scholarly journals On Omega Index and Average Degree of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Sadik Delen ◽  
Musa Demirci ◽  
Ahmet Sinan Cevik ◽  
Ismail Naci Cangul
Keyword(s):  
Social Sciences ◽  
Arithmetic Mean ◽  
Average Degree ◽  
Graph Invariant ◽  
Edge Deletion ◽  
Graph Operations ◽  
Vertex Deletion ◽  
Tricyclic Graphs ◽  
Cyclic Graphs ◽  
Vertex Degrees

Average degree of a graph is defined to be a graph invariant equal to the arithmetic mean of all vertex degrees and has many applications, especially in determining the irregularity degrees of networks and social sciences. In this study, some properties of average degree have been studied. Effect of vertex deletion on this degree has been determined and a new proof of the handshaking lemma has been given. Using a recently defined graph index called o m e g a index, average degree of trees, unicyclic, bicyclic, and tricyclic graphs have been given, and these have been generalized to k -cyclic graphs. Also, the effect of edge deletion has been calculated. The average degree of some derived graphs and some graph operations have been determined.

scholarly journals On Degree Properties of Crossing-Critical Families of Graphs

10.37236/7753 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Drago Bokal ◽  
Mojca Bračič ◽  
Marek Derňár ◽  
Petr Hliněný
Keyword(s):  
Average Degree ◽  
Critical Graphs ◽  
Odd Degree ◽  
Vertex Degrees ◽  
Open Question

Answering an open question from 2007, we construct infinite $k$-crossing-critical families of graphs that contain vertices of any prescribed odd degree, for any sufficiently large $k$. To answer this question, we introduce several properties of infinite families of graphs and operations on the families allowing us to obtain new families preserving those properties. This conceptual setup allows us to answer general questions on behaviour of degrees in crossing-critical graphs: we show that, for any set of integers $D$ such that $\min(D)\geq 3$ and $3,4\in D$, and for any sufficiently large $k$, there exists a $k$-crossing-critical family such that the numbers in $D$ are precisely the vertex degrees that occur arbitrarily often in (large enough) graphs of this family. Furthermore, even if both $D$ and some average degree in the interval $(3,6)$ are prescribed, $k$-crossing-critical families exist for any sufficiently large $k$.

A novel graph invariant: The third leap Zagreb index under several graph operations

2019 ◽  
Vol 11 (05) ◽  
pp. 1950054 ◽  
Author(s):  
Durbar Maji ◽  
Ganesh Ghorai
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Lexicographic Product ◽  
Graph Invariant ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
The Third ◽  
Graph Operations ◽  
Strong Product ◽  
Corona Product

The third leap Zagreb index of a graph [Formula: see text] is denoted as [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] and [Formula: see text] are the 2-distance degree and the degree of the vertex [Formula: see text] in [Formula: see text], respectively. The first, second and third leap Zagreb indices were introduced by Naji et al. [A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs, Commun. Combin. Optim. 2(2) (2017) 99–117] in 2017. In this paper, the behavior of the third leap Zagreb index under several graph operations like the Cartesian product, Corona product, neighborhood Corona product, lexicographic product, strong product, tensor product, symmetric difference and disjunction of two graphs is studied.

Functions on adjacent vertex degrees of trees with given degree sequence

2014 ◽  
Vol 12 (11) ◽  
Author(s):  
Hua Wang
Keyword(s):  
Symmetric Function ◽  
Degree Sequence ◽  
Connectivity Index ◽  
Adjacent Vertex ◽  
Randić Index ◽  
Graph Invariant ◽  
Harmonic Index ◽  
Randic Index ◽  
Special Cases ◽  
Vertex Degrees

AbstractIn this note we consider a discrete symmetric function f(x, y) where $$f(x,a) + f(y,b) \geqslant f(y,a) + f(x,b) for any x \geqslant y and a \geqslant b,$$ associated with the degrees of adjacent vertices in a tree. The extremal trees with respect to the corresponding graph invariant, defined as $$\sum\limits_{uv \in E(T)} {f(deg(u),deg(v))} ,$$ are characterized by the “greedy tree” and “alternating greedy tree”. This is achieved through simple generalizations of previously used ideas on similar questions. As special cases, the already known extremal structures of the Randic index follow as corollaries. The extremal structures for the relatively new sum-connectivity index and harmonic index also follow immediately, some of these extremal structures have not been identified in previous studies.

scholarly journals Extremal Values of Randić Index among Some Classes of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Ali Ghalavand ◽  
Ali Reza Ashrafi ◽  
Marzieh Pourbabaee
Keyword(s):  
Upper Bounds ◽  
Simple Graph ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Randic Index ◽  
Chemical Graphs ◽  
Cyclic Graphs ◽  
Vertex Degrees ◽  
Extremal Values

Suppose G is a simple graph with edge set E G . The Randić index R G is defined as R G = ∑ u v ∈ E G 1 / deg G u deg G v , where deg G u and deg G v denote the vertex degrees of u and v in G , respectively. In this paper, the first and second maximum of Randić index among all n − vertex c − cyclic graphs was computed. As a consequence, it is proved that the Randić index attains its maximum and second maximum on two classes of chemical graphs. Finally, we will present new lower and upper bounds for the Randić index of connected chemical graphs.

scholarly journals The effect of edge and vertex deletion on omega invariant

2020 ◽  
pp. 46-46 ◽  
Author(s):  
Sadik Delen ◽  
Muge Togan ◽  
Aysun Yurttas ◽  
Ugur Ana ◽  
Ismail Cangu
Keyword(s):  
Euler Characteristic ◽  
Degree Sequence ◽  
Graph Invariant ◽  
Combinatorial Properties ◽  
Number Of Components ◽  
Vertex Deletion ◽  
The Family

Recently the first and last authors defined a new graph characteristic called omega related to Euler characteristic to determine several topological and combinatorial properties of a given graph. This new characteristic is defined in terms of a given degree sequence as a graph invariant and gives a lot of information on the realizability, number of realizations, connectedness, cyclicness, number of components, chords, loops, pendant edges, faces, bridges etc. of the family of realizations. In this paper, the effect of the deletion of vertices and edges from a graph on omega invariant is studied.

scholarly journals F-Index of some graph operations

2016 ◽  
Vol 08 (02) ◽  
pp. 1650025 ◽  
Author(s):  
Nilanjan De ◽  
Sk. Md. Abu Nayeem ◽  
Anita Pal
Keyword(s):  
Electron Energy ◽  
Topological Index ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Molecular Graphs ◽  
Basic Properties ◽  
Graph Operations ◽  
Physico Chemical ◽  
Vertex Degrees

The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced to study the structure-dependency of total [Formula: see text]-electron energy. But this topological index was not further studied till then. Very recently, Furtula and Gutman [A forgotten topological index,J. Math. Chem. 53(4) (2015) 1184–1190.] reinvestigated the index and named it “forgotten topological index” or “F-index”. In that paper, they present some basic properties of this index and showed that this index can enhance the physico-chemical applicability of Zagreb index. Here, we study the behavior of this index under several graph operations and apply our results to find the F-index of different chemically interesting molecular graphs and nanostructures.

scholarly journals On a Conjecture about the Sombor Index of Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1830
Author(s):  
Kinkar Chandra Das ◽  
Ali Ghalavand ◽  
Ali Reza Ashrafi
Keyword(s):  
Graph Invariant ◽  
Graph Invariants ◽  
Zagreb Indices ◽  
Pendent Vertex ◽  
Vertex Set ◽  
Cyclic Graphs ◽  
Pendent Vertices

Let G be a graph with vertex set V(G) and edge set E(G). A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. The Sombor and reduced Sombor indices of G are two new graph invariants defined as SO(G)=∑uv∈E(G)dG(u)2+dG(v)2 and SOred(G)=∑uv∈E(G)dG(u)−12+dG(v)−12, respectively, where dG(v) is the degree of the vertex v in G. We denote by Hn,ν the graph constructed from the star Sn by adding ν edge(s), 0≤ν≤n−2, between a fixed pendent vertex and ν other pendent vertices. Réti et al. [T. Réti, T Došlić and A. Ali, On the Sombor index of graphs, Contrib. Math.3 (2021) 11–18] proposed a conjecture that the graph Hn,ν has the maximum Sombor index among all connected ν-cyclic graphs of order n, where 0≤ν≤n−2. In some earlier works, the validity of this conjecture was proved for ν≤5. In this paper, we confirm that this conjecture is true, when ν=6. The Sombor index in the case that the number of pendent vertices is less than or equal to n−ν−2 is investigated, and the same results are obtained for the reduced Sombor index. Some relationships between Sombor, reduced Sombor, and first Zagreb indices of graphs are also obtained.

scholarly journals Graphs with maximal irregularity

Filomat ◽  
2014 ◽  
Vol 28 (7) ◽  
pp. 1315-1322 ◽  
Author(s):  
Hosam Abdo ◽  
Nathann Cohen ◽  
Darko Dimitrov
Keyword(s):  
Upper Bound ◽  
Undirected Graph ◽  
Graph Invariant ◽  
Zagreb Index ◽  
Second Zagreb Index ◽  
Chemical Graph Theory ◽  
Degree Of A Vertex ◽  
Maximal Vertex ◽  
Vertex Degrees ◽  
General Graphs

Albertson [3] has defined the P irregularity of a simple undirected graph G = (V,E) as irr(G) =?uv?E |dG(u)- dG(v)|, where dG(u) denotes the degree of a vertex u ? V. Recently, this graph invariant gained interest in the chemical graph theory, where it occured in some bounds on the first and the second Zagreb index, and was named the third Zagreb index [12]. For general graphs with n vertices, Albertson has obtained an asymptotically tight upper bound on the irregularity of 4n3/27: Here, by exploiting a different approach than in [3], we show that for general graphs with n vertices the upper bound ?n/3? ?2n/3? (?2n/3? -1) is sharp. We also present lower bounds on the maximal irregularity of graphs with fixed minimal and/or maximal vertex degrees, and consider an approximate computation of the irregularity of a graph.

scholarly journals THE ROLE OF CHILDREN POEMS ON TV CHANNELS TO ACHIEVEMENT OF CHILDREN MORAL EDUCATION: (TIYOUR EL JANNA CHANNEL AS AN EXAMPLE)

2021 ◽  
Vol 03 (06) ◽  
pp. 236-322
Author(s):  
Elsiddig Abdelsadig Elbadawi BALLA ◽  
Amira Abdelheiy Hassan DRAR
Keyword(s):  
Social Sciences ◽  
Moral Education ◽  
Statistical Significance ◽  
Arithmetic Mean ◽  
Female Teacher ◽  
Validity And Reliability ◽  
Future Studies ◽  
Academic Qualification ◽  
Study Population

This study aimed to explore the role of children poems on achievement of moral education among children with special emphasis on Tiyour El Janna TV Channel, in addition to the likely differences that are of statistical significance among responses of nurseries teachers on the role of children poems broadcasted by Tiyour El Janna TV Channel and their role in moral education of children. The studywas conducted at Rufaa City SUDAN during the year 2020-2021. A descriptive analytical method was adopted, and a questionnaire was designed and tested for validity and reliability. The study sample consisted of (68) female teacher constituting 46% of the study population, randomly selected. The researcher used Statistical Package for Social Sciences (SPSS) for dada analysis. Key results revealed that children poems broadcasted by Tiyour El Janna TV Channel fulfills the objectives of Islamic moral education by an arithmetic mean of (4.37) very good, broadcasting of poems (Tiyour El Janna TV) with an arithmetic mean of (4.51) highly valued. The study revealed statistically significant differences on the role of children poems due to type of nursery for government nursery compared to private nursery. No statistically significant differences were shown dur to academic qualification or experience variables. Accordingly, the study made several recommendations: the importance of encouraging children to follow TV channels which promote their moral education. The study also suggested several future studies to be conducted.

scholarly journals On the Reformulated Second Zagreb Index of Graph Operations

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Durbar Maji ◽  
Ganesh Ghorai ◽  
Yaé Ulrich Gaba
Keyword(s):  
Biological Activities ◽  
Cartesian Product ◽  
Real Numbers ◽  
Zagreb Index ◽  
Second Zagreb Index ◽  
Graph Operations ◽  
Sum Of Products ◽  
Physical Attributes ◽  
Vertex Degrees ◽  
Corona Product

Topological indices (TIs) are expressed by constant real numbers that reveal the structure of the graphs in QSAR/QSPR investigation. The reformulated second Zagreb index (RSZI) is such a novel TI having good correlations with various physical attributes, chemical reactivities, or biological activities/properties. The RSZI is defined as the sum of products of edge degrees of the adjacent edges, where the edge degree of an edge is taken to be the sum of vertex degrees of two end vertices of that edge with minus 2. In this study, the behaviour of RSZI under graph operations containing Cartesian product, join, composition, and corona product of two graphs has been established. We have also applied these results to compute RSZI for some important classes of molecular graphs and nanostructures.

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