• Journal of applied mathematics & informatics
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• v.34 no.3_4
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• pp.227-237
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• 2016

In this paper we introduce a new graph labeling called k-prime cordial labeling. Let G be a (p, q) graph and 2 ≤ p ≤ k. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). f is called a k-prime cordial labeling of G if |v_f (i) − v_f (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |e_f (0) − e_f (1)| ≤ 1 where v_f (x) denotes the number of vertices labeled with x, e_f (1) and e_f (0) respectively denote the number of edges labeled with 1 and not labeled with 1. A graph with a k-prime cordial labeling is called a k-prime cordial graph. In this paper we investigate the k-prime cordial labeling behavior of a star and we have proved that every graph is a subgraph of a k-prime cordial graph. Also we investigate the 3-prime cordial labeling behavior of path, cycle, complete graph, wheel, comb and some more standard graphs.

• Journal of applied mathematics & informatics
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• v.42 no.3
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• pp.567-582
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• 2024

A total coloring of a graph G is an assignment of colors to the elements of a graphs G such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph G , denoted by χ''(G), is the minimum number of colors that suffice in a total coloring. In this paper, we proved the Behzad and Vizing conjecture for certain convex polytope graphs D^p_n, Q^p_n, R^p_n, E_n, S_n, G_n, T_n, U_n, C_n,respectively. This significant result in a graph G contributes to the advancement of graph theory and combinatorics by further confirming the conjecture's applicability to specific classes of graphs. The presented proof of the Behzad and Vizing conjecture for certain convex polytope graphs not only provides theoretical insights into the structural properties of graphs but also has practical implications. Overall, this paper contributes to the advancement of graph theory and combinatorics by confirming the validity of the Behzad and Vizing conjecture in a graph G and establishing its relevance to applied problems in sciences and engineering.

• Journal of the Korean Mathematical Society
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• v.45 no.6
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• pp.1647-1659
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• 2008

Let R be a ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. First, we investigate some connected conditions of the zero-divisor graph $\Gamma(R)$ of a noncommutative ring R as follows: (1) if $\Gamma(R)$ has no sources and no sinks, then $\Gamma(R)$ is connected and diameter of $\Gamma(R)$, denoted by diam($\Gamma(R)$) (resp. girth of $\Gamma(R)$, denoted by g($\Gamma(R)$)) is equal to or less than 3; (2) if X is a union of finite number of orbits under the left (resp. right) regular action on X by G, then $\Gamma(R)$ is connected and diam($\Gamma(R)$) (resp. g($\Gamma(R)$)) is equal to or less than 3, in addition, if R is local, then there is a vertex of $\Gamma(R)$ which is adjacent to every other vertices in $\Gamma(R)$; (3) if R is unit-regular, then $\Gamma(R)$ is connected and diam($\Gamma(R)$) (resp. g($\Gamma(R)$)) is equal to or less than 3. Next, we investigate the graph automorphisms group of $\Gamma(Mat_2(\mathbb{Z}_p))$ where $Mat_2(\mathbb{Z}_p)$ is the ring of 2 by 2 matrices over the galois field $\mathbb{Z}_p$ (p is any prime).

• Journal of the Korea Institute of Information and Communication Engineering
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• v.21 no.11
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• pp.2103-2108
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• 2017

For n points $p_i$ on the m-dimensional plane $R^m$ and a fixed range r, consider a set $T_i$ containing points the distances from $p_i$ of which are less than or equal to r. In case m=1, $T_i$ is an interval on a line, it is a circle on a plane when m=2. For the vertices corresponding to the sets $T_i$, there is an edge between the vertices if the two sets intersect. Then this graph is called an intersection graph G. For m=1 G is called a proper interval graph and for m=2, it is called an unit disk graph. In this paper, we are concerned in the intersection graph G(r) when r changes. In particular, we consider the problem to find the minimum r such that G(r)is connected. For this problem, we propose an O(n) algorithm for the proper interval graph and an $O(n^2{\log}\;n)$ algorithm for the unit disk graph. For the dynamic environment in which the points on a line are added or deleted, we give an O(log n) algorithm for the problem.

• Communications of the Korean Mathematical Society
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• v.28 no.1
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• pp.79-85
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• 2013

The structure of a poset P with smallest element 0 is looked at from two view points. Firstly, with respect to the Zariski topology, it is shown that Spec(P), the set of all prime semi-ideals of P, is a compact space and Max(P), the set of all maximal semi-ideals of P, is a compact $T_1$ subspace. Various other topological properties are derived. Secondly, we study the semi-ideal-based zero-divisor graph structure of poset P, denoted by $G_I$ (P), and characterize its diameter.

• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.703-722
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• 2021

Let G = (V, E) be a connected locally finite and weighted graph, ∆_p be the p-th graph Laplacian. Consider the p-th nonlinear equation -∆_pu + h|u|^p-2u = f(x, u) on G, where p > 2, h, f satisfy certain assumptions. Grigor'yan-Lin-Yang [24] proved the existence of the solution to the above nonlinear equation in a bounded domain Ω ⊂ V. In this paper, we show that there exists a strictly positive solution on the infinite set V to the above nonlinear equation by modifying some conditions in [24]. To the m-order differential operator 𝓛_m,p, we also prove the existence of the nontrivial solution to the analogous nonlinear equation.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.1-23
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• 2005

The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents an $O(n^2)$ time sequential algorithm and an $O(n^2/p+logn)$ time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, where p and n represent respectively the number of processors and the number of vertices of the circular-arc graph.

• The Pure and Applied Mathematics
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• v.12 no.1
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• pp.47-55
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• 2005

For topological spaces X, Y and the function f : X → Y, it induces a function gr(f) : X → X x Y defined as gr(f)(χ) = (χ, f(χ)), for every χ ∈ X. It deals with some preliminary investigations relating to the behavior of functions and its graph functions. It has also been found that continuous functions are homotopic if and only if their graph functions are homotopic.

• Journal of Biomedical Engineering Research
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• v.38 no.2
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• pp.82-88
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• 2017

Enlargement of the lateral ventricles have been identified as a surrogate marker of neurological disorders. Quantitative measure of the lateral ventricle from MRI would enable earlier and more accurate clinical diagnosis in monitoring disease progression. Even though it requires an automated or semi-automated segmentation method for objective quantification, it is difficult to define lateral ventricles due to insufficient contrast and brightness of structural imaging. In this study, we proposed a fully automated lateral ventricle segmentation method based on a graph cuts algorithm combined with atlas-based segmentation and connected component labeling. Initially, initial seeds for graph cuts were defined by atlas-based segmentation (ATS). They were adjusted by partial volume images in order to provide accurate a priori information on graph cuts. A graph cuts algorithm is to finds a global minimum of energy with minimum cut/maximum flow algorithm function on graph. In addition, connected component labeling used to remove false ventricle regions. The proposed method was validated with the well-known tools using the dice similarity index, recall and precision values. The proposed method was significantly higher dice similarity index ($0.860{\pm}0.036$, p < 0.001) and recall ($0.833{\pm}0.037$, p < 0.001) compared with other tools. Therefore, the proposed method yielded a robust and reliable segmentation result.

• Journal of the Korea Society of Computer and Information
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• v.24 no.7
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• pp.61-69
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• 2019

In this paper, a classification method of random graph models is proposed and it is based on centralities of the random graphs. Similarity between two random graphs is measured for the classification of random graph models. The similarity between two random graph models $G^{R_1}$ and $G^{R_2}$ is defined by the distance of $G^{R_1}$ and $G^{R_2}$, where $G^{R_2}$ is a set of random graph $G^{R_2}=\{G_1^{R_2},...,G_p^{R_2}\}$ that have the same number of nodes and edges as random graph $G^{R_1}$. The distance($G^{R_1},G^{R_2}$) is obtained by comparing centralities of $G^{R_1}$ and $G^{R_2}$. Through the computational experiments, we show that it is possible to compare random graph models regardless of the number of vertices or edges of the random graphs. Also, it is possible to identify and classify the properties of the random graph models by measuring and comparing similarities between random graph models.