• Kyungpook Mathematical Journal
• /
• 제59권4호
• /
• pp.591-601
• /
• 2019

Let ℤ_n be the ring of integers modulo n and Γ(ℤ_n) the zero-divisor graph of ℤ_n. In this paper, we study some properties of Γ(ℤ_n). More precisely, we completely characterize the diameter and the girth of Γ(ℤ_n). We also calculate the chromatic number of Γ(ℤ_n).

• 대한수학회보
• /
• 제53권4호
• /
• pp.1197-1211
• /
• 2016

Let R be a finite commutative ring with nonzero identity. We define ${\Gamma}(R)$ to be the graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u of R such that x + uy is a unit of R. This graph provides a refinement of the unit and unitary Cayley graphs. In this paper, basic properties of ${\Gamma}(R)$ are obtained and the vertex connectivity and the edge connectivity of ${\Gamma}(R)$ are given. Finally, by a constructive way, we determine when the graph ${\Gamma}(R)$ is Hamiltonian. As a consequence, we show that ${\Gamma}(R)$ has a perfect matching if and only if ${\mid}R{\mid}$ is an even number.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제11권3호
• /
• pp.65-75
• /
• 2007

A graph G is self-complementary (sc) if it is isomorphic to its complement. G is perfect if for all induced subgraphs H of G, the chromatic number of H (denoted ${\chi}$(H)) equals the number of vertices in the largest clique in H (denoted ${\omega}$(H)). An sc graph which is also perfect is known as sc perfect graph. A comparability graph is an undirected graph if it can be oriented into transitive directed graph. An sc comparability (scc) is clearly a subclass of sc perfect graph. In this paper we show that no two non-isomorphic scc graphs with n vertices each, (n<13) have same spectrum, and that the smallest positive integer for which there exists hyper-energetic scc graph is 13.

• Kyungpook Mathematical Journal
• /
• 제60권4호
• /
• pp.723-729
• /
• 2020

Let ℤ_n be the ring of integers modulo n and let ℤ_n[X]] be either ℤ_n[X] or ℤ_n[[X]]. Let 𝚪(Z_n[X]]) be the zero-divisor graph of ℤ_n[X]]. In this paper, we study some properties of 𝚪(ℤ_n[X]]). More precisely, we completely characterize the diameter and the girth of 𝚪(ℤ_n[X]]). We also calculate the chromatic number of 𝚪(ℤ_n[X]]).

• Journal of applied mathematics & informatics
• /
• 제40권3_4호
• /
• pp.729-740
• /
• 2022

Let ℤ be the ring of integers and let ℤ_n be the ring of integers modulo n. Let ℤ(+)ℤ_n be the idealization of ℤ_n in ℤ and let (ℤ(+)ℤ_n)[X]] be either (ℤ(+)ℤ_n)[X] or (ℤ(+)ℤ_n)[[X]]. In this article, we study the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]]. More precisely, we completely characterize the diameter and the girth of the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]]. We also calculate the chromatic number of the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]].

• 대한수학회보
• /
• 제61권3호
• /
• pp.797-811
• /
• 2024

In this paper, we define the torsion graph determined by equivalence classes of torsion elements and denote it by A_E(M). The vertex set of A_E(M) is the set of equivalence classes {[x] | x ∈ T(M)^*}, where two torsion elements x, y ∈ T(M)^* are equivalent if ann(x) = ann(y). Also, two distinct classes [x] and [y] are adjacent in A_E(M), provided that ann(x)ann(y)M = 0. We shall prove that for every torsion finitely generated module M over a Dedekind domain R, a vertex of A_E(M) has degree two if and only if it is an associated prime of M.

• 대한수학회논문집
• /
• 제34권2호
• /
• pp.415-427
• /
• 2019

Let R be an associative ring with nonzero identity. The annihilator ideal graph of R, denoted by ${\Gamma}_{Ann}(R)$, is a graph whose vertices are all nonzero proper left ideals and all nonzero proper right ideals of R, and two distinct vertices I and J are adjacent if $I{\cap}({\ell}_R(J){\cup}r_R(J)){\neq}0$ or $J{\cap}({\ell}_R(I){\cup}r_R(I)){\neq}0$, where ${\ell}_R(K)=\{b{\in}R|bK=0\}$ is the left annihilator of a nonempty subset $K{\subseteq}R$, and $r_R(K)=\{b{\in}R|Kb=0\}$ is the right annihilator of a nonempty subset $K{\subseteq}R$. In this paper, we assume that R is a semicommutative ring. We study the structure of ${\Gamma}_{Ann}(R)$. Also, we investigate the relations between the ring-theoretic properties of R and graph-theoretic properties of ${\Gamma}_{Ann}(R)$. Moreover, some combinatorial properties of ${\Gamma}_{Ann}(R)$, such as domination number and clique number, are studied.

• Journal of applied mathematics & informatics
• /
• 제39권1_2호
• /
• pp.13-29
• /
• 2021

For a graph G = (V, E), a vertex coloring (or, simply, a coloring) of G is a function C : V (G) → {1, 2, …, k} (using the non-negative integers {1, 2, …, k} as colors). We say that a coloring of a graph G is injective if for every vertex v ∈ V (G), all the neighbors of v are assigned with distinct colors. The injective chromatic number χi(G) of a graph G is the least k such that there is an injective k-coloring [6]. In this paper, we study a natural variation of the injective coloring problem: coloring the edges of a graph under the same constraints (alternatively, to investigate the injective chromatic number of line graphs), we define the k- injective edge coloring of a graph G as a mapping C : E(G) → {1, 2, …, k}, such that for every edge e ∈ E(G), all the neighbors edges of e are assigned with distinct colors. The injective chromatic index χ′in(G) of G is the least positive integer k such that G has k- injective edge coloring, exact values of the injective chromatic index of different families of graphs are obtained, some related results and bounds are established. Finally, we define the injective clique number ωin and state a conjecture, that, for any graph G, ωin ≤ χ′in(G) ≤ ωin + 2.

• 제어로봇시스템학회:학술대회논문집
• /
• 제어로봇시스템학회 2001년도 ICCAS
• /
• pp.49.3-49
• /
• 2001

In 1994 Adelman´s pioneer work demonstrated that deoxyribonucleic acid (DNA) could be used as a medium for computation to solve mathematical problems. He described the use of DNA based computational approach to solve the Hamiltonian Path Problem (HPP). Since then a number of combinatorial problems have been analyzed by DNA computation approaches including, for example: Maximum Independent Set (MIS), Maximal Clique and Satisfaction (SAT) Problems. In the present paper we propose a method of solving another classic combinatorial optimization problem - the eraph Coloring Problem (GCP), using specifically designed circular DNA plasmids as a computation tool. The task of the analysis is to color the graph so that no two nodes ...

• Journal of Communications and Networks
• /
• 제18권5호
• /
• pp.846-856
• /
• 2016

Wireless sensor networks (WSNs) are widely applied in monitoring and control of environment parameters. It is sometimes necessary to disseminate data through wireless links after they are deployed in order to adjust configuration parameters of sensors or distribute management commands and queries to sensors. Several approaches have been proposed recently for data dissemination in WSNs. However, none of these approaches achieves both high efficiency and low complexity simultaneously. To address this problem, cluster-tree based network architecture, which divides a WSN into hierarchies and clusters is proposed. Upon this architecture, data is delivered from base station to all sensors in clusters hierarchy by hierarchy. In each cluster, father broadcasts data to all his children with instantly decodable network coding (IDNC), and a novel scheme targeting to maximize total transmission gain (MTTG) is proposed. This scheme employs a new packet scheduling algorithm to select IDNC packets, which uses weight status feedback matrix (WSFM) directly. Analysis and simulation results indicate that the transmission efficiency approximate to the best existing approach maximum weight clique, but with much lower computational overhead. Hence, the energy efficiency achieves both in data transmission and processing.