76 Results Found

A graph is a connection of objects. These objects are often known as vertices or nodes and the connection or relation in these nodes are called arcs or edges. There are certain rules to allocate values to these vertices and edges. This allocation of...

Let G be a finite group. The prime graph $\mathsf{\Gamma }\left(G\right)$ of G is defined as follows: The set of vertices of $\mathsf{\Gamma }\left(G\right)$ is the set of prime divisors of $\left|G\right|$ and two distinct vertices p and $p^{\prime }$ are connected in $\mathsf{\&Gamma}$...

Let G be a finite group. In this paper, we study how certain arithmetical conditions on the conjugacy class lengths of real elements of G influence the structure of G. In particular, a new type of prime graph is introduced and studied. We obtain a se...

Let $G=(V,E)$ be a finite simple graph. A module of G is a subset $M\subseteq V$ such that every vertex in $V\setminus M$ is adjacent either to all vertices of M or to none of them. The sets ∅, V, and the singletons of V are called the trivial modules of G....

Let $n\ge 1$ be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of ${\mathbb{Z}}_n$, where $n=p^{\alpha },p^{2}q,p^2{q}^2,pqr,p^{3}q,$$p^{2}qr$, and $pqrs$ f...

Let G be a finite group. The vertex set of the prime-power graph of G is defined as $V\left(G\right)=\left\{p^{e_p\left(G\right)}|p\in \rho \left(G\right)\right\}$, where $\rho \left(G\right)$ is the set of all prime divisors of the degrees of all irreducible characters of G and $p^{e_p\left(G\right)}=max\left\{\psi {\left(1\right)}_p\mid \psi \&isi\right\}$...

Let $G=\left(V\right(G),E(G\left)\right)$ be a simple, finite and undirected graph of order n. Given a bijection $f:V\left(G\right)\to \{1,\dots ,n\}$, and every edge $uv$ in $E\left(G\right)$, let $S=f\left(u\right)+f\left(v\right)$ and $D=\left|f\right(u)-f(v\left)\right|$. The labeling f induces an edge labeling $f^{\prime }:E\left(G\right)\to \{0,1\}$ such that for an edge $uv$ in $E$...

In this paper, we consider a non-standard dynamical inverse problem for the wave equation on a metric tree graph. We assume that the so-called delta-prime matching conditions are satisfied at the internal vertices of the graph. Another specific featu...

A graph G consists of a possibly infinite set $V\left(G\right)$ of vertices with a collection $E\left(G\right)$ of unordered pairs of distinct vertices, called the set of edges of G. Such a graph is denoted by $\left(V\right(G),E(G\left)\right)$. Two distinct vertices u and v of a graph G are a...

The Zeta-Minimizer Theorem establishes that the Riemann zeta function $\zeta \left(s\right)$ and the primes arise variationally as unique minimizers of a phase functional defined on a symmetric measure space $\left(X,\mu ,G\right)$ equipped with helical operators. Three fundament...

A graph is edge-primitive if its automorphism group acts primitively on the edge set of the graph. Edge-primitive graphs form an important subclass of symmetric graphs. In this paper, edge-primitive graphs of order as a product of two distinct primes...

A graph $\Gamma$ is called G-basic if G is quasiprimitive or bi-quasiprimitive on the vertex set of $\Gamma$, where $G⩽\mathsf{Aut}(\Gamma )$. It is known that locally primitive vertex-transitive graphs are normal covers of basic ones. In this paper, a com...

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 c...

Prime structures are link chains with 0 DoF (degrees of freedom), not including subchains with 0 or fewer DoF, which are expected to be used in systematic kinematic and dynamic analyses of link mechanisms. This paper describes the exhaustive enumerat...

A graph is symmetric if its automorphism group is transitive on the arcs of the graph. Guo et al. determined all of the connected seven-valent symmetric graphs of order $8p$ for each prime p. We shall generalize this result by determining all of the co...

The category of directed graphs is isomorphic to a particular category whose objects are labeled undirected bipartite graphs and whose morphisms are undirected graph morphisms that respect the labeling. Based on this isomorphism, we begin by showing...

In this paper, we investigate the graph structures on hoop algebras. First, by using the quasi-filters and r-prime (one-prime) filters, we construct an implicative graph and show that it is connected and under which conditions it is a star or tree. B...

In this article, some properties of the zero-divisor graph $\Gamma \left(Z_n\right)$ are investigated when n is a square-free positive integer. It is shown that the zero-divisor graph $\Gamma \left(Z_n\right)$ of ring $Z_n$ is a $(2^k-2)$-partite graph when the prime decomposit...

The power graph ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ of ${\displaystyle {\mathbb{Z}}_n}$ for a finite cyclic group ${\displaystyle {\mathbb{Z}}_n}$ is a simple undirected connected graph such that two distinct nodes ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle {\mathbb{Z}}_n}$ are adjacent in ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ if and only if ${\displaystyle x\ne y}$ and ${\displaystyle x^i=y}$ or ${\displaystyle y^i=x}$ for some non-negative integer ${\displaystyle i}$. In this article,...

This paper studies a family of $(3,29)$-regular bipartite graphs introduced by Tanner for low-density parity-check (LDPC) codes. For a prime $p\equiv 1(\mathrm{mod}87)$, the graph has $3p$ left vertices and $29p$ right vertices. In the associated LDPC code, the right...

This paper is devoted to the study of the arithmetic graph of a composite number m, denoted by ${\mathcal{A}}_m$ . It has been observed that there exist different composite numbers for which the arithmetic graphs are isomorphic. It is proved that the maximu...

We study the undirected power congruence graph $G_n^k\left(\{p,q\}\right)$ on vertex set $V=\{0,1,\dots ,n-1\}$ in which two distinct vertices $a,b$ are adjacent whenever $a^k\equiv b^k\left(\mathrm{mod}p\right)$ or $a^k\equiv b^k\left(\mathrm{mod}q\right)$, where $p,q$ are distinct primes with $pq<n$ and $k\ge 1$...

Let R be a commutative ring with unity. The unit graph $G\left(R\right)$ is defined such that the vertex set of $G\left(R\right)$ is the set of all elements of R, and two distinct vertices are adjacent if their sum is a unit in $R.$ In this paper, we show that for each prime, $p$...

The Tutte polynomial is an isomorphism invariant of graphs that generalizes the chromatic and the flow polynomials. This two-variable polynomial with integral coefficients is known to carry important information about the properties of the graph. It...

Background: Biomedical Knowledge Graphs (KGs) such as PrimeKG and Hetionet offer vast structured repositories for clinical discovery; however, conventional Graph Neural Networks (GNNs) suffer from black-box opacity, while standard Retrieval-Augmented...

In this paper, we examine the interplay between the structural and spectral properties of the unit graph $G\left({\mathbb{Z}}_n\right)$ for $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$, where $p_1,p_2,\dots ,p_k$ are distinct primes and $k,r_1,r_2,\dots ,r_k$ are positive integers such that at least on...

Chemical graph theory is a branch of mathematical chemistry which deals with the non-trivial applications of graph theory to solve molecular problems. Graphs containing finite commutative rings also have wide applications in robotics, information and...

The zero-divisor graph of a commutative ring $\mathfrak{R}$ with a nonzero identity, denoted by $\Gamma \left(\mathfrak{R}\right)$, is an undirected graph where the vertex set $Z{\left(\mathfrak{R}\right)}^{*}$ consists of all nonzero zero-divisors of $\mathfrak{R}$. Two distinct vertices a and b in $\Gamma \left(\mathfrak{R}\right)$ are adjacent if a...

A graph is said to be symmetric if the action of its automorphism group on the set of arcs is transitive (arc-transitive). Building upon prior work regarding the classification of symmetric graphs of valency five, this article investigates connected...

This work explores the unit graph of the commutative ring ${\mathbb{Z}}_p$ for prime $p$ and explores the significance of the Sombor index and the Sombor polynomial in comprehending its structural details. We systematically investigate the unit graph and the identit...

This work is on the nature and properties of graphs which arise in the study of centered polygonal lacunary functions. Such graphs carry both graph-theoretic properties and properties related to the so-called p-sequences found in the study of centere...

Given a commutative ring R with identity $1\ne 0$, let the set $Z\left(R\right)$ denote the set of zero-divisors and let $Z^{*}\left(R\right)=Z\left(R\right)\setminus \left\{0\right\}$ be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by $\Gamma \left(R\right)$, is a simple graph whose vertex set is $Z^{*}\left(R\right)$...

In this paper, we study the way the symmetries of a given graph are reflected in its characteristic polynomials. Our aim is not only to find obstructions for graph symmetries in terms of its polynomials but also to measure how faithful these algebrai...

We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schrödinger Equation on graphs endowed with vertices where the matching condition, instead of being free (or Kirchhoff&rs...

This paper investigates the enumeration of orientable vertex-transitive embeddings of complete graphs ${\mathsf{K}}_{p^2}$, where $p\ge 5$ is a prime. Building on the classification framework established by Li, we derive a precise count of these embeddings up to non-is...

Magic squares have been widely studied, with publications of mathematical interest dating back over 100 years. Most studies construct and analyse specific subsets of magic squares, with some exploring links to puzzles, number theory, and graph theory...

Radio antennas use different frequency bands of Electromagnetic (EM) Spectrum for switching signals in the forms of radio waves. Regulatory authorities issue a unique number (unique identifying call sign) to each radio center, that must be used in al...

Assume that $\mathcal{G}$ is a finite group. The power graph $\mathcal{P}\left(\mathcal{G}\right)$ of $\mathcal{G}$ is a graph in which $\mathcal{G}$ is its node set, where two different elements are connected by an edge whenever one of them is a power of the other. A topological index is a number generated from a mol...

We construct several new families of directed strongly regular Cayley graphs (DSRCGs) over the metacyclic group $M_{4n}=⟨a,b|a^n=b^4=1,b^{-1}ab=a^{-1}⟩$ , some of which generalize those ea...

The spectrum of a graph $\mathrm{\Gamma }$, denoted by $Spec\left(\mathrm{\Gamma }\right)$, is the multiset of eigenvalues of its adjacency matrix. A Cayley graph $Cay(G,S)$ of a finite group G is called Cay-DS (Cayley graph determined by its spectrum) if, for any other Cayley graph $C$...

A bijective product k-cordial labeling f of a graph G with vertex set V and edge set E is a bijection from V to $\{1,2,\dots ,|V|\}$ such that the induced edge labeling $f^{\times }:E\left(G\right)\to {\mathbb{Z}}_k=\{i|0\le i\le k-1\}$ defined as $f^{\times }\left(uv\right)\equiv f\left(u\right)f(v$...

In this article, we introduce and investigate a novel class of graphs that are called Power Congruence Graph PCGs, which are defined over the vertex set V $=\{0,1,2,\dots ,n-1\}$ where two vertices $a,b\in V$ are adjacent if $a^k\equiv b^k(modm)$ for some modulus $m\in M_p$, wher...

Let $G\left(R\right)$ be the unit graph associated with a ring R. Let p be a prime number and let R be a finite ring of order p or $p^2$ and be one of the rings ${\mathbb{Z}}_p,{\mathbb{Z}}_{p^2},GF\left(p^2\right),{\mathbb{Z}}_p(+){\mathbb{Z}}_p$ or ${\mathbb{Z}}_p\times {\mathbb{Z}}_p$. We determine the geodetic number $g\left(G\right(R\left)\right)$ associated with each such...

Let us consider the finite commutative ring R, whose unity is $1\ne 0.$ Its weakly zero-divisor graph, represented as $W\Gamma \left(R\right)$, is a basic undirected graph with two distinct vertices, $c_1$ and $c_2$, that are adjacent if and only if there exist $r\in$&n...

Let $1\ne 0$ be the identity of the commutative ring R. The cozero-divisor graph of a ring R is an undirected simple graph, represented by ${\mathsf{\Gamma }}^{\prime }\left(R\right)$, where two different vertices g and h are adjacent if and only if $g\notin Rh$ and $h\notin Rg$. Th...

Let R be a commutative ring with identity, and $Z\left(R\right)$ be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by $W\Gamma \left(R\right)$ is an undirected (simple) graph with vertex set $Z{\left(R\right)}^{*}$, and two distinct vertices...

Variable-weight (also called multi-weight) optical orthogonal codes (VW-OOCs) with maximum correlation one admit a cyclic difference-packing formulation. We characterize the exact difference spectrum of an existing Chinese-remainder graph constructio...

A girth cycle refers to a cycle that has the minimum length within a graph. The graphs $D(k,q)$ form an important family of algebraically defined bipartite graphs over finite fields, and their short-cycle structure is closely related to questions in ex...

A generalization of Ding’s construction is proposed that employs as a defining set the collection of the sth powers ( $s\ge 2$ ) of all nonzero elements in $GF\left(p^m\right)$ , where $p\ge 2$ is prime. Some of the resulting code...

We present and generalize the basic ideas underlying recent work aimed at the construction of mutually unbiased bases in finite dimensional Hilbert spaces with the help of group and graph theoretical concepts. In this approach finite groups are used...