Stolarsky-Puebla index

We introduce a degree-based variable topological index inspired on the Stolarsky mean (known as the generalization of the logarithmic mean). We name this new index as the Stolarsky-Puebla index: $SP_\alpha(G) = \sum_{uv \in E(G)} d_u$, if $d_u=d_v$, and $SP_\alpha(G) = \sum_{uv \in E(G)} \left[\left( d_u^\alpha-d_v^\alpha\right)/\left( \alpha(d_u-d_v\right)\right]^{1/(\alpha-1)}$, otherwise. Here, $uv$ denotes the edge of the network $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $\alpha \in \mathbb{R} \backslash \{0,1\}$. Indeed, for given values of $\alpha$, the Stolarsky-Puebla index reproduces well-known topological indices such as the reciprocal Randic index, the first Zagreb index, and several mean Sombor indices. Moreover, we apply these indices to random networks and demonstrate that $\left$, normalized to the order of the network, scale with the corresponding average degree $\left$.

For given values of α, S α (x, y) reproduces known means including the logarithmic mean, when α → 0, and some cases of the power mean [3,4] P M α (x, y) = x α + y α 2 As examples, in Table 1 we show some expressions for S α (x, y) for selected values of α with their corresponding names, when available. Also, there is a well-known inequality relating the Stolarsky mean and the power mean, namely [2,5,6]: or more explicitely √ xy ≤ LogMean(x, y) ≤ x 1/3 + y 1/3 2 where the equality is attained when x = y.

Stolarsky-Puebla index
A large number of graph invariants of the form Table 1: Expressions for the Stolarsky mean S α (x, y) for selected values of α. α S α (x, y) name (when available) −∞ S α→−∞ (x, y) = min(x, y) minimum value, P M α→−∞ (x, y) 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]. Inspired by the Stolarsky mean and given a simple graph G = (V (G), E(G)), here we choose the function F (x, y) in Eq. (5) as the Stolarsky mean S α (x, y) and define the degree-based variable topological index 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 α ∈ R\{0, 1}. We name SP α (G) as the Stolarsky-Puebla index. Note, that for given values of α, SP α (G) is related to widely studied topological indices: [9], and SP 2 (G) = M 1 (G)/2, where M 1 (G) is the first Zagreb index [10]. Also, for selected values of α, SP α (G) reproduces several mean Sombor indices recently introduced in [11]. In Table 2 we report some expressions for SP α (G) for selected values of α that we identify with known topological indices, when applicable.

Computational study of SP α (G) on random networks
As a first test of the Stolarsky-Puebla index, here we apply it on two models of random networks: Erdös-Rényi (ER) networks and random geometric (RG) graphs. ER networks [12][13][14][15] G ER (n, p) are formed by n vertices connected independently with probability p ∈ [0, 1]. While RG graphs [16,17] G RG (n, r) consist of n vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidean distance is less or equal than the connection radius r ∈ [0, √ 2]. We stress that the computational study of the Stolarsky-Puebla index we perform here is justified by the random nature of the network models we want to explore. Since a given parameter set [(n, p) or (n, r)] represents an infinite-size ensemble of random [ER or RG] networks, the computation of SP α (G) on a single network is irrelevant. In contrast, the computation of the average value of SP α (G) on a large ensemble of random networks, all characterized by the same parameter set, may provide useful average information about the full ensemble. This statistical approach, well known in random matrix theory studies, has been recently applied to random networks by means of topological indices, see e.g. [18][19][20]. Moreover, it has been shown that average topological indices may serve as complexity measures equivalent to standard random matrix theory measures [21,22].

SP α (G) on Erdös-Rényi random networks
In what follows we present the average values of selected Stolarsky-Puebla indices. All averages are computed over ensembles of 10 7 /n ER networks characterized by the parameter pair (n, p).
In Fig. 1 we present the average Stolarsky-Puebla index SP α (G ER ) for α → −∞, α → 0, α → 1, and α → ∞ as a function of the probability p of ER networks of sizes n = {125, 250, 500, 1000}. From this figure we observe that the curves of SP α (G ER ) are monotonically increasing functions of p.
We note that in the dense limit, i.e. when np ≫ 1, we can approximate Thus, when np ≫ 1, we can approximate SP α (G ER ) as where we have used |E(G ER )| = n(n − 1)p/2. In Fig. 1, we show that Eq. (9) (dashed lines) indeed describes well the data (thick full curves) for large enough p; except for the case SP α→1 (G ER ) , see Fig. 1(c). We also verified that Eq. (9) describes well the data for other values of α, however we did not include them in Fig. 1 to avoid figure saturation. We also observed that the smaller the value of α the wider the range of p where the coincidence between Eq. (9) and the computational data is observed; compare for example Figs. 1(a) and 1(d), where it is clear that the correspondence of the computational data with Eq. (9) is much better in the case of α → −∞ than for α → ∞. In addition, it is relevant to note that Eq. (9) does not depend on α.
We also notice that in Fig. 1 we present average Stolarsky-Puebla indices as a function of the probability p of ER networks of four different sizes n. It is quite clear from these figures that the curves, characterized by the different network sizes, are very similar but displaced on both axes. This behavior suggests that the average Stolarsky-Puebla indices can be scaled, as will be shown below.  From Eq. (9) we observe that SP α (G ER ) ∝ nf [(n − 1)p)] or Therefore, in Fig. 2 we plot again the average Stolarsky-Puebla indices reported in Fig. 1 network sizes fall on top of each other. Moreover, we can rewrite Eq. (10) as In Fig. 2, we show that Eq. (11) (orange-dashed lines) indeed describe well the data (thick full curves) for d ≥ 10; except for SP α→1 (G ER ) , see Fig. 2

(c).
It is relevant to stress that even when Eq. (10) was expected to be valid in the dense limit (i.e. for d ≫ 1), it is indeed valid for any d as clearly seen in Fig. 2.

SP α (G) on random geometric graphs
As in the previous Subsection, here we present the average values of selected Stolarsky-Puebla indices. Again, all averages are computed over ensembles of 10 7 /n random graphs, each ensemble characterized by a fixed parameter pair (n, r).
In Fig. 3 we present the average Stolarsky-Puebla index SP α (G ER ) for α → −∞, α → 0, α → 1, and α → ∞ as a function of the connection radius r of RG graphs of sizes n = {125, 250, 500, 1000}. For comparison purposes, Fig. 3 is equivalent to Fig. 1. In fact, all the observations made in the previous Subsection for ER networks are also valid for RG graphs by just replacing G ER → G RG and p → g(r), with [23] g(r) = r 2 π − 8 3 r + 1 2 r 2 0 ≤ r ≤ 1 , As well as for ER networks, here, in the dense limit, when nr ≫ 1, we can approximate Therefore, in the dense limit, SP α (G RG ) is well approximated by: In Fig. 3, we show that Eq. (14) (dashed lines) indeed describes well the data (thick full curves) for large enough r; except for the case SP α→1 (G RG ) , see Fig. 3(c). It is quite remarkable to note that by substituting the average degree of Eq. (13) into Eq. (14) we get exactly the same expression of Eq. (11): So, in Fig. 4 we plot again the average Stolarsky-Puebla indices reported in Fig. 3 for RG graphs, but now normalized to n, as a function of d showing that all curves are now properly scaled. Also, in Fig. 4

Discussion and conclusions
We have introduced a degree-based variable topological index inspired on the Stolarsky mean, known as the generalization of the logarithmic mean. We named this new index as the Stolarsky-Puebla index SP α (G), see Eq. (6). For given values of α, the Stolarsky-Puebla index is related to well-known topological indices, in particular it reproduces several mean Sombor indices mSO α (G), see Eq. (7). We want to add that the inequality of Eq. (4) can be straightforwardly used to state inequalities for the indices SP α (G) and mSO α (G), as well as for related indices: or which sets bounds for the logarithmic-mean topological index with respect to the reciprocal Randic index, the mean Sombor index with α = 1/3, and the first Zagreb index.
Since there are not many degree-based topological indices including logarithmic functions (as well-known exceptions we can mention the logarithms of the three multiplicative Zagreb indices [7] and the Adriatic indices [24,25]) we want to highlight the release of the logarithmic-mean topological index LogMean(G) of Eq. (18) as well as the identric-mean index corresponding to SP α→0 (G) and SP α→1 (G), respectively. We have also applied the Stolarsky-Puebla index SP α (G) to Erdös-Rényi (ER) networks and random geometric (RG) graphs and within a statistical random matrix theory approach we demonstrated that SP α (G) , normalized to the order of the network, scales with the corresponding average degree d . However, it is fair to recognize that, for both random network models, SP α→1 (G) = idLogMean(G) did not scale; so we believe that the identric-mean index deserves further investigation.
In addition, from Eq. (16) we are able to write an equivalent inequality but for the corresponding average values: Indeed, we verified that (20) is satisfied for both, ER random networks and RG graphs (not shown here). Moreover, we computationally found that idLogMean(G) ≤ SP α =1 (G) , for the two random network models we study here (not explicitelly shown here but partially observed in Figs. 1 and 3). The equalities in Eqs. (20) and (21) are attained when p = 1 and r = √ 2, for ER random networks and RG graphs, respectively. Finally, we want to recall that through a quantitative structure property relationship (QSPR) analysis it was shown [11] that mSO α→±∞ (G) are good predictors of the standard enthalpy of vaporization, the enthalpy of vaporization, and the heat of vaporization at 25 • C of octane isomers. Furthermore, since SP α→±∞ (G) = mSO α→±∞ (G), we can conclude that SP α→±∞ (G) correlate well with the aforementioned physicochemical properties of octane isomers.
In future works we plan to explore mathematical and computational properties of SP α (G), as well as finding optimal bounds and new relationships with known topological indices.