• Kyungpook Mathematical Journal
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• v.61 no.2
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• pp.441-453
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• 2021

In this work, we define the concept of a mixed G-monotone mapping on a modular metric space endowed with a graph, and prove some fixed point theorems for this new class of mappings. Results of this paper extend coupled fixed point theorems from partially ordered metric spaces into the modular metric spaces endowed with a graph. An example is presented to illustrate the new result.

• Korean Journal of Mathematics
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• v.27 no.3
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• pp.563-580
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• 2019

In this paper, we study the strong metric dimension of zero-divisor graph ${\Gamma}(R)$ associated to a ring R. This is done by transforming the problem into a more well-known problem of finding the vertex cover number ${\alpha}(G)$ of a strong resolving graph $G_{sr}$. We find the strong metric dimension of zero-divisor graphs of the ring ${\mathbb{Z}}_n$ of integers modulo n and the ring of Gaussian integers ${\mathbb{Z}}_n$[i] modulo n. We obtain the bounds for strong metric dimension of zero-divisor graphs and we also discuss the strong metric dimension of the Cartesian product of graphs.

• Kyungpook Mathematical Journal
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• v.63 no.1
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• pp.123-129
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• 2023

For an ordered set W = {w₁, w₂, . . . , w_k} of vertices and a vertex v in a connected graph G, the k-vector r(v|W) = (d(v, w₁), d(v, w₂), . . . , d(v, w_k)) is called the metric representation of v with respect to W, where d(x, y) is the distance between the vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct metric representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension dim(G), and a resolving set of minimum cardinality is a basis of G. The corona product, G ⊙ H of graphs G and H is obtained by taking one copy of G and n(G) copies of H, and by joining each vertex of the ith copy of H to the ith vertex of G. In this paper, we obtain bounds for dim(G ⊙ K₁), characterize all graphs G with dim(G ⊙ K₁) = dim(G), and prove that dim(G ⊙ K₁) = n - 1 if and only if G is the complete graph K_n or the star graph K_1,n-1.

• East Asian mathematical journal
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• v.17 no.2
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• pp.191-195
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• 2001

The purpose of this note is to introduce various metrics and to prove the properties of given metric spaces.

• Journal of applied mathematics & informatics
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• v.41 no.4
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• pp.883-892
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• 2023

In this research, under some specific situations, we precisely derive new coupled fixed point theorems in a complete b-metric space endowed with the graph. We also use the concept of coupled fixed points to ensure the solution of differential equations for the system of impulse effects.

• Nonlinear Functional Analysis and Applications
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• v.26 no.5
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• pp.1105-1114
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• 2021

In this paper, we prove a fixed point theorem for G-contractive type non-self mapping in cone metric space endowed with a graph. Our result generalizes many results in the literature and provide a new pavement for solving nonlinear functional equations.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.2
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• pp.81-86
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• 2001

In this paper, we present the proof of the property for interior metric space and geodesic space.

• Journal of applied mathematics & informatics
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• v.39 no.5_6
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• pp.689-701
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• 2021

The purpose of this paper is to optimize the number of source places required for the unique representation of the destination using the tools of graph theory. A subset S of vertices of a graph G is called a rational resolving set of G if for each pair u, v ∈ V - S, there is a vertex s ∈ S such that d(u/s) ≠ d(v/s), where d(x/s) denotes the mean of the distances from the vertex s to all those y ∈ N[x]. A rational resolving set is called minimal rational resolving set if no proper subset of it is a rational resolving set. In this paper we study varieties of minimal rational resolving sets defined on the basis of its complements and compute the minimum and maximum cardinality of such sets, respectively called as lower and upper rational metric dimensions for power of a path P_n analysing various possibilities.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1167-1182
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• 2016

Let M be an R-module, where R is a commutative ring with identity 1 and let G(V,E) be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three simple graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ to M called full annihilating, semi-annihilating and star-annihilating graph. When M is finite over R, we investigate metric dimensions in $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$. We show that M over R is finite if and only if the metric dimension of the graph $ann_f({\Gamma}(M_R))$ is finite. We further show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if M is a prime-multiplication-like R-module. We investigate the case when M is a free R-module, where R is an integral domain and show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if $$M{\sim_=}R$$. Finally, we characterize all the non-simple weakly virtually divisible modules M for which Ann(M) is a prime ideal and Soc(M) = 0.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.26 no.1
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• pp.128-133
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• 2022

This paper deals with the graph in which the weights of edges are given the distances between two end vertices on a metric space. In particular, we will study about a path P with n vertices for these graphs. We obtain a new graph $\bar{P}$ by augmenting an edge to P. Then the length of the shortest path between two vertices on $\bar{P}$ is considered and we focus on the maximum of these lengths. This maximum is called the diameter of the graph $\bar{P}$. We wish to find the augmented edge to minimize the diameter of $\bar{P}$. Especially, for an arbitrary real number λ > 0, we should determine whether the diameter of $\bar{P}$ is less than or equal to λ and we propose an O(n)-time algorithm for this problem, which improves on the time complexity O(nlogn) previously known. Using this decision algorithm, for the length D of P, we provide an O(nlogD)-time algorithm to find the minimum of the diameter of $\bar{P}$.