On the variable inverse sum deg index

Several important topological indices studied in mathematical chemistry are expressed in the following way Puv∈E(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/(dua + dva), 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 hydrocarbon

Computational and analytical studies of the harmonic index on Erdös-Rényi models

A main topic in the study of topological indices is to find bounds of the indices involving several parameters and/or other indices. In this paper we perform statistical (numerical) and analytical studies of the harmonic index H(G), and other topological indices of interest, on Erdos-Rényi (ER) graphs G(n, p) characterized by n vertices connected independently with probability p ∈ (0, 1). Particularly, in addition to H(G), we study here the (−2) sum-connectivity index χ−2(G), the modified Zagreb index MZ(G), the inverse degree index ID(G) and the Randic index R(G). First, to perform the statistical study of these indices, we define the averages of the normalized indices to their maximum value: {H(G)}, {χ−2(G)}, {MZ(G)}, {ID(G)}, {R(G)}. Then, from a detailed scaling analysis, we show that the averages of the normalized indices scale with the product ξ ≈ np. Moreover, we find two different behaviors. On the one hand, hH(G)i and hR(G)i, as a function of the probability p, show a smooth transition from zero to n/2 as p increases from zero to one. Indeed, after scaling, it is possible to define three regimes: a regime of mostly isolated vertices when ξ 10 (H(G), R(G) ≈ n/2). On the other hand, hχ−2(G)i, hMZ(G)i and hID(G)i increase with p until approaching their maximum value, then they decrease by further increasing p. Thus, after scaling the curves corresponding to these indices display bell-like shapes in log scale, which are symmetric around ξ ≈ 1; i.e. the percolation transition point of ER graphs. Therefore, motivated by the scaling analysis, we analytically (i) obtain new relations connecting the topological indices H, χ−2, MZ, ID and R that characterize graphs which are extremal with respect to the obtained relations and (ii) apply these results in order to obtain inequalities on H, χ−2, MZ, ID and R for graphs in ER models.J.A.M.-B. acknowledges financial support from FAPESP (Grant No. 2019/ 06931-2), Brazil, CONACyT (Grant No. 2019-000009-01EXTV-00067) and PRODEP-SEP (Grant No. 511-6/2019.-11821), Mexico. J.M.R. and J.M.S. acknowledge financial support from Agencia Estatal de Investigación (PID2019-106433GB-I00/AEI/ 10.13039/501100011033), Spain

Bond Additive Modeling 1. Adriatic Indices

Some of the most famous molecular descriptors are bond additive, i.e. they are calculated as the sum of edge contributions (Randić-type indices, Balaban-type indices, Wiener index and its modifications, Szeged index...). In this paper, the methods of calculations of bond contributions of these descriptors are analyzed. The general concepts are extracted, and based on these concepts a large class of molecular descriptors is defined. These descriptors are named Adriatic indices. An especially interesting subclass of these descriptors consists of 148 discrete Adriatic indices. They are analyzed on the testing sets provided by the International Academy of Mathematical Chemistry, and it has been shown that they have good predictive properties in many cases. They can be easily encoded in the computer and it may be of interest to incorporate them in the existing software packages for chemical modeling. It is possible that they could improve various QSAR and QSPR studies

Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures

We consider a class of self-adjoint extensions using the boundary triple technique. Assuming that the associated Weyl function has the special form M(z)=\big(m(z)\Id-T\big) n(z)^{-1} with a bounded self-adjoint operator $T$ and scalar functions $m,n$ we show that there exists a class of boundary conditions such that the spectral problem for the associated self-adjoint extensions in gaps of a certain reference operator admits a unitary reduction to the spectral problem for $T$. As a motivating example we consider differential operators on equilateral metric graphs, and we describe a class of boundary conditions that admit a unitary reduction to generalized discrete laplacians.Comment: 19 page

Mean Sombor index

A Special Volume on Chemical Graph Theory in Memory of Nenad TrinajsticWe introduce a degree–based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: SOα(G) = P uv∈E(G) [(d α u + d α v ) /2]1/α. Here, uv denotes the edge of the graph G connecting the vertices u and v, du is the degree of the vertex u, and α ∈ R\{0}. We also consider the limit cases mSOα→0(G) and SOα→±∞(G). Indeed, for given values of α, the mean Sombor index is related to well-known opological 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 mSOα(G) correlates well with several physicochemical properties of octane isomers. Some mathematical properties of the mean Sombor index as well as bounds and new relationships with known topological indices are also discussed.J.A.M.-B. acknowledges financial support from CONACyT (Grant No. A1-S-22706) and BUAP (Grant No. 100405811VIEP2021) .E.D.M. and J.M.R. were supported by a grant from Agencia Estatal de Investigación (PID 2019-106433GBI00 / AEI / 10.13039 / 501100011033), Spain. J.M.R. was supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the VPRICIT (Regional Programme of Research and Technological Innovation)

RiffleScrambler - a memory-hard password storing function

We introduce RiffleScrambler: a new family of directed acyclic graphs and a corresponding data-independent memory hard function with password independent memory access. We prove its memory hardness in the random oracle model. RiffleScrambler is similar to Catena -- updates of hashes are determined by a graph (bit-reversal or double-butterfly graph in Catena). The advantage of the RiffleScrambler over Catena is that the underlying graphs are not predefined but are generated per salt, as in Balloon Hashing. Such an approach leads to higher immunity against practical parallel attacks. RiffleScrambler offers better efficiency than Balloon Hashing since the in-degree of the underlying graph is equal to 3 (and is much smaller than in Ballon Hashing). At the same time, because the underlying graph is an instance of a Superconcentrator, our construction achieves the same time-memory trade-offs.Comment: Accepted to ESORICS 201

On the Differential Privacy of Bayesian Inference

We study how to communicate findings of Bayesian inference to third parties, while preserving the strong guarantee of differential privacy. Our main contributions are four different algorithms for private Bayesian inference on proba-bilistic graphical models. These include two mechanisms for adding noise to the Bayesian updates, either directly to the posterior parameters, or to their Fourier transform so as to preserve update consistency. We also utilise a recently introduced posterior sampling mechanism, for which we prove bounds for the specific but general case of discrete Bayesian networks; and we introduce a maximum-a-posteriori private mechanism. Our analysis includes utility and privacy bounds, with a novel focus on the influence of graph structure on privacy. Worked examples and experiments with Bayesian na{\"i}ve Bayes and Bayesian linear regression illustrate the application of our mechanisms.Comment: AAAI 2016, Feb 2016, Phoenix, Arizona, United State