• 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 $\bar{P}$.

• Journal of applied mathematics & informatics
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• v.8 no.3
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• pp.889-897
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• 2001

Based on the prime graph of a finite simple group, its order is the product of its order components (see[4]). It is known that Suzuki-Ree groups [6], $PSL_2(q)$ [8] and $E_8(q)$ [7] are uniquely deternubed by their order components. In this paper we prove that the simple groups $A_p$ are also unipuely determined by their order components, where p and p-2 are primes.

• Journal of applied mathematics & informatics
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• v.33 no.3_4
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• pp.365-377
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• 2015

For a (molecular) graph, the first and second Zagreb indices (M₁ and M₂) are two well-known topological indices, first introduced in 1972 by Gutman and Trinajstić. The first Zagreb index M₁ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M₂ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Let $K_{n_1,n_2}^{P}$ with n₁ $\leq$ n₂, n₁ + n₂ = n and p < n₁ be the set of bipartite graphs obtained by deleting p edges from complete bipartite graph K_n1,n2. In this paper, we determine sharp upper and lower bounds on Zagreb indices of graphs from $K_{n_1,n_2}^{P}$ and characterize the corresponding extremal graphs at which the upper and lower bounds on Zagreb indices are attained. As a corollary, we determine the extremal graph from $K_{n_1,n_2}^{P}$ with respect to Zagreb coindices. Moreover a problem has been proposed on the first and second Zagreb indices.

• Journal of Applied and Pure Mathematics
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• v.5 no.1_2
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• pp.41-53
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• 2023

Let G = (V, E) be a (p, q) graph. Define $${\rho}=\{{\frac{2}{p}},\;{\text{{\qquad} if p is even}}\\{\frac{2}{p-1}},\;{{\text{if p is odd}}$$ and L = {±1, ±2, ±3, … , ±ρ} called the set of labels. Consider a mapping f : V ⟶ L by assigning different labels in L to the different elements of V when p is even and different labels in L to p-1 elements of V and repeating a label for the remaining one vertex when p is odd.The labeling as defined above is said to be a pair difference cordial labeling if for each edge uv of G there exists a labeling |f(u) - f(v)| such that ${\mid}{\Delta}_{f_1}-{\Delta}_{f^c_1}{\mid}{\leq}1$, where ${\Delta}_{f_1}$ and ${\Delta}_{f^c_1}$ respectively denote the number of edges labeled with 1 and number of edges not labeled with 1. A graph G for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate pair difference cordial labeling behaviour of Petersen graphs P(n, k) like P(n, 2), P(n, 3), P(n, 4).

• Journal of the Korean Institute of Gas
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• v.16 no.6
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• pp.41-47
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• 2012

There are PFD(Process Flow Diagram) and P&ID(Piping and Instrument Diagram) for designing and managing chemical process efficiently. They provide the operation condition and equipment specifications of chemical process, but they do not provide the reliability of chemical process. Therefore, in this study, Reliability Flow Diagram(RFD) which provide the cycle and time of preventive maintenance has been developed using Directed Graph Analysis methodology. Directed Graph Analysis methodology is capable of assessing the reliability of chemical process. It models chemical process into Directed Graph with nodes and arcs and assesses the reliability of normal operation of chemical process by assessing Directed Graph sequential. In this paper, the chemical process reliability transition according to operation time was assessed. And then, Reliability Flow Diagram has been developed by inserting the result into P&ID. Like PFD and P&ID, Reliability Flow Diagram provide valuable and useful information for the design and management of chemical process.

• Korean Journal of Mathematics
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• v.31 no.4
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• pp.485-494
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• 2023

The degree of hesitancy of a vertex in a hesitancy fuzzy graph depends on the degree of membership and non-membership of the vertex. We define a new class of hesitancy fuzzy graph, the intuitionistic hesitancy fuzzy graph in which the degree of hesitancy of a vertex is independent of the degree of its membership and non-membership. We introduce the idea of β-product of a pair of hesitancy fuzzy graphs and intuitionistic hesitancy fuzzy graphs and prove certain results based on this product.

• Bulletin of the Korean Mathematical Society
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• v.52 no.6
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• pp.2025-2034
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• 2015

In this paper, we characterize some PGL(2, q) by their orders and maximum element orders. We also prove that PSL(2, p) with $p{\geqslant}3$ a prime can be determined by their orders and maximum element orders. Moreover, we show that, in general, if $q=p^n$ with p a prime and n > 1, PGL(2, q) can not be uniquely determined by their orders and maximum element orders. Several known results are generalized.

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.133-147
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• 2024

Let G = (V, E) be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on G $${\Delta}u={\lambda}e^u(e^u-1)^5+4{\pi}\sum_{s=1}^{N}\delta_{ps}$$, where λ > 0, δ_ps is the Dirac mass at the vertex p_s, and p₁, p₂, . . . , p_N are arbitrarily chosen distinct vertices on the graph. We show that there exists a critical value $\hat{\lambda}$ such that when λ > $\hat{\lambda}$, the generalized Chern-Simons equation has at least two solutions, when λ = $\hat{\lambda}$, the generalized Chern-Simons equation has a solution, and when λ < $\hat{\lambda}$, the generalized Chern-Simons equation has no solution.

• Communications of the Korean Mathematical Society
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• v.35 no.1
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• pp.1-12
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• 2020

A pathos block line cut-vertex graph of a tree T, written P BL_c(T), is a graph whose vertices are the blocks, cut-vertices, and paths of a pathos of T, with two vertices of P BL_c(T) adjacent whenever the corresponding blocks of T have a vertex in common or the edge lies on the corresponding path of the pathos or one corresponds to a block B_i of T and the other corresponds to a cut-vertex c_j of T such that c_j is in B_i; two distinct pathos vertices P_m and P_n of P BL_c(T) are adjacent whenever the corresponding paths of the pathos P_m(v_i, v_j) and P_n(v_k, v_l) have a common vertex. We study the properties of P BL_c(T) and present the characterization of graphs whose P BL_c(T) are planar; outerplanar; maximal outerplanar; minimally nonouterplanar; eulerian; and hamiltonian. We further show that for any tree T, the crossing number of P BL_c(T) can never be one.

• Bulletin of the Korean Mathematical Society
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• v.37 no.4
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• pp.649-657
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• 2000

The 2-step competition graph of D has the same vertex set as D and an edge between vertices x and y if and only if there exist (x, z)-walk of length 2 and (y, z)-walk of length 2 for some vertex z in D. The 2-step competition number of a graph G is the smallest number k such that G together with k isolated vertices is the 2-step competition graph of an acyclic digraph. Cho, et al. showed that the 2-step competition number of a path of length at least two is two. In this paper, we characterize all the minimal acyclic digraphs whose 2-step competition graphs are paths of length n with two isolated vertices and construct all such digraphs.