}}\le ID(G)\le \frac{M_1^{\alpha }(G)}{\delta ^{\alpha +1}}, \quad&\text {if }\, -1 \le \alpha < 0. \end{aligned} \end{aligned}$$

The equality is attained in each inequality for some \(\alpha \ne -1\) if and only if G is regular. If \(\alpha = -1\), then each inequality is an equality for every graph.

Proof

If \(\alpha \le -1\), then

$$\begin{aligned} \begin{aligned} M_1^{\alpha }(G)&= \sum _{u\in V(G)} d_u^{\alpha +1} d_u^{-1} \le \delta ^{\alpha +1} \sum _{u\in V(G)} d_u^{-1} = \delta ^{\alpha +1} ID(G), \\ M_1^{\alpha }(G)&= \sum _{u\in V(G)} d_u^{\alpha +1} d_u^{-1} \ge \Delta ^{\alpha +1} \sum _{u\in V(G)} d_u^{-1} = \Delta ^{\alpha +1} ID(G). \end{aligned} \end{aligned}$$

If \(-1 \le \alpha < 0\), then

$$\begin{aligned} \begin{aligned} M_1^{\alpha }(G)&= \sum _{u\in V(G)} d_u^{\alpha +1} d_u^{-1} \le \Delta ^{\alpha +1} \sum _{u\in V(G)} d_u^{-1} = \Delta ^{\alpha +1} ID(G), \\ M_1^{\alpha }(G)&= \sum _{u\in V(G)} d_u^{\alpha +1} d_u^{-1} \ge \delta ^{\alpha +1} \sum _{u\in V(G)} d_u^{-1} = \delta ^{\alpha +1} ID(G). \end{aligned} \end{aligned}$$

The equality is attained in each inequality for some \(\alpha \ne -1\) if and only if we have either \(d_u=\delta \) for every \(u \in V(G)\) or \(d_u=\Delta \) for every \(u \in V(G)\), i.e., G is regular. If \(\alpha = -1\), then \(M_1^{-1}(G) = ID(G)\) and each inequality is an equality for every graph. \(\square \)

Note that the second inequality in Theorem 4 also holds for \(\alpha > 0\), but it is trivial in this case.

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

Theorem 5

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

$$\begin{aligned} \begin{aligned} ID(G)\le \frac{M_2^{\alpha }(G)+m\Delta ^{2\alpha }}{\Delta ^{2\alpha +2}}, \quad&\text {if }\, \alpha \le -2, \\ ID(G)\le \frac{M_2^{\alpha }(G)+m\delta ^{2\alpha }}{\delta ^{2\alpha +2}}, \quad&\text {if }\, \alpha \ge -2. \end{aligned} \end{aligned}$$

If \(\alpha \ne -2,\) then the equality is attained if and only if G is regular. If \(\alpha = -2,\) then the equality holds in the first (respectively, second) inequality if and only if every edge has a vertex with maximum (respectively, minimum) degree.

Proof

We have for every \(\alpha \in \mathbb {R}\)

$$\begin{aligned} \begin{aligned}&(\Delta ^\alpha - d_u^\alpha )(\Delta ^\alpha - d_v^\alpha )\ge 0,\\&\Delta ^{2\alpha }-\Delta ^\alpha (d_u^\alpha + d_v^\alpha )+(d_u d_v)^\alpha \ge 0,\\&\sum _{uv\in E(G)}\Delta ^{2\alpha }-\sum _{uv\in E(G)}\Delta ^\alpha (d_u^\alpha + d_v^\alpha )+\sum _{uv\in E(G)}(d_u d_v)^\alpha \ge 0,\\&m\Delta ^{2\alpha }+ M_2^{\alpha }(G) \ge \sum _{u\in V(G)}\Delta ^\alpha d_u^{\alpha +1} = \Delta ^\alpha M_1^{\alpha +1}(G). \end{aligned} \end{aligned}$$

Since Theorem 4 gives \(ID(G) \le \frac{M_1^{\alpha +1}(G)}{\Delta ^{\alpha +2}}\) if \(\alpha \le -2\), we obtain

$$\begin{aligned} m\Delta ^{2\alpha }+ M_2^{\alpha }(G)\ge \Delta ^{2\alpha +2}ID(G). \end{aligned}$$

In a similar way, we have for every \(\alpha \in \mathbb {R}\)

$$\begin{aligned} \begin{aligned}&(\delta ^\alpha -d_u^\alpha )(\delta ^\alpha -d_v^\alpha ) \ge 0, \\&\delta ^{2\alpha }-\delta ^\alpha (d_u^\alpha + d_v^\alpha )+(d_u d_v)^\alpha \ge 0, \\&\sum _{uv\in E(G)}\delta ^{2\alpha }-\sum _{uv\in E(G)}\delta ^\alpha (d_u^\alpha + d_v^\alpha )+\sum _{uv\in E(G)}(d_u d_v)^\alpha \ge 0, \\&m\delta ^{2\alpha }+ M_2^{\alpha }(G) \ge \sum _{u\in V(G)}\delta ^\alpha d_u^{\alpha +1} = \delta ^\alpha M_1^{\alpha +1}(G). \end{aligned} \end{aligned}$$

Since Theorem 4 gives \(ID(G)\le \frac{M_1^{\alpha +1}(G)}{\delta ^{\alpha +2}}\) if \(\alpha \ge -2\), we obtain

$$\begin{aligned} m\delta ^{2\alpha }+ M_2^{\alpha }(G)\ge \delta ^{2\alpha +2}ID(G). \end{aligned}$$

If the graph is regular, then both bounds are the same, and they are equal to ID(G). If the equality is attained for some \(\alpha \ne -2\), then the equality is attained also in Theorem 4, and thus G is regular.

If \(\alpha = -2,\) then \(ID(G) =\frac{M_1^{\alpha +1}(G)}{\Delta ^{\alpha +2}} =\frac{M_1^{\alpha +1}(G)}{\delta ^{\alpha +2}}\) for every graph G. Thus, the equality holds in the first inequality if and only if \((\Delta ^\alpha - d_u^\alpha )(\Delta ^\alpha - d_v^\alpha ) =0\) for every \(uv \in E(G)\), i.e., every edge has a vertex with maximum degree. The same argument gives that the equality holds in the second inequality if and only if every edge has a vertex with minimum degree. \(\square \)

The atom-bond connectivity index (ABC-index) is a useful topological index employed in studying the stability of alkanes and the strain energy of cycloalkanes. The atom-bond connectivity index of a graph G was defined in [15] as

$$\begin{aligned} ABC(G) = \sum _{uv\in E(G)} \sqrt{\frac{d_u + d_v - 2}{d_u d_v}} \;, \end{aligned}$$

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.

The generalized atom-bond connectivity index was defined in [17] as

$$\begin{aligned} ABC_{\alpha }(G) = \sum _{uv\in E(G)} \Big ( \, \frac{d_u + d_v - 2}{d_u d_v} \, \Big )^{\alpha }. \end{aligned}$$

for any \(\alpha \in \mathbb {R}\setminus \{0\}\). Note that \(ABC_{1/2}\) is the ABC-index and \(ABC_{-3}\) is the augmented Zagreb index.

There are a lot of papers studying the ABC and \(ABC_{\alpha }\) indices (see, e.g., [3, 11, 12, 17, 22]).

Our next result relates the generalized atom-bond connectivity and the inverse degree indices. Recall that an isolated edge is a graph with just two vertices and an edge.

Theorem 6

Let G be a graph with maximum degree \(\Delta \), minimum degree \(\delta \), and \(\alpha \le -1\).

(1) If \(\delta > 1\), then

$$\begin{aligned} \begin{aligned}&2\left( \frac{2\Delta -2}{\Delta } \right) ^\alpha ABC_{-\alpha }(G) \le M_1^{\alpha +1}(G) \\&\quad \le \max \left\{ 2\left( \frac{2\delta -2}{\delta } \right) ^\alpha , \,(\Delta +\delta -2)^\alpha (\Delta ^{-\alpha }+\delta ^{-\alpha }) \right\} ABC_{-\alpha }(G)\,. \end{aligned} \end{aligned}$$

The equality in the first bond is attained if and only if G is a regular graph.

(2) If \(\delta = 1\) and G does not contain isolated edges, then

$$\begin{aligned} \begin{aligned} 2\left( \frac{2\Delta -2}{\Delta } \right) ^\alpha ABC_{-\alpha }(G) \le M_1^{\alpha +1}(G) \le (1+2^{-\alpha }) ABC_{-\alpha }(G)\,. \end{aligned} \end{aligned}$$

The equality in the second bond is attained if and only if G is a union of path graphs \(P_3\).

Proof

Assume first that \(\delta > 1\).

We are going to compute the minimum and maximum values of

$$\begin{aligned} f(x, y)=\left( \frac{x+y-2}{x y}\right) ^{\alpha } (x^\alpha +y^\alpha ) = \left( \frac{x+y-2}{y} \right) ^{\alpha } + \left( \frac{x+y-2}{x} \right) ^{\alpha } \,, \end{aligned}$$

on \([\delta ,\Delta ] \times [\delta ,\Delta ]\). By symmetry we can assume that \(\{\delta \le x \le y \le \Delta \}\). We have

$$\begin{aligned} \begin{aligned} \frac{\partial f}{\partial x}(x,y)&= \alpha \, \frac{1}{y}\left( \frac{x+y-2}{y} \right) ^{\alpha -1} + \alpha \, \frac{2-y}{x^2}\left( \frac{x+y-2}{x} \right) ^{\alpha -1} \\&= \alpha \left( x+y-2\right) ^{\alpha -1} \left( \frac{1}{y^\alpha } + \frac{2-y}{x^{\alpha +1}} \right) \\&= \alpha \left( x+y-2\right) ^{\alpha -1} \left( \frac{x^{\alpha +1}-y^{\alpha +1}}{x^{\alpha +1}y^\alpha } +\frac{2}{x^{\alpha +1}}\right) \le 0 \,, \end{aligned} \end{aligned}$$

since \(\alpha \le -1\). Thus, f(x, y) is decreasing on \(x \in [\delta , y]\) for every fixed \(\delta \le y\le \Delta \), and so \(f(y, y) \le f(x, y) \le f(\delta , y)\). Let us consider

$$\begin{aligned} g_1(y)=f(y, y)=2\left( \frac{2y-2}{y} \right) ^\alpha \,, \end{aligned}$$

then

$$\begin{aligned} g_1^{\prime }(y)=\frac{4\alpha }{y^2}\left( \frac{2y-2}{y} \right) ^{\alpha -1} <0 \,, \end{aligned}$$

so \(g_1(y)\) is decreasing on \(y\in [\delta ,\Delta ]\). Thus, we have \(g_1(\Delta )\le g_1(y) \le f(x,y)\) and the equality holds if and only if \(x=y=\Delta \). Then for each \(uv\in E(G)\), we have

$$\begin{aligned} 2\left( \frac{2\Delta -2}{\Delta } \right) ^\alpha \left( \frac{d_u+d_v-2}{d_u d_v}\right) ^{-\alpha } \le d_u^\alpha +d_v^\alpha \,, \end{aligned}$$

and the equality is attained if and only if \(d_u=d_v=\Delta \). Hence,

$$\begin{aligned} 2\left( \frac{2\Delta -2}{\Delta } \right) ^\alpha ABC_{-\alpha }(G) \le M_1^{\alpha +1}(G) \end{aligned}$$

and the equality is attained if and only if \(d_u=d_v=\Delta \) for every \(uv\in E(G)\), i.e. G is a regular graph.

Let us consider now

$$\begin{aligned} g_2(y)=f(\delta ,y)= \left( \frac{\delta +y-2}{\delta } \right) ^{\alpha } + \left( \frac{\delta +y-2}{y} \right) ^{\alpha }. \end{aligned}$$

Thus,

$$\begin{aligned} g_2^{\prime }(y)= \alpha \left( y+\delta -2\right) ^{\alpha -1} \left( \frac{y^{\alpha +1}-\delta ^{\alpha +1}}{y^{\alpha +1}\delta ^\alpha } +\frac{2}{y^{\alpha +1}}\right) . \end{aligned}$$

Note that this formula also holds if \(\alpha = -1\), since \(g_2^{\prime }(y)= -2\left( y+\delta -2\right) ^{-2}\) in this case.

If \(\alpha = -1\), then \(g_2^{\prime }(y) \ne 0\) on \([\delta ,\Delta ]\).

If \(\alpha < -1\) and \(g_2^{\prime }(y)=0\), then

$$\begin{aligned} \frac{y^{\alpha +1}-\delta ^{\alpha +1}+2\delta ^\alpha }{y^{\alpha +1}\delta ^\alpha }=0 \quad \Rightarrow \quad y= \delta \left( 1-\frac{2}{\delta }\right) ^{1/(\alpha +1)} \end{aligned}$$

(recall that \(\delta \ge 2\)). Therefore, there exists at most a zero of \(g_2^{\prime }\) in \([\delta ,\Delta ]\). Since \(g_2^{\prime }(\delta )<0\), it is clear that for every \(y \in [\delta ,\Delta ]\)

$$\begin{aligned} \begin{aligned} g_2(y)&\le \max \left\{ g_2(\delta ), \,g_2(\Delta ) \right\} = \max \left\{ f(\delta ,\delta ), \,f(\delta ,\Delta ) \right\} \\&= \max \left\{ 2\left( \frac{2\delta -2}{\delta } \right) ^\alpha , \,(\Delta +\delta -2)^\alpha (\Delta ^{-\alpha }+\delta ^{-\alpha }) \right\} . \end{aligned} \end{aligned}$$

Then for each \(uv\in E(G)\), we have

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

and so,

$$\begin{aligned} M_1^{\alpha +1}(G) \le \max \left\{ 2\left( \frac{2\delta -2}{\delta } \right) ^\alpha , \,(\Delta +\delta -2)^\alpha (\Delta ^{-\alpha }+\delta ^{-\alpha }) \right\} ABC_{-\alpha }(G)\,. \end{aligned}$$

Assume now that G does not contain isolated edges and \(\delta = 1\). If \(2\le d_u,d_v \le \Delta \), then we have proved that

$$\begin{aligned} \begin{aligned}&2\left( \frac{2\Delta -2}{\Delta } \right) ^\alpha \left( \frac{d_u+d_v-2}{d_u d_v}\right) ^{-\alpha } \le d_u^\alpha + d_v^\alpha \\&\quad \le \max \left\{ 2, \,\Delta ^\alpha (\Delta ^{-\alpha }+2^{-\alpha }) \right\} \left( \frac{d_u+d_v-2}{d_u d_v}\right) ^{-\alpha }. \end{aligned} \end{aligned}$$

Let us consider now the case \(d_u=1\). Since G does not contain isolated edges, we have \(2\le d_v \le \Delta \).

Consider on \([2,\Delta ]\) the function

$$\begin{aligned} g_3(y) = f(1,y) = \left( \frac{y-1}{y}\right) ^{\alpha } \left( 1+y^\alpha \right) = \left( y-1 \right) ^{\alpha } \left( 1+y^{-\alpha } \right) . \end{aligned}$$

Hence,

$$\begin{aligned} \begin{aligned} g_3'(y)&= \alpha \left( y-1 \right) ^{\alpha -1} \left( 1+y^{-\alpha } \right) -\alpha \left( y-1 \right) ^{\alpha } y^{-\alpha -1} \\&= \alpha \left( y-1 \right) ^{\alpha -1} \left( 1+y^{-\alpha } - ( y-1 ) y^{-\alpha -1} \right) \\&= \alpha \left( y-1 \right) ^{\alpha -1} \left( 1+ y^{-\alpha -1} \right) < 0. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \left( \frac{\Delta -1}{\Delta }\right) ^{\alpha } \left( 1+\Delta ^\alpha \right) = g_3(\Delta ) \le f(1,y) \le g_3(2) = 1+2^{-\alpha }. \end{aligned}$$

We conclude that

$$\begin{aligned} \begin{aligned}&\min \left\{ 2\left( \frac{2\Delta -2}{\Delta } \right) ^\alpha , \,\left( \frac{\Delta -1}{\Delta }\right) ^{\alpha } \left( 1+\Delta ^\alpha \right) \right\} \left( \frac{d_u+d_v-2}{d_u d_v}\right) ^{-\alpha } \le d_u^\alpha + d_v^\alpha \\&\quad \le \max \left\{ 2, \, 1 +\left( \frac{\Delta }{2} \right) ^{\alpha },\, 1+2^{-\alpha } \right\} \left( \frac{d_u+d_v-2}{d_u d_v}\right) ^{-\alpha } \end{aligned} \end{aligned}$$

for every \(uv \in E(G)\).

Since \(\alpha \le -1\), we have \(1 +(\Delta /2)^{\alpha } \le 2 < 1+2^{-\alpha }\) and so,

$$\begin{aligned} \max \left\{ 2, \, 1 +\left( \frac{\Delta }{2} \right) ^{\alpha },\, 1+2^{-\alpha } \right\} = 1+2^{-\alpha }. \end{aligned}$$

Since \(2^{\alpha +1} \le 1 < 1+\Delta ^{\alpha }\), we obtain

$$\begin{aligned} \min \left\{ 2\left( \frac{2\Delta -2}{\Delta } \right) ^\alpha , \,\left( \frac{\Delta -1}{\Delta }\right) ^{\alpha } \left( 1+\Delta ^\alpha \right) \right\} = 2\left( \frac{2\Delta -2}{\Delta } \right) ^\alpha . \end{aligned}$$

Hence,

$$\begin{aligned} \begin{aligned} 2\left( \frac{2\Delta -2}{\Delta } \right) ^\alpha \left( \frac{d_u+d_v-2}{d_u d_v}\right) ^{-\alpha } \le d_u^\alpha + d_v^\alpha \le \left( 1+2^{-\alpha }\right) \left( \frac{d_u+d_v-2}{d_u d_v}\right) ^{-\alpha } \end{aligned} \end{aligned}$$

for every \(uv \in E(G)\), and the equality in the upper bound is attained if and only if \(\{d_u, d_v\}=\{1,2\}\). So, we conclude

$$\begin{aligned} 2\left( \frac{2\Delta -2}{\Delta } \right) ^\alpha ABC_{-\alpha }(G) \le M_1^{\alpha +1}(G) \le (1+2^{-\alpha }) ABC_{-\alpha }(G), \end{aligned}$$

and the equality in the second bond is attained if and only if G is a union of path graphs \(P_3\). \(\square \)

3 Inequalities involving exponential extensions

Given a function \(a:[1,\infty )\rightarrow \mathbb {R}\), we consider the general topological indices

$$\begin{aligned} A(G)=\displaystyle \sum _{u\in V(G)} a(d_u), \quad e^A(G)=\displaystyle \sum _{u\in V(G)} e^{a(d_u)}. \end{aligned}$$

Also, given a symmetric function \(b:[1,\infty ) \times [1,\infty ) \rightarrow \mathbb {R}\), we consider the general topological indices

$$\begin{aligned} B(G)=\displaystyle \sum _{uv\in E(G)} b(d_u,d_v), \quad e^B(G)=\displaystyle \sum _{uv\in E(G)} e^{b(d_u,d_v)}. \end{aligned}$$

In this section we discuss some inequalities involving these general topological indices.

Proposition 7

Let G be a graph with n vertices, and \(A_1(G) = \sum _{u\in V(G)} a_1(d_u)\), \(A_2(G) = \sum _{u\in V(G)} a_2(d_u)\) topological indices. If there exists a constant \(c \ge 1\) such that \(c\, a_1(d_u) \le a_2(d_u)\) for each \(u\in V(G)\), then

$$\begin{aligned} n^{1-c} \left( e^{A_1}(G) \right) ^{c} \le e^{A_2}(G). \end{aligned}$$

If \(c > 1\), then the equality in the bound is attained if and only if there exists a constant k such that \(a_2(d_u)=c\,a_1(d_u)=c\,k\) for every \(u\in V(G)\). If \(c = 1\), then the equality in the bound is attained if and only if \(a_2(d_u)=a_1(d_u)\) for every \(u\in V(G)\).

Proof

The function \(a(x)=x^{c}\) is convex and so, Jensen’s inequality gives

$$\begin{aligned} \left( \frac{1}{n} \displaystyle \sum _{u\in V(G)} e^{a_1(d_u)} \right) ^c \le \frac{1}{n} \sum _{u\in V(G)} \left( e^{ a_1(d_u)} \right) ^c = \frac{1}{n} \sum _{u\in V(G)} e^{ c\,a_1(d_u)}. \end{aligned}$$

When \(c > 1\) the equality holds if and only if \(a_1(d_u)= a_1(d_v)\) for every \(u\in V(G)\). Otherwise, if \(c=1\), the equality holds for every graph G. Since \(c\, a_1(d_u) \le a_2(d_u)\) for each \(u \in V(G)\), we have

$$\begin{aligned} \sum _{u\in V(G)} e^{ c\,a_1(d_u)} \le \sum _{u\in V(G)} e^{ a_2(d_u)} \end{aligned}$$

and the equality is attained if and only if \(c\, a_1(d_u) = a_2(d_u)\) for each \(u \in V(G)\). Hence, we have

$$\begin{aligned} \left( \frac{1}{n} \, e^{A_1}(G) \right) ^c \le \frac{1}{n} \, e^{A_2}(G) \end{aligned}$$

and the equality is attained when \(c > 1\) if and only if there exists a constant k such that \(a_2(d_u)=c\,a_1(d_u)=c\,k\) for each \(u\in V(G)\).

If \(c = 1\), then the equality in the bound is attained if and only if \(a_2(d_u)=a_1(d_u)\) for each \(u\in V(G)\). \(\square \)

By changing the roles of \(a_1\) and \(a_2\) in Proposition 7, and taking \(c=1/C\), we obtain the following:

Corollary 8

Let G be a graph with n vertices, and \(A_1(G) = \sum _{u\in V(G)} a_1(d_u)\), \(A_2(G) = \sum _{u\in V(G)} a_2(d_u)\) topological indices. If there exists a constant \(0< C \le 1\) such that \(a_2(d_u) \le C a_1(d_u)\) for each \(u\in V(G)\), then

$$\begin{aligned} e^{A_2}(G) \le n^{1-C} \left( e^{A_1}(G) \right) ^{C}. \end{aligned}$$

If \(0< C < 1\), then the equality holds if and only if there exists a constant k such that \(a_2(d_u)=C\,a_1(d_u)=C\,k\) for each \(u\in V(G)\). If \(C = 1\), then the equality holds if and only if \(a_2(d_u)=a_1(d_u)\) for each \(u\in V(G)\).

Proposition 9

Let us consider a symmetric function \(b:[1,\infty ) \times [1,\infty ) \rightarrow [0,1]\) and G any graph with m edges. Then

$$\begin{aligned} B(G)+m \le e^B(G) \le (e-1)B(G)+m. \end{aligned}$$

If \(b:[1,\infty ) \times [1,\infty ) \rightarrow (0,1]\), then the upper bound is attained for a graph G if and only if \(b(d_u,d_v)=1\) for every \(uv \in E(G)\).

Proof

Since the exponential function is a strictly convex function, it is greater or equal than its tangent line at 0, and less or equal than its secant line at 0 and 1, since f takes values on [0, 1]. Thus, \(x+1 \le e^x \le (e-1)x+1\) for every \(x \in [0,1]\) and

$$\begin{aligned} \begin{aligned} b(d_u,d_v)+1&\le e^{b(d_u,d_v)} \le (e-1)b(d_u,d_v)+1, \\ B(G)+m&\le e^B(G) \le (e-1)B(G)+m. \end{aligned} \end{aligned}$$

Assume now that \(b:[1,\infty ) \times [1,\infty ) \rightarrow (0,1]\). Since the exponential function is a strictly convex function, \(e^x = (e-1)x+1\) for some \(x \in (0,1]\) if and only if \(x=1\). Thus, \(e^B(G) = (e-1)B(G)+m\) for a graph G if and only if \(b(d_u,d_v)=1\) for every \(uv \in E(G)\). \(\square \)

The argument in the proof of Proposition 9 also gives the following result.

Proposition 10

Let us consider a function \(a:[1,\infty ) \rightarrow [0,1]\) and G any graph with n vertices. Then

$$\begin{aligned} A(G)+n \le e^A(G) \le (e-1)A(G)+n. \end{aligned}$$

If \(a:[1,\infty ) \rightarrow (0,1]\), then the upper bound is attained for a graph G if and only if \(a(d_u)=1\) for every \(u \in V(G)\).

Corollary 11

Let G be any graph with n vertices and m edges, and \(\alpha < 0\). Then:

(1)

$$\begin{aligned} M_2^\alpha (G)+m \le e^{M_2^\alpha }(G) \le (e-1)M_2^\alpha (G)+m. \end{aligned}$$

The upper bound is attained for a graph G if and only if G is a union of isolated edges.

(2)

$$\begin{aligned} M_1^\alpha (G)+n \le e^{M_1^\alpha }(G) \le (e-1)\,M_1^\alpha (G)+n. \end{aligned}$$

The upper bound is attained for a graph G if and only if G is a union of isolated edges.

Proof

Since \(\alpha < 0\), the following function satisfies

$$\begin{aligned} 0< b(x,y) = ( xy)^\alpha \le 1 \end{aligned}$$

for every \(x,y \ge 1\). Thus, Proposition 9 gives the inequalities

$$\begin{aligned} M_2^\alpha (G)+m \le e^{M_2^\alpha }(G) \le (e-1)M_2^\alpha (G)+m, \end{aligned}$$

and the upper bound is attained if and only if \((d_u d_v)^{\alpha }=1\) for every \(uv \in E(G)\), i.e., \(d_u = d_v=1\) for every \(uv \in E(G)\), and this happens if and only if G is a union of isolated edges.

Since \(\alpha < 0\), then the following function satisfies

$$\begin{aligned} 0< a(x) = x^\alpha \le 1 \end{aligned}$$

for every \(x,y \ge 1\). Thus, Proposition 10 gives the inequalities

$$\begin{aligned} M_1^\alpha (G)+n \le e^{M_1^\alpha }(G) \le (e-1)\,M_1^\alpha (G)+n, \end{aligned}$$

and the upper bound is attained if and only if \(d_u^{\alpha }=1\) for every \(u \in V(G)\), i.e., \(d_u =1\) for every \(u \in V(G)\), and this holds if and only if G is a union of isolated edges. \(\square \)

The argument in the proof of Proposition 9 also gives the following results, since it is easy to check that \(e^x \le 1+x+ x^2\) for every \(x \in [0,1]\).

Proposition 12

Let G be any graph with n vertices and m edges, and \(\alpha < 0\). Then

$$\begin{aligned} \begin{aligned} m+M_2^\alpha (G)+\frac{1}{2} M_2^{2\alpha }(G)&\le e^{M_2^\alpha }(G) \le m+M_2^\alpha (G)+ M_2^{2\alpha }(G) \\ n+M_1^\alpha (G)+\frac{1}{2} M_1^{2\alpha }(G)&\le e^{M_1^\alpha }(G) \le n+M_1^\alpha (G)+ M_1^{2\alpha }(G). \end{aligned} \end{aligned}$$

Proposition 13

Let G be any graph with n vertices and m edges, \(k \in \mathbb {Z}^+\) and \(\alpha \in \mathbb {R}\). Then

$$\begin{aligned} e^{M_2^\alpha }(G) \ge m+ \frac{1}{k!} M_2^{k\alpha }(G) \qquad e^{M_1^\alpha }(G) \ge n+ \frac{1}{k!} M_1^{k\alpha }(G). \end{aligned}$$

4 QSPR study of ID and \(e^{ID}\) on octane isomers

In this section we perform a QSPR study of the inverse degree index and the exponential inverse degree index to develop linear models of some physicochemical properties of octane isomers. We select the following properties for the study: acentric factor (AcentFact), heat capacity at P constant (CP), standard enthalpy of formation (DHFORM), standard enthalpy of vaporization (DHVAP), enthalpy of vaporization (HVAP) and entropy (S).

In Figs. 1 and 2 we plot respectively ID and \(e^{ID}\) vs. physicochemical properties of octane isomers. Moreover. In Fig. 1 we tested the following linear regression model

$$\begin{aligned} {\mathcal {P}} = m (ID) + c, \end{aligned}$$
(1)

where \({\mathcal {P}}\) is the physicochemical property. Therefore, the following linear QSPR models were obtained (see the red lines in Fig. 1):

$$\begin{aligned} \begin{aligned} \text{ AcentFac }= & {} -0.095&\, ID+ & {} \ 0.876\\ \text{ CP }= & {} 2.43&\, ID+ & {} \ 12.653\\ \text{ DHFORM }= & {} 0.591&\, ID+ & {} \ 0.467\\ \text{ DHVAP }= & {} -0.996&\, ID+ & {} \ 14.808\\ \text{ HVAP }= & {} -5.025&\, ID+ & {} \ 97.822\\ \text{ S }= & {} -11.643&\, ID+ & {} \ 171.821 . \end{aligned} \end{aligned}$$
Fig. 1
figure 1

Inverse degree index vs. the physicochemical properties of octane isomers a AccenFact, b CP, c DHFORM, d DHVAP, e HVAP, f S. Red lines are the linear QSPR models of Eq. (1) (Color figure online)

Full size image

In addition, in Table 1 we resume the regression and statistical parameters of the linear QSPR models above.

In Fig. 2 we tested the following linear regression model

$$\begin{aligned}{} & {} {\mathcal {P}} = m (e^{ID}) + c,\nonumber \\{} & {} \begin{aligned}{} & {} \text{ AcentFac } = -0.04 \, e^{ID} + \ 1.013\\{} & {} \text{ CP } = 1.027 \, e^{ID} + \ 9.024\\{} & {} \text{ DHFORM } = 0.251 \, e^{ID} - \ 0.442 \\{} & {} \text{ DHVAP } = -0.419 \, e^{ID} + \ 16.262\\{} & {} \text{ HVAP } = -2.116 \, e^{ID} + \ 105.193\\{} & {} \text{ S } = -4.879 \, e^{ID} + \ 188.471\\ \end{aligned} \end{aligned}$$
(2)
Fig. 2
figure 2

Exponential inverse degree index vs. the physicochemical properties of octane isomers a AccenFact, b CP, c DHFORM, d DHVAP, e HVAP, f S. Red lines are the linear QSPR models of Eq. (2) (Color figure online)

Full size image

In addition, in Table 2 we resume the regression and statistical parameters of the linear QSPR models above.

Table 1 Parameters of the linear QSPR models of Eq. (1)
Full size table
Table 2 Parameters of the linear QSPR models of Eq. (2)
Full size table

5 Conclusion

In this work we studied the ID index, its generalizations and its exponential extension. Optimal inequalities were obtained taking into account some graph invariants such as the number of vertices, number of edges, minimum degree, maximum degree and other topological indexes.

In addition, a QSPR study was performed to test the predictive power of the ID and \(e^{ID}\) indices. From this study, we can conclude that these indices provides a good predictive power for some of the properties studied, in particular for AcentFac, DHVAP, HVAP and S for which the correlation coefficients (absolute values) are closer or higher than 0.9. However, it could be appreciated that the ID index, for the octane isomers and the studied properties, presents correlation coefficient values slightly better than its exponential extension. Also, for the CP and DHFORM properties, these indices does not present a good predictive power, with low values of the correlation coefficients.

The possibility of obtaining similar results and their applications for other families of topological indices is an interesting topic to consider for future investigations. It would be valuable to consider replicating these in other contexts, and to discuss the possible implications of these results for other families of topological indices. This could provide valuable insights and contribute to expand the scope and impact of these topics.