scholarly journals New Results on the Geometric-Arithmetic Index

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Akbar Jahanbani ◽  
Maryam Atapour ◽  
Zhibin Du
Keyword(s):  
Upper Bounds ◽  
Topological Indices ◽  
Lower And Upper Bounds ◽  
Degree Of Vertex ◽  
Vertex Set

Let G be a graph with vertex set V G and edge set E G . Let d u denote the degree of vertex u ∈ V G . The geometric-arithmetic index of G is defined as GA G = ∑ u v ∈ E G 2 d u d v / d u + d v . In this paper, we obtain some new lower and upper bounds for the geometric-arithmetic index and improve some known bounds. Moreover, we investigate the relationships between geometric-arithmetic index and several other topological indices.

scholarly journals On the first Zagreb index and multiplicative Zagreb coindices of graphs

2016 ◽  
Vol 24 (1) ◽  
pp. 153-176 ◽  
Author(s):  
Kinkar Ch. Das ◽  
Nihat Akgunes ◽  
Muge Togan ◽  
Aysun Yurttas ◽  
I. Naci Cangul ◽  
...  
Keyword(s):  
Degree Sequence ◽  
Molecular Graph ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Irregularity Index ◽  
Vertex Set ◽  
The First Zagreb Index

AbstractFor a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as, where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariantsandnamed as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) =. The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

scholarly journals The Bounds of the Edge Number in Generalized Hypertrees

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 2 ◽  
Author(s):  
Ke Zhang ◽  
Haixing Zhao ◽  
Zhonglin Ye ◽  
Yu Zhu ◽  
Liang Wei
Keyword(s):  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Number Of Components ◽  
Uniform Hypergraphs ◽  
Vertex Set

A hypergraph H = ( V , ε ) is a pair consisting of a vertex set V , and a set ε of subsets (the hyperedges of H ) of V . A hypergraph H is r -uniform if all the hyperedges of H have the same cardinality r . Let H be an r -uniform hypergraph, we generalize the concept of trees for r -uniform hypergraphs. We say that an r -uniform hypergraph H is a generalized hypertree ( G H T ) if H is disconnected after removing any hyperedge E , and the number of components of G H T − E is a fixed value k   ( 2 ≤ k ≤ r ) . We focus on the case that G H T − E has exactly two components. An edge-minimal G H T is a G H T whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r -uniform G H T on n vertices has at least 2 n / ( r + 1 ) edges and it has at most n − r + 1 edges if r ≥ 3   and   n ≥ 3 , and the lower and upper bounds on the edge number are sharp. We then discuss the case that G H T − E has exactly k   ( 2 ≤ k ≤ r − 1 ) components.

scholarly journals On the $P_3$-Hull Number of Kneser Graphs

10.37236/9903 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Luciano N. Grippo ◽  
Adrián Pastine ◽  
Pablo Torres ◽  
Mario Valencia-Pabon ◽  
Juan C. Vera
Keyword(s):  
Upper Bounds ◽  
Minimum Size ◽  
Lower And Upper Bounds ◽  
Kneser Graph ◽  
Graph Homomorphisms ◽  
The Family ◽  
Vertex Set ◽  
Kneser Graphs

This paper considers an infection spreading in a graph; a vertex gets infected if at least two of its neighbors are infected. The $P_3$-hull number is the minimum size of a vertex set that eventually infects the whole graph. In the specific case of the Kneser graph $K(n,k)$, with $n\ge 2k+1$, an infection spreading on the family of $k$-sets of an $n$-set is considered. A set is infected whenever two sets disjoint from it are infected. We compute the exact value of the $P_3$-hull number of $K(n,k)$ for $n>2k+1$. For $n = 2k+1$, using graph homomorphisms from the Knesser graph to the Hypercube, we give lower and upper bounds.

Signed domination number of hypercubes

2021 ◽  
Author(s):  
B. ShekinahHenry ◽  
Y. S. Irine Sheela
Keyword(s):  
Domination Number ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Signed Domination ◽  
Vertex Set ◽  
Number Formula ◽  
Signed Domination Number

The [Formula: see text]-cube graph or hypercube [Formula: see text] is the graph whose vertex set is the set of all [Formula: see text]-dimensional Boolean vectors, two vertices being joined if and only if they differ in exactly one co-ordinate. The purpose of the paper is to investigate the signed domination number of this hypercube graphs. In this paper, signed domination number [Formula: see text]-cube graph for odd [Formula: see text] is found and the lower and upper bounds of hypercube for even [Formula: see text] are found.

scholarly journals On the Distinguishing Number of Functigraphs

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 332 ◽  
Author(s):  
Muhammad Fazil ◽  
Muhammad Murtaza ◽  
Zafar Ullah ◽  
Usman Ali ◽  
Imran Javaid
Keyword(s):  
Upper Bounds ◽  
Complete Graphs ◽  
Lower And Upper Bounds ◽  
Distinguishing Number ◽  
Vertex Set

Let G 1 and G 2 be disjoint copies of a graph G and g : V ( G 1 ) → V ( G 2 ) be a function. A functigraph F G consists of the vertex set V ( G 1 ) ∪ V ( G 2 ) and the edge set E ( G 1 ) ∪ E ( G 2 ) ∪ { u v : g ( u ) = v } . In this paper, we extend the study of distinguishing numbers of a graph to its functigraph. We discuss the behavior of distinguishing number in passing from G to F G and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs.

Bounds for signed double Roman k-domination in trees

2019 ◽  
Vol 53 (2) ◽  
pp. 627-643 ◽  
Author(s):  
Hong Yang ◽  
Pu Wu ◽  
Sakineh Nazari-Moghaddam ◽  
Seyed Mahmoud Sheikholeslami ◽  
Xiaosong Zhang ◽  
...  
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Finite Graph ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Vertex Set ◽  
Dominating Function

Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V(G). A signed double Roman k-dominating function (SDRkDF) on a graph G is a function f:V(G) → {−1,1,2,3} such that (i) every vertex v with f(v) = −1 is adjacent to at least two vertices assigned a 2 or to at least one vertex w with f(w) = 3, (ii) every vertex v with f(v) = 1 is adjacent to at least one vertex w with f(w) ≥ 2 and (iii) ∑u∈N[v]f(u) ≥ k holds for any vertex v. The weight of a SDRkDF f is ∑u∈V(G) f(u), and the minimum weight of a SDRkDF is the signed double Roman k-domination number γksdR(G) of G. In this paper, we investigate the signed double Roman k-domination number of trees. In particular, we present lower and upper bounds on γksdR(T) for 2 ≤ k ≤ 6 and classify all extremal trees.

scholarly journals The general Albertson irregularity index of graphs

2021 ◽  
Vol 7 (1) ◽  
pp. 25-38
Author(s):  
Zhen Lin ◽  
◽  
Ting Zhou ◽  
Xiaojing Wang ◽  
Lianying Miao ◽  
...  
Keyword(s):  
Real Number ◽  
Connected Graph ◽  
Positive Real Number ◽  
Upper Bounds ◽  
Calculation Formula ◽  
Lower And Upper Bounds ◽  
Positive Real ◽  
Irregularity Index ◽  
Degree Of Vertex ◽  
Extremal Value

<abstract><p>We introduce the general Albertson irregularity index of a connected graph $ G $ and define it as $ A_{p}(G) = (\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}} $, where $ p $ is a positive real number and $ d(v) $ is the degree of the vertex $ v $ in $ G $. The new index is not only generalization of the well-known Albertson irregularity index and $ \sigma $-index, but also it is the Minkowski norm of the degree of vertex. We present lower and upper bounds on the general Albertson irregularity index. In addition, we study the extremal value on the general Albertson irregularity index for trees of given order. Finally, we give the calculation formula of the general Albertson index of generalized Bethe trees and Kragujevac trees.</p></abstract>

scholarly journals On the Randić Index of Corona Product Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Ismael G. Yero ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Degree Sequence ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Regular Graphs ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Randic Index ◽  
Product Graphs ◽  
Vertex Set ◽  
Corona Product

Let G be a graph with vertex set V=(v1,v2,…,vn). Let δ(vi) be the degree of the vertex vi∈V. If the vertices vi1,vi2,…,vih+1 form a path of length h≥1 in the graph G, then the hth order Randić index Rh of G is defined as the sum of the terms 1/δ(vi1)δ(vi2)⋯δ(vih+1) over all paths of length h contained (as subgraphs) in G. Lower and upper bounds for Rh, in terms of the vertex degree sequence of its factors, are obtained for corona product graphs. Moreover, closed formulas are obtained when the factors are regular graphs.

scholarly journals On inverse degree and topological indices of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2111-2120 ◽  
Author(s):  
Kinkar Das ◽  
Kexiang Xu ◽  
Jinlan Wang
Keyword(s):  
Maximum Degree ◽  
Minimum Degree ◽  
Upper Bounds ◽  
Topological Indices ◽  
Simple Graph ◽  
Lower And Upper Bounds ◽  
Inverse Degree ◽  
Abc Index

Let G=(V,E) be a simple graph of order n and size m with maximum degree ? and minimum degree ?. The inverse degree of a graph G with no isolated vertices is defined as ID(G) = ?n,i=1 1/di, where di is the degree of the vertex vi?V(G). In this paper, we obtain several lower and upper bounds on ID(G) of graph G and characterize graphs for which these bounds are best possible. Moreover, we compare inverse degree ID(G) with topological indices (GA1-index, ABC-index, Kf-index) of graphs.

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 .

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