Prime Graph Research Articles

Prime Labeling of Union of Some Graphs

A prime labeling of a graph G is a map from the vertex set of G to the set {1, 2, ..., |V (G)|} such that any two adjacent vertices in the graph G have labels that are relatively prime. In this paper, we discuss when the disjoint union of some graphs is a prime graph.

Weak Randomness in Graphons and Theons

ABSTRACTCall a hereditary family of graphs strongly persistent if there exists a graphon such that in all subgraphons of , is precisely the class of finite graphs that have positive density in . Our first result is a complete characterization of the hereditary families of graphs that are strongly persistent as precisely those that are closed under substitutions. We call graphons with the self‐similarity property above weakly random. A hereditary family is said to have the weakly random Erdős–Hajnal property () if every graphon that is a limit of graphs in has a weakly random subgraphon. Among families of graphs that are closed under substitutions, we completely characterize the families that belong to as those with “few” prime graphs. We also extend some of the results above to structures in finite relational languages by using the theory of theons.

Prime graphs on the set of vanishing class sizes

Prime graphs on the set of vanishing class sizes

Subring in Graph Theory

In our study, we delved into the intricacies of graph theory by exploring the properties of subrings within various types of graphs. By focusing on prime graphs and simple graphs, we unraveled the complex relationship between subring-prime graphs. Additionally, we delved into the concept of homomorphism within both simple subring graphs and prime subring graphs, adding depth to our analysis.

The Wiener Index of Prime Graph $PG(\mathbb{Z}_n)$

Let G = (V, E) be a simple graph and (R, +, ·) be a ring with zero element 0R. The Wiener index of G, denoted by W(G), is defined as the sum of distances of every vertex u and v, or half of the sum of all entries of its distance matrix. The prime graph of R, denoted by P G(R), is defined as a graph with V (P G(R)) = R such that uv ∈ E(P G(R)) if and only if uRv = {0R} or vRu = {0R}. In this article, we determine the Wiener index of P G(Zn) in some cases n by constructing its distance matrix. We partition the set Zn into three types of sets, namely zero sets, nontrivial zero divisor sets, and unit sets. There are two objectives to be achieved. Firstly,we revise the Wiener index formula of P G(Zn) for n = p2 and n = p3 for prime number p and we compare this results with the results carried out by previous researchers. Secondly, we determine the Wiener index formula of P G(Zn) for n = pq, n = p2q, n =p 2q2, and n = pqr for distinct prime numbers p, q, and r.

Laplacian Spectrum and Energy of the Cyclic Order Product Prime Graph of Semi-dihedral Groups

This paper introduces cyclic order product prime graphs to examine the spectral graph properties of semi-dihedral groups, specifically focusing on the Laplacian spectrum and energy. Combining principles from cyclic graphs and order product prime graphs enhances understanding of group algebraic structures. In this context, the cyclic order product prime graph for a finite group is defined as one where two distinct vertices are adjacent if their generated subgroup is cyclic, and the product of their orders is a prime power. Our methodological approach begins with establishing a general presentation for these graphs within semi-dihedral groups. This foundational step is essential for deriving some properties, such as vertex degrees, the number of edges, and Laplacian characteristic polynomials. This information subsequently facilitates the determination of the Laplacian spectrum, characterized by seven eigenvalues of various multiplicities, and the computation of their Laplacian energy. The semi-dihedral group of order 16 is a particular example to illustrate the practicality and generality of our theorems.

On Distance Average Degree Independent Resolving Sets of Some Algebraic Graphs

Objectives: This study introduces the concept of distance average degree independent resolving sets and its dimensions for identity graphs of finite groups and order prime graphs of finite groups as a theorem in detail. Methods: The methodology involves first constructing the subset from the given graph, then finding the distance between each and every vertice of the graph and the subset, and also finding the average of the distance such that each and every vertice should receive a distinct code. The minimum number of vertices that satisfy this condition, their cardinality, is called the dimension. And then check various conditions like adjacency, equitability, independence, etc. in order to find various types of dimensions for the considered graphs. Findings: In this article, the concept of distance average degree resolving sets has been found, and further distance average degree independent resolving sets have been found, with their corresponding dimensions also found for algebraic graphs like identity graphs of finite groups and order prime graphs of finite groups. Novelty: The novelty in this article is the concept of distance-average-degree independent resolving sets. Particularly, this concept has been applied to the algebraic graphs, namely the identity graph and the order prime graph of a finite group. In addition to that, the concept of distance has been added to this resolving sets area, which is totally new to this area of graph theory. AMS Subject Classification: 05C12, 05C50. Keywords: Resolving set, Independent set, Distance average degree independent resolving set, Identity graphs, Order prime graphs

Extending Undirected Graph Techniques to Directed Graphs via Category Theory

We use Category Theory to construct a ‘bridge’ relating directed graphs with undirected graphs, such that the notion of direction is preserved. Specifically, we provide an isomorphism between the category of simple directed graphs and a category we call ‘prime graphs category’; this has as objects labeled undirected bipartite graphs (which we call prime graphs), and as morphisms undirected graph morphisms that preserve the labeling (which we call prime graph morphisms). This theoretical bridge allows us to extend undirected graph techniques to directed graphs by converting the directed graphs into prime graphs. To give a proof of concept, we show that our construction preserves topological features when applied to the problems of network alignment and spectral graph clustering.

Independence Number of Order Prime Graph

Objective: To explore the algebraic structures inherent in groups and their relationship to graph theory, particularly focusing on the process of converting a group into a graph. Method: Let be a group. The Order Prime Graph of is a graph with and two distinct vertices and are adjacent in if and only if In this paper, we characterize the structure of order prime graphs of some groups and also we find the independence number and clique number of the order prime graph. Finding: In the present article, we characterized the structure of order prime graphs of some groups and also we found the independence number and clique number of the order prime graph. Novelty: Introducing innovative methods for converting groups into graphs, departing from existing approaches, could be a key novelty. This might involve novel ways of encoding group elements as vertices and group operations as edges, potentially leveraging insights from other mathematical fields or computational techniques. Keywords: Order Prime graph, Finite group, Clique number, Independence number, Prime number

The adjacency spectra of some families of minimally connected prime graphs

The adjacency spectra of some families of minimally connected prime graphs

Two results on character codegrees

Let [Formula: see text] be a finite group and [Formula: see text] be the set of irreducible characters of [Formula: see text]. The codegree of an irreducible character [Formula: see text] of the group [Formula: see text] is defined as [Formula: see text]. In this paper, we study two topics related to the character codegrees. The first result is related to the prime graph of character codegrees and we show that the codegree prime graphs of several classes of groups can be characterized only by graph theoretical terms. The second result is about the [Formula: see text]-parts of the codegrees and character degrees.

Prime labeling of graphs constructed from wheel graph

A prime labeling of a simple undirected graph G is to assign unique integer labels from the set {1,2,...,|V(G)|} to each vertex such that any two adjacent vertices in the graph have labels that are relatively prime. Studying prime labeling in graphs can help us understand the structure and properties of graphs and prime labeling has potential applications in cryptography and network security. In this paper we investigate when some graphs that are constructed from wheels are prime graphs.

Eigenvalues of Relatively Prime Graphs Connected with Finite Quasigroups

Eigenvalues of Relatively Prime Graphs Connected with Finite Quasigroups

The prime graphs of groups with arithmetically small composition factors

The prime graphs of groups with arithmetically small composition factors

Prime vertex-minors of a prime graph

A graph is prime if it does not admit a partition (A,B) of its vertex set such that min{|A|,|B|}≥2 and the rank of the A×B submatrix of its adjacency matrix is at most 1. A vertex v of a graph is non-essential if at least two of the three kinds of vertex-minor reductions at v result in prime graphs.In 1994, Allys proved that every prime graph with at least four vertices has a non-essential vertex unless it is locally equivalent to a cycle graph. We prove that every prime graph with at least four vertices has at least two non-essential vertices unless it is locally equivalent to a cycle graph. As a corollary, we show that for a prime graph G with at least six vertices and a vertex x, there is a vertex v≠x such that G∖v or G∗v∖v is prime, unless x is adjacent to all other vertices and G is isomorphic to a particular graph on odd number of vertices.Furthermore, we show that a prime graph with at least four vertices has at least three non-essential vertices, unless it is locally equivalent to a graph consisting of at least two internally-disjoint paths between two fixed distinct vertices having no common neighbors. We also prove analogous results for pivot-minors.

Prime Graph Generation through Single Edge Addition: Characterizing a Class of Graphs

A graph G consists of a finite set V (G) of vertices with a collection E(G) of unordered pairs of distinct vertices called edge set of G. Let G be a graph. A set M of vertices is a module of G if, for vertices x and y in M and each vertex z outside M, {z, x} ∈ E(G) ⇐⇒ {z, y} ∈ E(G). Thus, a module of G is a set M of vertices indistinguishable by the vertices outside M. The empty set, the singleton sets and the full set of vertices represent the trivial modules. A graph is indecomposable if all its modules are trivial, otherwise it is decomposable. Indecomposable graphs with at least four vertices are prime graphs. The introduction and the study of the construction of prime graphs obtained from a given decomposable graph by adding one edge constitue the central points of this paper.

Applications on color (distance) signless laplacian energy of annihilator monic prime graph of commutative rings

In this study, we define the structure formation of the annihilator monic prime graph of commutative rings, whose distinct vertices X and J satisfies a condition annXJ≠annX⋃ann(J), graph is denoted by AMPG(Zn[x]/〈fx〉). The chromatic number of the annihilator monic prime graph is determined to establish the color (signless laplacian) energy of the graph. Also, we introduce color based energy of the annihilator monic prime graph, namely color distance signless Laplacian energy. Diag(Tr)+D(G)+Aχ, where Diag(Tr) denotes the diagonal matrix of the vertex transmissions in the graph, D(G) is the distance matrix of vertices whose diagonals are zero, the entries of the matrix Aχ are 1 if ηi and ηj are adjacent, −1 if ηi and ηj are non-adjacent with different colors, otherwise 0. The eigenvalues of color distance signless Laplacian matrix will be denoted by ∂1χ+≤∂2χ+≤∂3χ+≤⋯≤∂nχ+, then DSLχ+ of annihilator monic prime graph are define as EDSLχ+Znx〈fx〉=∑1n′∂iχ+-tG where t(G) is the transmission of the graph. Based on the eigenvalues, we can study some more properties of graphs. Applications of color based energy are also discuss in this paper.

PRIME LABELLING FOR SOME BIPARTIATE RELATED GRAPHS

A graph G =(V,E) with ‘n’ vertices is said to have a prime labeling if its vertices are labelled with distinct positive integers not exceeding n such that for each pair of adjacent vertices are relatively prime. A graph G which admits prime labeling is called a prime graph. In this paper, we investigate prime labeling for some bipartiate and cycle related graphs. We also discuss the prime labeling of some graph operation namely joint sum and path joining of bipartiate and cycle graphs

A classification of the prime graphs of pseudo-solvable groups

Abstract The prime graph Γ ⁢ ( G ) \Gamma(G) of a finite group 𝐺 (also known as the Gruenberg–Kegel graph) has as its vertices the prime divisors of | G | \lvert G\rvert , and p ⁢ - ⁢ q p\textup{-}q is an edge in Γ ⁢ ( G ) \Gamma(G) if and only if 𝐺 has an element of order p ⁢ q pq . Since their inception in the 1970s, these graphs have been studied extensively; however, completely classifying the possible prime graphs for larger families of groups remains a difficult problem. For solvable groups, such a classification was found in 2015. In this paper, we go beyond solvable groups for the first time and characterize the prime graphs of a more general class of groups we call pseudo-solvable. These are groups whose composition factors are either cyclic or isomorphic to A 5 A_{5} . The classification is based on two conditions: the vertices { 2 , 3 , 5 } \{2,3,5\} form a triangle in Γ ̄ ⁢ ( G ) \overline{\Gamma}(G) or { p , 3 , 5 } \{p,3,5\} form a triangle for some prime p ≠ 2 p\neq 2 . The ideas developed in this paper also lay the groundwork for future work on classifying and analyzing prime graphs of more general classes of finite groups.

Prime Graph over Cartesian Product over Rings and Its Complement

Prime Graph over Cartesian Product over Rings and Its Complement