• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.141-154
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• 2007

The connectivity problem is a fundamental problem in graph theory. The best known algorithm to solve the connectivity problem on general graphs with n vertices and m edges takes $O(K(G)mn^{1.5})$ time, where K(G) is the vertex connectivity of G. In this paper, an efficient algorithm is designed to solve vertex connectivity problem, which takes $O(n^2)$ time and O(n) space for a trapezoid graph.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.339-358
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• 2021

Zagreb indices and their modified versions of a molecular graph are important descriptors which can be used to characterize the structural properties of organic molecules from different aspects. In this work, we investigate some properties of the modified second multiplicative Zagreb index of graphs with given connectivity. In particular, we obtain the maximum values of the modified second multiplicative Zagreb index with fixed number of cut edges, or cut vertices, or edge connectivity, or vertex connectivity of graphs. Furthermore, we characterize the corresponding extremal graphs.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.20 no.2
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• pp.415-420
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• 2016

A graph G=(V,E) is called an interval graph with a set V of vertices representing intervals on a line such that there is an edge $(i,j){\in}E$ if and only if intervals i and j intersect. In this paper, we are concerned in the vertex connectivity, one of various characteristics of the graph. Specifically, the vertex connectivity of an interval graph is represented by the overlapping of intervals. Also we propose an efficient algorithm to compute the vertex connectivity on the fully dynamic environment in which the vertices or the edges are inserted or deleted. Using a special kind of interval tree, we show how to compute the vertex connectivity and to maintain the tree in O(logn) time when a new interval is added or an existing interval is deleted.

• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.61-74
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• 2021

Let G = (V, E) be a connected graph. The eccentric connectivity index of G is defined by ��C (G) = ∑u∈V (G) deg(u)e(u), where deg(u) and e(u) denote the degree and eccentricity of the vertex u in G, respectively. In this paper, we introduce a new formulation of ��C that will be called the distance eccentric connectivity index of G and defined by $${\xi}^{De}(G)\;=\;{\sum\limits_{u{\in}V(G)}}\;deg^{De}(u)e(u)$$ where degDe(u) denotes the distance eccentricity degree of the vertex u in G. The aim of this paper is to introduce and study this new topological index. The values of the eccentric connectivity index is calculated for some fundamental graph classes and also for some graph operations. Some inequalities giving upper and lower bounds for this index are obtained.

• Proceedings of the IEEK Conference
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• 2000.07a
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• pp.462-465
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• 2000

It is important to compress three dimensional (3D) data efficiently, since 3D data are too large to store or transmit in general. In this paper, we propose a lossless compression algorithm of the 3D mesh connectivity, based on the vertex degree. Most techniques for the 3D mesh compression treat the connectivity and the geometric separately, but our approach attempts to exploit the geometric information for compressing the connectivity information. We use the geometric angle constraint of the vertex fanout pattern to predict the vertex degree, so the proposed algorithm yields higher compression efficiency than the conventional algorithms.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.31-46
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• 2021

Let a and b be positive integers, and let V (G), ��(G), and ��2(G) be the vertex set of a graph G, the minimum degree of G, and the minimum degree sum of two non-adjacent vertices in V (G), respectively. An even [a, b]-factor of a graph G is a spanning subgraph H such that for every vertex v ∈ V (G), dH(v) is even and a ≤ dH(v) ≤ b, where dH(v) is the degree of v in H. Matsuda conjectured that if G is an n-vertex 2-edge-connected graph such that $n{\geq}2a+b+{\frac{a^2-3a}{b}}-2$, ��(G) ≥ a, and ${\sigma}_2(G){\geq}{\frac{2an}{a+b}}$, then G has an even [a, b]-factor. In this paper, we provide counterexamples, which are highly connected. Furthermore, we give sharp sufficient conditions for a graph to have an even [a, b]-factor. For even an, we conjecture a lower bound for the largest eigenvalue in an n-vertex graph to have an [a, b]-factor.

• Journal of KIISE:Computer Systems and Theory
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• v.29 no.9
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• pp.493-502
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• 2002

The connectivity and edge-connectivity have been the prime deterministic measure of fault tolerance in multicomputer networks. These parameters have a problem that they do not differentiate the different types of disconnected graphs which result from removing the disconnecting vertices or disconnecting edges. To compensate for this shortcoming, one can utilize generalized measures of connectedness such as superconnectivity, toughness, scattering number, vertex-integrity, binding number, and restricted connectivity. In this paper, we analyze such deterministic measures of fault tolerance in recursive circulants and hypercubes, and compare them in terms of fault tolerance.

• Journal of applied mathematics & informatics
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• v.38 no.1_2
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• pp.189-200
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• 2020

Let G be a simple connected graph with n vertices and m edges. Denote by d₁ ≥ d₂ ≥ ⋯ ≥ d_n > 0 and d(e₁) ≥ d(e₂) ≥ ⋯ ≥ d(e_m) sequences of vertex and edge degrees, respectively. If vertices v_i and v_j are adjacent, we write i ~ j. The general sum-connectivity index is defined as 𝒳_α(G) = ∑_i~j(d_i + d_j)^α, where α is an arbitrary real number. Firstly, we determine a relation between 𝒳_α(G) and 𝒳_α-1(G). Then we use it to obtain some new bounds for 𝒳_α(G).

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1197-1211
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• 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.

• Proceedings of the IEEK Conference
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• 2006.06a
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• pp.745-746
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• 2006

In this paper, we propose a new predictive coding scheme for color data of three-dimensional (3-D) mesh models. We exploit connectivity and geometry information to improve coding efficiency. After ordering all vertices in a 3-D mesh model with a vertex traversal technique, we employ a geometry predictor to compress the color data. The predicted color can be acquired by a weighted sum of reconstructed colors for adjacent vertices using both angles and distances between the current vertex and adjacent vertices.