scholarly journals Lower and Upper Bounds for Hyper-Zagreb Index of Graphs

2020 ◽  
pp. 1401-1406
Author(s):  
G. H. SHIRDEL ◽  
H. REZAPOUR ◽  
R. NASIRI
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Chemical Compounds ◽  
Upper Bounds ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Simple Connected Graph

The topological indices are functions on the graph that do not depend on the labeling of their vertices. They are used by chemists for studying the properties of chemical compounds.  Let  be a simple connected graph. The Hyper-Zagreb index of the graph ,  is defined as  ,where  and  are the degrees of vertex  and , respectively. In this paper, we study the Hyper-Zagreb index and give upper and lower bounds for .

scholarly journals Relations between ordinary and multiplicative degree-based topological indices

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 3031-3042 ◽  
Author(s):  
Ivan Gutman ◽  
Igor Milovanovic ◽  
Emina Milovanovic
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Vertex Degrees ◽  
Simple Connected Graph

Let G be a simple connected graph with n vertices and m edges, and sequence of vertex degrees d1 ? d2 ?...? dn > 0. If vertices i and j are adjacent, we write i ~ j. Denote by ?1, ?*1, Q? and H? the multiplicative Zagreb index, multiplicative sum Zagreb index, general first Zagreb index, and general sumconnectivity index, respectively. These indices are defined as ?1 = ?ni=1 d2i, ?*1 = ?i~j(di+dj), Q? = ?n,i=1 d?i and H? = ?i~j(di+dj)?. We establish upper and lower bounds for the differences H?-m (?1*)?/m and Q?-n(?1)?/2n . In this way we generalize a number of results that were earlier reported in the literature.

scholarly journals Sharp Upper and Lower Bounds of VDB Topological Indices of Digraphs

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1903
Author(s):  
Juan Monsalve ◽  
Juan Rada
Keyword(s):  
Lower Bounds ◽  
Topological Index ◽  
General Setting ◽  
Upper Bounds ◽  
Topological Indices ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Extremal Values

A vertex-degree-based (VDB, for short) topological index φ induced by the numbers φij was recently defined for a digraph D, as φD=12∑uvφdu+dv−, where du+ denotes the out-degree of the vertex u,dv− denotes the in-degree of the vertex v, and the sum runs over the set of arcs uv of D. This definition generalizes the concept of a VDB topological index of a graph. In a general setting, we find sharp lower and upper bounds of a symmetric VDB topological index over Dn, the set of all digraphs with n non-isolated vertices. Applications to well-known topological indices are deduced. We also determine extremal values of symmetric VDB topological indices over OTn and OG, the set of oriented trees with n vertices, and the set of all orientations of a fixed graph G, respectively.

scholarly journals Multiplicative Zagreb indices of cacti

2016 ◽  
Vol 08 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Connected Graph ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Material Chemistry ◽  
Cactus Graph ◽  
Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

scholarly journals Computation of General Zagreb Index of Nanotubes Covered by C5 and C7

2020 ◽  
Vol 11 (1) ◽  
pp. 8001-8008
Keyword(s):  
Carbon Nanotubes ◽  
Connected Graph ◽  
Molecular Graph ◽  
Topological Indices ◽  
Topological Descriptors ◽  
Chemical Bonds ◽  
Graph Invariant ◽  
Zagreb Index ◽  
Physical And Chemical ◽  
Simple Connected Graph

A molecular graph is hydrogen deleted simple connected graph in which vertices and edges are represented by atoms and chemical bonds, respectively. Topological indices are numerical parameters of a molecular graph which characterize its topology and are usually graph invariant. In Mathematical chemistry, topological descriptors play an important role in modeling different physical and chemical activities of molecules. In this study, the generalized Zagreb index for three types of carbon nanotubes is computed. By putting some particular values to the parameters, some important degree-based topological indices are also derived.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (4) ◽  
pp. 929-932 ◽  
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (04) ◽  
pp. 929-932
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

scholarly journals New Bounds for Topological Indices on Trees through Generalized Methods

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1097 ◽  
Author(s):  
Álvaro Martínez-Pérez ◽  
José M. Rodríguez
Keyword(s):  
Large Class ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Chemical Compounds ◽  
Wiener Index ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
Unified Approach ◽  
Wide Family ◽  
New Inequalities

Topological indices are useful for predicting the physicochemical behavior of chemical compounds. A main problem in this topic is finding good bounds for the indices, usually when some parameters of the graph are known. The aim of this paper is to use a unified approach in order to obtain several new inequalities for a wide family of topological indices restricted to trees and to characterize the corresponding extremal trees. The main results give upper and lower bounds for a large class of topological indices on trees, fixing or not the maximum degree. This class includes the first variable Zagreb, the Narumi–Katayama, the modified Narumi–Katayama and the Wiener index.

scholarly journals Spaces with high topological complexity

2014 ◽  
Vol 144 (4) ◽  
pp. 761-773
Author(s):  
Aleksandra Franc ◽  
Petar Pavešić
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Topological Complexity ◽  
Lower And Upper Bounds ◽  
Cw Complex ◽  
The Difference

By a formula of Farber, the topological complexity TC(X) of a (p − 1)-connected m-dimensional CW-complex X is bounded above by (2m + 1)/p + 1. We show that the same result holds for the monoidal topological complexity TCM(X). In a previous paper we introduced various lower bounds for TCM(X), such as the nilpotency of the ring H*(X × X, Δ(X)), and the weak and stable (monoidal) topological complexity wTCM(X) and σTCM(X). In general, the difference between these upper and lower bounds can be arbitrarily large. In this paper we investigate spaces with topological complexity close to the maximal value given by Farber's formula. We show that in these cases the gap between the lower and upper bounds is narrow and TC(X) often coincides with the lower bounds.

scholarly journals A short note on hyper Zagreb index

2017 ◽  
Vol 37 (2) ◽  
pp. 51-58
Author(s):  
Suresh Elumalai ◽  
Toufik Mansour ◽  
Mohammad Ali Rostami ◽  
Gnanadhass Britto Antony Xavier
Keyword(s):  
Maximum Degree ◽  
Minimum Degree ◽  
Short Note ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Second Zagreb Index ◽  
Counter Example ◽  
Inverse Degree

In this paper, we present and analyze the upper and lower bounds on the Hyper Zagreb index $\chi^2(G)$ of graph $G$ in terms of the number of vertices $(n)$, number of edges $(m)$, maximum degree $(\Delta)$, minimum degree $(\delta)$ and the inverse degree $(ID(G))$. In addition, we give a counter example on the upper bound  of the second Zagreb index for Theorems 2.2 and  2.4 from \cite{ranjini}. Finally, we present lower and upper bounds on $\chi^2(G)+\chi^2(\overline G)$, where $\overline G$ denotes the complement of $G$.

scholarly journals First and Second Zagreb Polynomials of VC5C7[p,q] and HC5C7[p,q]Nanotubes

2014 ◽  
Vol 31 ◽  
pp. 56-62 ◽  
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Topological Indices ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
Connectivity Indices ◽  
Anti Inflammatory ◽  
Simple Connected Graph

Let G = (V,E) be a simple connected graph. The sets of vertices and edges of G are denoted by V = V(G) and E = E(G), respectively. There exist many topological indices and connectivity indices in graph theory. The First and Second Zagreb indices were first introduced by Gutman and Trinajstić In1972. It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. In this paper, we focus on the structure of ”G = VC5C7[p,q]”and ”H = HC5C7[p,q]” nanotubes and counting first Zagreb index Zg1(G) = ∑veVdv2 and Second Zagreb index Zg2(G) =∑e=uveE(G)(du·dv) of G and H, as well as First Zagreb polynomial Zg1(G,x ) =∑e=uveE(G)xdu+dv and Second Zagreb Polynomial Zg2(G,x) = ∑e=uveE(G)xdu·dv

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