111 research outputs found
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
Asymptotic distribution of degree--based topological indices
Topological indices play a significant role in mathematical chemistry. Given
a graph with vertex set and edge
set , let be the degree of node . The degree-based
topological index is defined as , where 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 for a constant exhibits a
phase change at
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
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
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
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
and scalar functions 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 . 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
Geometry and complexity of O'Hara's algorithm
In this paper we analyze O'Hara's partition bijection. We present three type
of results. First, we show that O'Hara's bijection can be viewed geometrically
as a certain scissor congruence type result. Second, we obtain a number of new
complexity bounds, proving that O'Hara's bijection is efficient in several
special cases and mildly exponential in general. Finally, we prove that for
identities with finite support, the map of the O'Hara's bijection can be
computed in polynomial time, i.e. much more efficiently than by O'Hara's
construction.Comment: 20 pages, 4 figure
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