Estimating the Vertex PI Index

2010 ◽  
Vol 65 (3) ◽  
pp. 240-244 ◽  
Author(s):  
Kinkar Ch Das ◽  
Ivan Gutman
Keyword(s):  
Molecular Structure ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Pi Index ◽  
Structure Descriptor

The vertex PI index is a distance-based molecular structure descriptor, that recently found numerous chemical applications. Lower and upper bounds for PI are obtained, as well as results of Nordhaus-Gaddum type. Also a relation between the Szeged and vertex PI indices is established

scholarly journals Some Extremal Graphs with Respect to Sombor Index

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1202
Author(s):  
Kinkar Chandra Das ◽  
Yilun Shang
Keyword(s):  
Molecular Structure ◽  
Chromatic Number ◽  
Clique Number ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Graph Parameters ◽  
Structure Descriptor

Let G be a graph with set of vertices V(G)(|V(G)|=n) and edge set E(G). Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by SO(G) and is defined as SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of the vertex vi in G. In this paper we present some lower and upper bounds on the Sombor index of graph G in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs.

Estimating the Laplacian Energy-Like Molecular Structure Descriptor

2012 ◽  
Vol 67 (6-7) ◽  
pp. 403-406 ◽  
Author(s):  
Ivan Gutman
Keyword(s):  
Molecular Structure ◽  
Spanning Trees ◽  
Molecular Graph ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Laplacian Energy ◽  
Number Of Spanning Trees ◽  
Structure Descriptor ◽  
Better Than

Lower and upper bounds for the Laplacian energy-like (LEL) molecular structure descriptor are obtained, better than those previously known. These bonds are in terms of number of vertices and edges of the underlying molecular graph and of graph complexity (number of spanning trees)

scholarly journals The Bounds of Vertex Padmakar–Ivan Index on k-Trees

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 324 ◽  
Author(s):  
Shaohui Wang ◽  
Zehui Shao ◽  
Jia-Bao Liu ◽  
Bing Wei
Keyword(s):  
Molecular Structure ◽  
Topological Index ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Structure Descriptor

The Padmakar–Ivan ( P I ) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges u v of a graph such that these vertices are not equidistant from u and v. In this paper, we explore the results of P I -indices from trees to recursively clustered trees, the k-trees. Exact sharp upper bounds of PI indices on k-trees are obtained by the recursive relationships, and the corresponding extremal graphs are given. In addition, we determine the P I -values on some classes of k-trees and compare them, and our results extend and enrich some known conclusions.

scholarly journals Some basic properties of Sombor indices

2021 ◽  
Vol 4 (1) ◽  
pp. 1-3
Author(s):  
Ivan Gutman ◽  
Keyword(s):  
Molecular Structure ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Lower And Upper Bounds ◽  
Basic Properties ◽  
Special Case

The recently introduced class of vertex-degree-based molecular structure descriptors, called Sombor indices (\(SO\)), are examined and a few of their basic properties established. Simple lower and upper bounds for \(SO\) are determined. It is shown that any vertex-degree-based descriptor can be viewed as a special case of a Sombor-type index.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

scholarly journals Relationship between Age and Value of Information for a Noisy Ornstein–Uhlenbeck Process

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 940
Author(s):  
Zijing Wang ◽  
Mihai-Alin Badiu ◽  
Justin P. Coon
Keyword(s):  
Value Of Information ◽  
Markov Models ◽  
Upper Bounds ◽  
Distribution Functions ◽  
Lower And Upper Bounds ◽  
Uhlenbeck Process ◽  
Source Data ◽  
Non Linear ◽  
Queue Model ◽  
Selection Of

The age of information (AoI) has been widely used to quantify the information freshness in real-time status update systems. As the AoI is independent of the inherent property of the source data and the context, we introduce a mutual information-based value of information (VoI) framework for hidden Markov models. In this paper, we investigate the VoI and its relationship to the AoI for a noisy Ornstein–Uhlenbeck (OU) process. We explore the effects of correlation and noise on their relationship, and find logarithmic, exponential and linear dependencies between the two in three different regimes. This gives the formal justification for the selection of non-linear AoI functions previously reported in other works. Moreover, we study the statistical properties of the VoI in the example of a queue model, deriving its distribution functions and moments. The lower and upper bounds of the average VoI are also analysed, which can be used for the design and optimisation of freshness-aware networks. Numerical results are presented and further show that, compared with the traditional linear age and some basic non-linear age functions, the proposed VoI framework is more general and suitable for various contexts.

The Lower and Upper Bounds of Turán Number for Odd Wheels

2021 ◽  
Vol 37 (3) ◽  
pp. 919-932
Author(s):  
Byeong Moon Kim ◽  
Byung Chul Song ◽  
Woonjae Hwang
Keyword(s):  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Turán Number
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