Energy and skew-energy of a modified graph

2021 ◽  
Vol 30 (1) ◽  
pp. 41-48
Author(s):  
V. LOKESHA ◽  
Y. SHANTHAKUMARI ◽  
K. ZEBA YASMEEN
Keyword(s):  
Complete Graph ◽  
Scientific Community ◽  
Undirected Graph ◽  
Molecular Chemistry ◽  
Graph Operations ◽  
Skew Energy ◽  
Direct Applicability

Graph energies draw the greater attention of the scientific community due to their direct applicability in molecular chemistry. In this paper, we establish the energy of a graph obtained by the means of some graph operations. The energy of the product of graph Kn × G, where Kn is a complete graph and G is a simple undirected graph and energy of the corresponding digraph are estimated. Further, the duplication graph DG is considered and proved that the energy E(DG) = 2E(G) and E(DGσ) = 2E(Gσ).

A ZERO-DIVISOR GRAPH FOR MODULES WITH RESPECT TO THEIR (FIRST) DUAL

2012 ◽  
Vol 12 (02) ◽  
pp. 1250151 ◽  
Author(s):  
M. BAZIAR ◽  
E. MOMTAHAN ◽  
S. SAFAEEYAN
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Complete Graph ◽  
Projective Module ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Covariant Functor ◽  
Zero Divisor Graph

Let M be an R-module. We associate an undirected graph Γ(M) to M in which nonzero elements x and y of M are adjacent provided that xf(y) = 0 or yg(x) = 0 for some nonzero R-homomorphisms f, g ∈ Hom (M, R). We observe that over a commutative ring R, Γ(M) is connected and diam (Γ(M)) ≤ 3. Moreover, if Γ(M) contains a cycle, then gr (Γ(M)) ≤ 4. Furthermore if ∣Γ(M)∣ ≥ 1, then Γ(M) is finite if and only if M is finite. Also if Γ(M) = ∅, then any nonzero f ∈ Hom (M, R) is monic (the converse is true if R is a domain). For a nonfinitely generated projective module P we observe that Γ(P) is a complete graph. We prove that for a domain R the chromatic number and the clique number of Γ(M) are equal. When R is self-injective, we will also observe that the above adjacency defines a covariant functor between a subcategory of R-MOD and the Category of graphs.

scholarly journals On r-dynamic coloring of some graph operations

2016 ◽  
Vol 1 (1) ◽  
pp. 22 ◽  
Author(s):  
Ika Hesti Agustin ◽  
D. Dafik ◽  
A. Y. Harsya
Keyword(s):  
Natural Number ◽  
Chromatic Number ◽  
Undirected Graph ◽  
Simple Observation ◽  
Graph Operations

Let $G$ be a simple, connected and undirected graph. Let $r,k$ be natural number. By a proper $k$-coloring  of a graph $G$, we mean a map $ c : V (G) \rightarrow S$, where $|S| = k$, such that any two adjacent vertices receive different colors. An $r$-dynamic $k$-coloring is a proper $k$-coloring $c$ of $G$ such that $|c(N (v))| \geq min\{r, d(v)\}$ for each vertex $v$ in $V(G)$, where $N (v)$ is the neighborhood of $v$ and $c(S) = \{c(v) : v \in S\}$ for a vertex subset $S$ . The $r$-dynamic chromatic number, written as $\chi_r(G)$, is the minimum $k$ such that $G$ has an $r$-dynamic $k$-coloring. Note that the $1$-dynamic chromatic number of graph is equal to its chromatic number, denoted by $\chi(G)$, and the $2$-dynamic chromatic number of graph has been studied under the name a dynamic chromatic number, denoted by $\chi_d(G)$. By simple observation it is easy to see that $\chi_r(G)\le \chi_{r+1}(G)$, however $\chi_{r+1}(G)-\chi_r(G)$ can be arbitrarily large, for example $\chi(Petersen)=2, \chi_d(Petersen)=3$, but $\chi_3(Petersen)=10$. Thus, finding an exact values of $\chi_r(G)$ is significantly useful. In this paper, we will show some exact values of $\chi_r(G)$ when $G$ is an operation of special graphs.

scholarly journals Signless Laplacian determinations of some graphs with independent edges

2018 ◽  
Vol 10 (1) ◽  
pp. 185-196 ◽  
Author(s):  
R. Sharafdini ◽  
A.Z. Abdian
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Degree Matrix

Let $G$ be a simple undirected graph. Then the signless Laplacian matrix of $G$ is defined as $D_G + A_G$ in which $D_G$ and $A_G$ denote the degree matrix and the adjacency matrix of $G$, respectively. The graph $G$ is said to be determined by its signless Laplacian spectrum (DQS, for short), if any graph having the same signless Laplacian spectrum as $G$ is isomorphic to $G$. We show that $G\sqcup rK_2$ is determined by its signless Laplacian spectra under certain conditions, where $r$ and $K_2$ denote a natural number and the complete graph on two vertices, respectively. Applying these results, some DQS graphs with independent edges are obtained.

scholarly journals On a Question of Prime Labeling of Graphs

2021 ◽  
pp. 87-93
Author(s):  
A. M. C. U. M. Athapattu ◽  
P. G. R. S. Ranasinghe
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Undirected Graph ◽  
Prime Labeling

In the field of graph theory, the complete graph  of  vertices is a simple undirected graph such that every pair of distinct vertices is connected by a unique edge. In the present work, we introduce planar subgraph  of  with maximal number of edges . Generally,  does not admit prime labeling. We present an algorithm to obtain prime-labeled subgraphs of  . We conclude the paper by stating two conjectures based on labeling of . In particular, the planar subgraph admits anti-magic labeling but does not admit edge magic total labeling.

The zero-divisor graph of a ring with involution

2018 ◽  
Vol 17 (03) ◽  
pp. 1850050 ◽  
Author(s):  
Avinash Patil ◽  
B. N. Waphare
Keyword(s):  
Complete Graph ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Sufficient Conditions ◽  
Left Zero ◽  
Star Graph ◽  
Artinian Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

For a *-ring [Formula: see text], we associate a simple undirected graph [Formula: see text] having all nonzero left zero-divisors of [Formula: see text] as vertices and, two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. In case of Artinian *-rings and Rickart *-rings, characterizations are obtained for those *-rings having [Formula: see text] a complete graph or a star graph, and sufficient conditions are obtained for [Formula: see text] to be connected and also for [Formula: see text] to be disconnected. For a Rickart *-ring [Formula: see text], we characterize the girth of [Formula: see text] and prove a sort of Beck’s conjecture.

scholarly journals Los límites del reduccionismo en Química

2019 ◽  
Vol 3 (1) ◽  
pp. 01
Author(s):  
Martín Labarca
Keyword(s):  
Quantum Mechanics ◽  
Scientific Community ◽  
Chemistry Education ◽  
Historical Perspective ◽  
Scientific Practice ◽  
Close Contact ◽  
Molecular Chemistry ◽  
Scientific Disciplines ◽  
The Relationship

Resumen: La química y la física son disciplinas científicas que abarcan un conjunto específico de teorías, cada una de ellas con sus propios conceptos y leyes. Aunque ambas disciplinas mantienen un estrecho contacto en varios campos, el problema de la relación entre ellas se manifiesta principalmente en los vínculos entre la química molecular y la mecánica cuántica: es en este caso donde generalmente se asume que las entidades químicas, cuando se las analiza en profundidad, no son más que entidades físicas muy complejas. El propósito de este trabajo es mostrar que este supuesto ampliamente compartido no sólo en la comunidad cientifica, sino también, y en general, entre los educadores en química encuentra serios obstáculos cuando se lo analiza a partir de casos concretos de estudio, tanto desde una perspectiva histórica como desde la propia práctica científica.Palabras clave: Reduccionismo; Química molecular; Mecánica cuántica; Educación química. The limits of reductionism in Chemistry Abstract: Chemistry and physics are scientific disciplines that encompass a specific set of theories, each with its own concepts and laws. Although both disciplines maintain a close contact in several fields, the problem of the relationship between them is manifested mainly in the links between molecular chemistry and quantum mechanics: it is in this case that it is generally assumed that chemical entities, when analyzed in depth, are nothing else than extremely complex physical entities. The aim of this work is to show that this assumption widely shared not only in the scientific community, but also, and in general, among chemistry educators, encounters serious obstacles when it is analyzed from concrete cases of study, both from a historical perspective as from the scientific practice itself. Keywords: : Reductionism; Molecular chemistry; Quantum mechanics; Chemistry education.

GRAPHS DETERMINED BY THEIR -GAIN SPECTRA

2020 ◽  
pp. 1-9
Author(s):  
SAI WANG ◽  
DEIN WONG ◽  
FENGLEI TIAN
Keyword(s):  
Complete Graph ◽  
Arbitrary Order ◽  
Undirected Graph ◽  
Gain Spectrum ◽  
Gain Graphs ◽  
Gain Spectra ◽  
Gain Graph

An undirected graph $G$ is determined by its $T$ -gain spectrum (DTS) if every $T$ -gain graph cospectral to $G$ is switching equivalent to $G$ . We show that the complete graph $K_{n}$ and the graph $K_{n}-e$ obtained by deleting an edge from $K_{n}$ are DTS, the star $K_{1,n}$ is DTS if and only if $n\leq 2$ , and an odd path $P_{2m+1}$ is not DTS if $m\geq 2$ . We give an operation for constructing cospectral $T$ -gain graphs and apply it to show that a tree of arbitrary order (at least $5$ ) is not DTS.

scholarly journals On the Skew Spectra of Cartesian Products of Graphs

10.37236/2864 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Cui Denglan ◽  
Hou Yaoping
Keyword(s):  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Bipartite Graphs ◽  
Cartesian Products ◽  
Product Graph ◽  
Underlying Graph ◽  
Skew Energy ◽  
Skew Adjacency Matrix

An oriented graph ${G^{\sigma}}$ is a simple undirected graph $G$ with an orientation, which assigns to each edge of $G$ a direction so that ${G^{\sigma}}$ becomes a directed graph. $G$ is called the underlying graph of ${G^{\sigma}}$ and we denote by $S({G^{\sigma}})$ the skew-adjacency matrix of ${G^{\sigma}}$ and its spectrum $Sp({G^{\sigma}})$ is called the skew-spectrum of ${G^{\sigma}}$. In this paper, the skew spectra of two orientations of the Cartesian products are discussed, as applications, new families of oriented bipartite graphs ${G^{\sigma}}$ with $Sp({G^{\sigma}})={\bf i} Sp(G)$ are given and the orientation of a product graph with maximum skew energy is obtained.

scholarly journals Related graphs of the conjugacy classes of a 3-generator 5-group

2018 ◽  
Vol 14 ◽  
pp. 454-456
Author(s):  
Alia Husna Mohd Noor ◽  
Nor Haniza Sarmin ◽  
Hamisan Rahmat
Keyword(s):  
Graph Theory ◽  
Conjugacy Class ◽  
Group Action ◽  
Equivalence Relation ◽  
Complete Graph ◽  
Undirected Graph ◽  
Conjugacy Classes ◽  
Group Presentation ◽  
Class Graph

The study on conjugacy class has started since 1968. A conjugacy class is defined as an equivalence class under the equivalence relation of being conjugate. In this research, let be a 3-generator 5-group and the scope of the graphs is a simple undirected graph. This paper focuses on the determination of the conjugacy classes of where the set omega is the subset of all commuting elements in the group. The elements of the group with order 5 are identified from the group presentation. The pair of elements are formed in the form of  which is of size two where  and  commute. In addition, the results on conjugacy classes of are applied into graph theory. The determination of the set omega is important in the computation of conjugacy classes in order to find the generalized conjugacy class graph and orbit graph. The group action that is considered to compute the conjugacy classes is conjugation action. Based on the computation, the generalized conjugacy class graph and orbit graph turned out to be a complete graph.

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