Mean Sombor index

We introduce a degree-based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: $mSO_\alpha(G) = \sum_{uv \in E(G)} \left[\left( d_u^\alpha+d_v^\alpha \right) /2 \right]^{1/\alpha}$. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $\alpha \in \mathbb{R} \backslash \{0\}$. We also consider the limit cases $mSO_{\alpha\to 0}(G)$ and $mSO_{\alpha\to\pm\infty}(G)$. Indeed, for given values of $\alpha$, the mean Sombor index is related to well-known topological indices such as the inverse sum indeg index, the reciprocal Randic index, the first Zagreb index, the Stolarsky--Puebla index and several Sombor indices. Moreover, through a quantitative structure property relationship (QSPR) analysis we show that $mSO_\alpha(G)$ correlates well with several physicochemical properties of octane isomers. Some mathematical properties of mean Sombor indices as well as bounds and new relationships with known topological indices are also discussed.


Preliminaries
For two positive real numbers x, y the power mean or generalized mean P M α (x, y) with exponent α ∈ R\{0} is given as P M α (x, y) = x α + y α 2 see e.g. [2,3]. P M α (x, y) is also known as Hölder mean. For given values of α, P M α (x, y) reproduces well-known mean values. As examples, in Table 1 we show some expressions for P M α (x, y) for selected values of α with their corresponding names, when available. There is a well known inequality for the power mean, namely [4][5][6]: For any α 1 < α 2 , where the equality is attained for x = y. * Corresponding author (jmendezb@ifuap.buap.mx) Table 1: Expressions for the generalized mean P M α (x, y) for selected values of α. α P M α (x, y) name (when available) −∞ P M α→−∞ (x, y) = min(x, y) minimum value −1 P M −1 (x, y) = 2xy x + y harmonic mean 0 P M α→0 (x, y) = √ xy geometric mean 1/2 P M 1/2 (x, y) = √ x + √ y 2 are currently been studied in mathematical chemistry; where uv denotes the edge of the graph G connecting the vertices u and v, d u is the degree of the vertex u, and F (x, y) is an appropriate chosen function, see e.g. [7][8][9]. Inspired by the power mean and given a simple graph G = (V (G), E(G)), here we choose the function F (x, y) in Eq. (3) as the power mean P M α (x, y) and define the degree-based variable topological index where α ∈ R\{0}. We name mSO α (G) as the mean Sombor index. Note, that for given values of α, the mean Sombor index is related to known topological indices: mSO −1 (G) = 2ISI(G), where ISI(G) is the inverse sum indeg index [10,11], mSO α→0 (G) = R −1 (G), where R −1 (G) is the reciprocal Randic index [12], and mSO 1 (G) = M 1 (G)/2, where M 1 (G) is the first Zagreb index [13]. Also, it is relevant to stress that the mean Sombor index is related to several Sombor indices: [15], and mSO α (G) [16]. In addition, the limit cases mSO α→±∞ (G) correspond with the limit cases SP α→±∞ (G) of the recently introduced Stolarsky-Puebla index [17].
In Table 2 we report some expressions for mSO α (G) for selected values of α that we identify with known topological indices.

QSPR study of mSO α (G) on octane isomers
As a first application of mean Sombor indices, here we perform a quantitative structure property relationship (QSPR) study of mSO α (G) to model some physicochemical properties of octane isomers. Here we choose to study the following properties: acentric factor (AcentFac), boiling point (BP), heat capacity at constant pressure (HCCP), critical temperature (CT), relative density (DENS), standard enthalpy of formation (DHFORM), standard enthalpy of vaporization (DHVAP), enthalpy of formation (HFORM), heat of vaporization (HV) at 25 • C, enthalpy of vaporization (HVAP), and entropy (S).
The experimental values of the physicochemical properties of the octane isomers were kindly provided by Dr. S. Mondal, see Table 2 in Ref. [18].     Table 3.
In Fig. 1 we plot mSO α (G) vs. the physicochemical properties of octane isomers for the values of α that maximize the absolute value of Pearson's correlation coefficient r; see Table 3. Moreover, in Fig. 1 we tested the following linear regression model where P represents a given physicochemical property. In Table 3 we resume the regression and statistical parameters of the linear QSPR models (see the red dashed lines in Fig. 1) given by Eq. (5). From Table 3 we can conclude that mSO α (G) provides good predictions of AcentFac, BP, HCCP, DHVAP, HFORM, HV, HVAP, and S for which the correlation coefficients (absolute values) are closer or higher than 0.9. Note that for all these physicochemical properties of octane isomers the statistical significance of the linear model of Eq. (5) is far below 5%. Also notice that the mean Sombor index that better correlates (linearly) with the AcentFac is mSO α→0 (G), which indeed coincides with the reciprocal Randic index. Moreover, we found that |r| is maximized when α → ∞, for DHVAP and HVAP, and when α → −∞ for HV. This means that the limiting cases mSO α→±∞ (G) are also relevant from an application point of view.  Fig. 1: values of α that maximize the absolute value of Pearson's correlation coefficient r. c 2 , c 1 , SE, F , and SF are the intercept, slope, standard error, F -test, and statistical significance, respectively, of the linear QSPR models of Eq. (5

Inequalities involving mSO α (G)
Equation (2) can be straightforwardly used to state a monotonicity property for the mSO α (G) index, as well as inequalities for related indices. That is, if α 1 < α 2 we have, which implies, for the the first (a, b) − KA index, that and moreover Note that this last inequality involves the inverse sum indeg index, the reciprocal Randic index, the (a, b) − KA index, the first Zagreb index, and the Sombor index. It is fair to acknowledge that the very last inequality in (8) was already included in the Theorem 3.1 of [19].
In what follows we will state bounds for the mean Sombor index as well as new relationships with known topological indices.
We will use the following particular case of Jensen's inequality.
Lemma 4.1. If g is a convex function on R and x 1 , . . . , x m ∈ R, then If g is strictly convex, then the equality is attained in the inequality if and only if x 1 = · · · = x m .
Theorem 4.1. Let G be a graph with m edges and α ∈ R; if α > 1 then

and the equality in each bound is attained for a connected graph G if and only if G is regular or biregular.
Proof. Assume first that α > 1 then, for x ≥ 0, x 1/α is a concave function and by Lemma 4.1 we have Assume now that α < 1 and α = 0, then x 1/α is a convex function and by Jensen's inequality we obtain If G is regular or biregular, with maximum and minimum degrees ∆ and δ, respectively, If any of these equalities hold, for every uv, so all the neighbors of a vertex u ∈ V (G) have the same degree. Thus, since G is a connected graph, G is regular or biregular.
In order to prove the next result we need an additional technical result. In [1, Theorem 3] appears a converse of Hölder inequality, which in the discrete case can be stated as follows [1, Corollary 2]. Lemma 4.2. If 1 < p, q < ∞ with 1/p + 1/q = 1, x j , y j ≥ 0 and ay q j ≤ x p j ≤ by q j for 1 ≤ j ≤ k and some positive constants a, b, then: If x j > 0 for some 1 ≤ j ≤ k, then the equality in the bound is attained if and only if a = b and x p j = ay q j for every 1 ≤ j ≤ k.
Theorem 4.2. Let G be a graph with m edges, maximum degree ∆ and minimum degree δ, let 0 < α < 1, then

the equality holds if and only if G is a regular graph.
Proof. For each uv ∈ E(G) we have If we take x j = d α u , y j = d α v and p = 1/α by Lemma 4.2 we have , if 1 2 < α < 1 , and the equality holds if and only if δ = ∆, i.e., G is regular.
The following inequalities are known for x, y > 0: x a + y a < (x + y) a ≤ 2 a −1 (x a + y a ) if a > 1, 2 a−1 (x a + y a ) ≤ (x + y) a < x a + y a if 0 < a < 1, if a < 0, (9) and the equality in the second, third or fifth bound is attained for each a if and only if x = y.
Proposition 4.1. Let G be a graph and α ∈ R\{0}, then Proof. If we divide each one of the inequalities in (9) by 2 a we obtain If we take x = d α u , y = d α v and a = 2/α; then the previous inequalities give if α < 0, and the equality in the second, third or fifth bounds are attained for each a if and only if d u = d v . From this we obtain The following result appears in [4].
Proposition 4.2. If G is a graph with m edges, then Proof. If we take a i = d α u + d a v and r = β, by Lemma 4.3 we have Given a graph G, let us define the mean Sombor matrix mSM α (G) with entries One can easily check the following result about the trace of the matrix mSM α (G) 2 : Denote by σ 2 the variance of the sequence of the terms  Proof. By the definition of σ 2 , we have then using the expression (11) we have , and the result follows from this equality. Proof. Let be δ, ∆ the minimum and maximum degree of G, respectively. Let's analyze the behavior of the function so f is a decreasing function for each y. Thus, we have f (x, y) ≥ f (y, y) = 0, so and the equality is attained if and only if x = y. Therefore for any uv ∈ E(G), 2 and the equality is attained if and only if d u = d v . The desired result is obtained by adding up for each uv ∈ E(G).

Discussion and conclusions
We have introduced a degree-based variable topological index inspired on the power mean (also known as generalized mean and Hölder mean). We named this new index as the mean Sombor index mSO α (G), see Eq. (4). For given values of α, the mean Sombor index is related to well-known topological indices, in particular with several Sombor indices.
In addition, through a QSPR study, we showed that mean Sombor indices are suitable to model acentric factor, boiling point, heat capacity at constant pressure, standard enthalpy of vaporization, enthalpy of formation, heat of vaporization at 25 • C, enthalpy of vaporization, and entropy of octane isomers; see Section 3.
We have also discussed some mathematical properties of mean Sombor indices as well as stated bounds and new relationships with known topological indices; see Section 4, where the mean Sombor matrix was also introduced in Eq. (10).
Finally, we would like to remark that, in addition to all the known indices that the mean Sombor index reproduces, we discover the indices and which, from the QSPR study of Section 3, were shown to be good predictors of the standard enthalpy of vaporization, the enthalpy of vaporization, and the heat of vaporization at 25 • C of octane isomers. It is fair to mention that several known topological indices include the min/max functions; among them we can mention the min-max (and max-min) rodeg index, the min-max (and max-min) sdi index, and the min-max (and max-min) deg index, introduced in Ref. [10]. However, to the best of our knowledge, the indices mSO ±∞ (G) have not been theoretically studied before (for an exception where the equivalent Stolarsky-Puebla indices have been computationally applied to random networks see [17]). Thus, we believe that a theoretical study of these two new indices is highly pertinent.