Distinct Vertices Research Articles

On divisor graphs of valuation domains and local Artinian principal ideal rings

For an element $x$ of a commutative ring $R$, let $\Gamma_x(R)$ be the graph whose vertices are the elements of $R$ that divide $x$ such that distinct vertices $r$ and $s$ are adjacent if and only if $rs=x$. If $x$ is a nonzero nonirreducible nonunit of an integral domain $R$, then $\Gamma_x^\mathcal{C}(R)$ is the graph whose vertices are the associate classes of divisors of $x$ that are neither units nor associates of $x$ such that distinct vertices $A$ and $B$ are adjacent if and only if $rs$ divides $x$ for some (and hence every) $r\in A$ and $s\in B$. The graphs $\Gamma_x(R)$ are considered when $R$ is a local Artinian principal ideal ring, and $\Gamma_x^\mathcal{C}(R)$ is examined when $R$ is a valuation domain. For example, it is shown that a finite local ring $R$ is a principal ideal ring if and only if there exists $x\in R$ such that every connected component of $\Gamma_x(R)$ is a star graph of a prescribed cardinality. Moreover, it is proved that an integral domain $R$ is a discrete valuation ring if and only if its collection of graphs $\Gamma_x^\mathcal{C}(R)$ consists precisely of a single-vertex graph, along with every (up to isomorphism) graph that is realizable as the compressed $0$-divisor graph of a local Artinian principal ideal ring. Certain graphs associated with partially ordered abelian groups have an essential role in the work.

On the generalized distance spectrum of some connected graphs of diameter at most two

Let G be a simple, undirected connected graph. The generalized distance matrix of a graph G, denoted by D_\alpha(G) is the convex combination of the distance matrix D(G) and the diagonal matrix of the vertex transmissions Tr(G). It is of the form D_\alpha(G) = \alpha Tr(G) + (1-\alpha)D(G). In this paper, we determine the spectrum of the generalized distance matrix of the join of regular graphs of diameter at most two, viz. the weakly zero divisor graph and the zero divisor graph on the ring Z_n, for some values of n. For a commutative ring R with unity, the zero-divisor graph, denoted by \Gamma(R), is an undirected simple graph whose vertex set is the set of all non zero and zero-divisors of R and two distinct vertices x and y are adjacent if and only if xy =0. The weakly zero-divisor graph of R denoted by W \Gamma (R) is the simple undirected graph whose vertices are the non zero zero-divisors of R and two distinct vertices x and y are adjacent if and only if there exist r in ann(x) and s in ann(y) such that rs = 0.

Metric and strong metric dimension in TI-power graphs of finite groups

<p>Given a finite group $ G $ with the identity $ e $, the TI-power graph (trivial intersection power graph) of $ G $ is an undirected graph with vertex set $ G $, in which two distinct vertices $ x $ and $ y $ are adjacent if $ \langle x\rangle\cap \langle y\rangle = \{e\} $. In this paper, we obtain closed formulas for the metric and strong metric dimensions of the TI-power graph of a finite group. As applications, we compute the metric and strong metric dimensions of the TI-power graph of a cyclic group, a dihedral group, a generalized quaternion group, and a semi-dihedral group.</p>

Groups Whose Common Divisor Graph on p-Regular Classes has Diameter Three

AbstractLet G be a finite p-separable group, for some fixed prime p. Let $$\Gamma _p(G)$$ Γ p ( G ) be the common divisor graph built on the set of non-central conjugacy classes of p-regular elements of G: this is the graph whose vertices are the conjugacy classes of those non-central elements of G such that p does not divide their orders, and two distinct vertices are adjacent if and only if the greatest common divisor of their lengths is strictly greater than one. The aim of this paper is twofold: to positively answer an open question concerning the maximum possible distance in $$\Gamma _p(G)$$ Γ p ( G ) between a vertex with maximal cardinality and any other vertex, and to study the p-structure of G when $$\Gamma _p(G)$$ Γ p ( G ) has diameter three.

On the Planar Property of an Ideal-Based Weakly Zero-Divisor Graph

Let R be a commutative ring with a nonzero identity and Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R, denoted by WГ(R), is the graph with the vertex set 〖Z(R)〗^*=Z(R)\\{0}, where two distinct vertices a and b form an edge if ar=bs=rs=0 for r,s∈R\\{0}. For an ideal I of R, the ideal-based zero-divisor graph of R, denoted by Г_I (R), has vertices {a∈R\I:ab∈I for some b∈R\I} and edges {(a,b):ab∈I,a,b∈R\I,a≠b}. In this article, an ideal-based weakly zero-divisor graph of R, denoted by 〖WГ〗_I (R), is introduced which contains Г_I (R) as a subgraph and is identical to the graph WГ(R) when I={0}. The relationship between the graphs WГ_I (R) and WГ(R/I) is investigated and the planar property of 〖WГ〗_I (R) is studied. The results show that WГ(R/I) is isomorphic to a subgraph of 〖WГ〗_I (R). For 〖WГ〗_I (R) to be planar, some restraints are provided on the size of the ideal I and girth of 〖WГ〗_I (R). In conclusion, the results suggest that WГ_I (R) and WГ(R/I) are strongly related and establish necessary and sufficient conditions for WГ_I (R) to be planar. In addition, rings R with planar WГ_I (R) are classified.

Some properties of a graph associated with a non-quasi-local weakly factorial domain and a supergraph of it

Let [Formula: see text] be a weakly factorial domain which admits at least two maximal ideals. We use [Formula: see text] to denote the set of all primary elements of [Formula: see text] and denote [Formula: see text] by [Formula: see text]. Let [Formula: see text]. We denote the subset of [Formula: see text] consisting of all [Formula: see text] such that [Formula: see text] does not belong to the Jacobson radical of [Formula: see text] by [Formula: see text]. With [Formula: see text], in this paper, we associate an undirected graph denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In Secs. 2 and 3 of this paper, we discuss results regarding the connectedness, the girth, and the clique number of [Formula: see text] and study the interplay between graph properties of [Formula: see text] and the properties of [Formula: see text]. In Secs. 4 and 5 of this paper, we consider a supergraph of [Formula: see text], denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and study some graph properties of [Formula: see text].

Metric and strong metric dimension in intersection power graphs of finite groups

Let G be a finite group and let e be its identity element. The intersection power graph of G is the undirected graph with vertex set G, in which two distinct vertices x and y are adjacent if either 〈 x 〉 ∩ 〈 y 〉 ≠ { e } or one of x and y is e. In this paper, we first compute the strong metric dimension of the intersection power graph of a cyclic group, a dihedral group, and a generalized quaternion group. We also characterize all finite groups G whose intersection power graph has strong metric dimension | G | − 2 . Then, we give the sharp upper and lower bounds for the metric dimension of the intersection power graph of a finite group. As applications, we obtain the metric dimension of the intersection power graph of a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.

Wiener and Harary index of the zero-divisor graph of ℤn

The zero-divisor graph of a ring [Formula: see text] is a graph whose vertex set consists of all the elements of [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. We study the zero-divisor graph [Formula: see text], for [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct primes. Topological indices such as Wiener and Harary indices of [Formula: see text] are determined.

On the annihilator graphs of partial transformation semigroups

Let X n = { 1 , 2 , … , n } and P n be a partial transformation semigroup on X n . Obviously, the empty set ∅ is a zero element of P n and denoted by 0. Let P n * = P n \\ { 0 } . An element α ∈ P n is a zero divisor of P n if there exists β ∈ P n * such that α β = 0 = β α . The set Ann ( α ) = { β ∈ P n : α β = 0 = β α } is called the annihilator of α in P n . It is clear that 0 ∈ Ann ( α ) for all α ∈ P n . Let Z ( P n ) be the set of all zero divisors of P n and Z ( P n ) * = Z ( P n ) \\ { 0 } . Generally, if α ∈ Z ( P n ) , then there exists β ∈ Z ( P n ) * in which β ∈ Ann ( α ) . In this paper, we construct the annihilator graph Γ n of P n which is an undirected simple graph with vertex set Z ( P n ) * and two distinct vertices α and β are adjacent if and only if Ann ( α ) ∩ Ann ( β ) ≠ { 0 } . Furthermore, we prove some basic structural properties of Γ n and determine invariants for Γ n such as the diameter, girth, clique, domination number, and independence number.

Edge‐arc‐disjoint paths in semicomplete mixed graphs

AbstractThe so‐called weak‐2‐linkage problem asks for a given digraph and distinct vertices of whether has arc‐disjoint paths so that is an ‐path for . This problem is NP‐complete for general digraphs but the first author showed that the problem is polynomially solvable and that all exceptions can be characterized when is a semicomplete digraph, that is, a digraph with no pair of nonadjacent vertices. In this paper we extend these results to paths which are both edge‐disjoint and arc‐disjoint in semicomplete mixed graphs, that is, a mixed graph in which every pair of distinct vertices has either an arc, an edge, or both an arc and an edge between them. We give a complete characterization of the negative instances and explain how this gives rise to a polynomial algorithm for the problem.

On locating-chromatic number of helm graph H_m FOR 10≤m≤28

Let G = (V,E) be a connected graph and c be a k-coloring of G. The color class S_i of G is a set of vertices given color i, for 1 ≤ i ≤ k. Let  = {S_1,S_2,…,S_k} be an ordered partition of V(G). The color code of a vertex v $in$ (element) V(G) is defined as the ordered k--tuplec_Π (v)=(d(v,S_1),d(v,S_2),...,d(v,S_k)),where d(v,S_i) = min{d(v,x)| x $in$ (element) S_i} for 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then c is called a locating-coloring of G. The locating-chromatic number χ_L (G) is the minimum number of colors in a locating-coloring of G. This paper discusses the locating-chromatic number of helm graph H_m for 10 ≤ m ≤ 28. Helm graph H_m is constructed by adding some leaves to the corresponding vertices of wheels W_m, for m ≥ 3.

The First General Zagreb Index of the Zero Divisor Graph for the Ring Zpqk

This study investigates the application of graph theory in analyzing the zero divisor graph of a commutative ring, with a specific focus on its connection to the topological index. For an undirected graph Γ with consists of a non-empty set of vertices, V , and a set of edges, E, the first general Zagreb index is defined as a graph invariant that measures the sum of the degree of each vertex to the power of α= 0. Meanwhile, the zero divisor graph Γ of the commutative ring, R is the (undirected) graph with vertices the zero-divisors of R, and distinct vertices a and b are adjacent if and only if ab = 0. In this paper, the general formulas of the first general Zagreb index of the zero divisor graph for the ring of integers modulo pqk are computed for the cases δ = 1, 2, and 3. This research focuses on the ring defined as the integers modulo pqk, where k is a positive integer, p and q are primes p < q. Two examples are given to demonstrate the main f indings.

Parameterized complexity of vertex splitting to pathwidth at most 1

Motivated by the planarization of 2-layered straight-line drawings, we consider the problem of modifying a graph such that the resulting graph has pathwidth at most 1. The problem Pathwidth-One Vertex Explosion (POVE) asks whether such a graph can be obtained using at most k vertex explosions, where a vertex explosion replaces a vertex v by deg⁡(v) degree-1 vertices, each incident to exactly one edge that was originally incident to v. For POVE, we give an FPT algorithm with running time O(4k⋅m) and an O(k2) kernel, thereby improving over the O(k6)-kernel by Ahmed et al. [2] in a more general setting. Similarly, a vertex split replaces a vertex v by two distinct vertices v1 and v2 and distributes the edges originally incident to v arbitrarily to v1 and v2. Analogously to POVE, we define the problem variant Pathwidth-One Vertex Splitting (POVS) that uses the split operation instead of vertex explosions. Here we obtain a linear kernel and an algorithm with running time O((6k+12)k⋅m). This answers an open question by Ahmed et al. [2].Finally, we consider the problem Π-VertexSplitting (Π-VS), which generalizes the problem POVS and asks whether a given graph can be turned into a graph of a specific graph class Π using at most k vertex splits. For graph classes Π that can be defined in monadic second-order graph logic (MSO2), we show that the problem Π-VS can be expressed as an MSO2 formula, resulting in an FPT algorithm for Π-VS parameterized by k if Π additionally has bounded treewidth. We obtain the same result for the problem variant using vertex explosions.

On Graphs of a Finite Full Transformation Semigroup

For n ≥ 2 , let T n denote the full transformation semigroup on X n = { 1 , … , n } , and let π denote the constant map defined by x π = 1 for all x ∈ X n . Consider the sets: L = { α ∈ T n : α β = π for some β ∈ T n ∖ { π } } , R = { α ∈ T n : γ α = π for some γ ∈ T n ∖ { π } } , T = L ∩ R and Z = L ∪ R . We define two undirected graphs, denoted by Γ ( T n ) and Ω ( T n ) , as simple graphs whose vertex sets are T ∖ { π } and Z ∖ { π } , and whose two distinct vertices α and β are adjacent if and only if α β = π or β α = π . We then investigate some properties of both Γ ( T n ) and Ω ( T n ) .

Numerical Study of Shock Wave Interaction with V-Shaped Heavy/Light Interface

This paper investigates numerically the shock wave interaction with a V-shaped heavy/light interface. For numerical simulations, we choose six distinct vertex angles (θ=40∘,60∘,90∘,120∘,150∘, and 170∘), five distinct shock wave strengths (Ms=1.12,1.22,1.30,1.60, and 2.0), and three different Atwood numbers (At=−0.32,−0.77, and −0.87). A two-dimensional space of compressible two-component Euler equations are solved using a third-order modal discontinuous Galerkin approach for the simulations. The present findings demonstrate that the vertex angle has a crucial influence on the shock wave interaction with the V-shaped heavy/light interface. The vertex angle significantly affects the flow field, interface deformation, wave patterns, spike generation, and vorticity production. As the vertex angle decreases, the vorticity production becomes more dominant. A thorough analysis of the vertex angle effect identifies the factors that propel the creation of vorticity during the interaction phase. Notably, smaller vertex angles lead to stronger vorticity generation due to a steeper density gradient, while larger angles result in weaker, more dispersed vorticity and a less complex interaction. Moreover, kinetic energy and enstrophy both dramatically rise with decreasing vortex angles. A detailed analysis is also carried out to analyze the vertex angle effects on the temporal variations of interface features. Finally, the impacts of different Mach and Atwood numbers on the V-shaped interface are briefly presented.

Degree-based topological indices of the idempotent graph of the ring [formula omitted]

Let R be a finite commutative ring with a non-zero identity, and Id(R) be the set of idempotent elements of R. The idempotent graph of R, denoted by GId(R), is a simple undirected graph with all elements of R as vertices, and two distinct vertices u, v are adjacent if and only if u+v∈Id(R). In this paper, we consider the idempotent graph of the ring Zn and investigate some degree-based topological indices, such as the general sum-connectivity index, the general Randić index, the general Zagreb index, and the Sombor index of that graph by considering the M-polynomial of the graph.

Fault Tolerant Metric Dimension of Arithmetic Graphs

For a graph G , two vertices x , y ∈ G are said to be resolved by a vertex s ∈ G , if d ( x | s ) ≠ d ( y | s ) . The minimum cardinality of such a resolving set R in G is called its metric dimension. A resolving set R is said to be fault-tolerant, if for every p ∈ R , R − p preserves the property of being a resolving set. A fault-tolerant metric dimension of G is a minimal possible order fault-tolerant resolving set. A wide variety of situations, in which connection, distance, and connectivity are important aspects, call for the utilisation of metric dimension. The structure and dynamics of complex networks, as well as difficulties connected to robotics network design, navigation, optimisation, and facility positioning, are easier to comprehend as a result of this. As a result of the relevant concept of metric dimension, robots are able to optimise their methods of localization and navigation by making use of a limited number of reference locations. As a consequence of this, numerous applications of robotics, including collaborative robotics, autonomous navigation, and environment mapping, have become more precise, efficient, and resilient. The arithmetic graph A l is defined as the graph with its vertex set as the set of all divisors of l , where l is a composite number and l = p γ 1 1 p η 2 2 , … , p n n , where p n ≥ 2 and the p i ’s are distinct primes. Two distinct divisors x , y of l are said to have the same parity if they have the same prime factors (i.e., x = p 1 p 2 and y = p 2 1 p 3 2 have the same parity). Further, two distinct vertices x , y ∈ A l are adjacent if and only if they have different parity and gcd ( x , y ) = p i (greatest common divisor) for some i ∈ { 1 , 2 , … , t } . This article is dedicated to the investigation of the arithmetic graph of a composite number l , which will be referred to throughout the text as A l . In this study, we compute the fault-tolerant resolving set and the fault-tolerant metric dimension of the arithmetic graph A l , where l is a composite number.

D-Index of Certain Ladder Graphs

For any two distinct vertices , of a connected graph , the -distance , in which the minimum is taken over all paths, and is the length of the path . The -index of is defined as . In this paper, we obtained a formula for -index or Wiener -index , where is the ladder graph, . Also, we obtained the Wiener -index and Wiener index of semi-Ladder graph .

On the k-Spanning Cyclability of 4-Valent Cayley Graphs on Abelian Groups

A graph X is k -spanning cyclable if for any subset S of k distinct vertices there is a 2-factor of X consisting of k cycles such that each vertex in S belongs to a distinct cycle. In this paper, we examine the k -spanning cyclability of 4-valent Cayley graphs on Abelian groups.

Multiplicative Distance-Location Number of Graphs

For a set \( S \) of vertices in a connected graph \( G \), the multiplicative distance of a vertex \( v \) with respect to \( S \) is defined by \(d_{S}^{*}(v) = \prod\limits_{x \in S, x \neq v} d(v,x).\) If \( d_{S}^{*}(u) \neq d_{S}^{*}(v) \) for each pair \( u,v \) of distinct vertices of \( G \), then \( S \) is called a multiplicative distance-locating set of \( G \). The minimum cardinality of a multiplicative distance-locating set of \( G \) is called its multiplicative distance-location number \( loc_{d}^{*}(G) \). If \( d_{S}^{*}(u) \neq d_{S}^{*}(v) \) for each pair \( u,v \) of distinct vertices of \( G-S \), then \( S \) is called an external multiplicative distance-locating set of \( G \). The minimum cardinality of an external multiplicative distance-locating set of \( G \) is called its external multiplicative location number \( loc_{e}^{*}(G) \). We prove the existence or non-existence of multiplicative distance-locating sets in some well-known classes of connected graphs. Also, we introduce a family of connected graphs such that \( loc_{d}^{*}(G) \) exists. Moreover, there are infinite classes of connected graphs \( G \) for which \( loc_{d}^{*}(G) \) exists as well as infinite classes of connected graphs \( G \) for which \( loc_{d}^{*}(G) \) does not exist. A lower bound for the multiplicative distance-location number of a connected graph is established in terms of its order and diameter.