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

Topological indices for the antiregular graphs

We determine some classical distance-based and degree-based topo- logical indices of the connected antiregular graphs (maximally irregular graphs). More precisely, we obtain explicitly the k-Wiener index, the hyper-Wiener index, the degree distance, the Gutman index, the first, sec- ond and third Zagreb index, the reduced first and second Zagreb index, the forgotten Zagreb index, the hyper-Zagreb index, the refined Zagreb index, the Bell index, the min-deg index, the max-deg index, the symmet- ric division index, the harmonic index, the inverse sum indeg index, the M-polynomial and the Zagreb polynomial

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)

Asymptotic distribution of degree--based topological indices

Topological indices play a significant role in mathematical chemistry. Given a graph $\mathcal{G}$ with vertex set $\mathcal{V}=\{1,2,\dots,n\}$ and edge set $\mathcal{E}$, let $d_i$ be the degree of node $i$. The degree-based topological index is defined as $\mathcal{I}_n=$ $\sum_{\{i,j\}\in \mathcal{E}}f(d_i,d_j)$, where $f(x,y)$ is a symmetric function. In this paper, we investigate the asymptotic distribution of the degree-based topological indices of a heterogeneous Erd\H{o}s-R\'{e}nyi random graph. We show that after suitably centered and scaled, the topological indices converges in distribution to the standard normal distribution. Interestingly, we find that the general Randi\'{c} index with $f(x,y)=(xy)^{\tau}$ for a constant $\tau$ exhibits a phase change at $\tau=-\frac{1}{2}$

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