• Journal of applied mathematics & informatics
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• v.42 no.5
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• pp.1121-1135
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• 2024

In this paper, we introduced a new distance-based index called the minimum degree Wiener index, which is the sum of distances between all unordered pairs of vertices with the minimum degree. Additionally, a matrix related to this index was introduced, and it was discovered that the sum of entries in each row was the same for some classes of graphs, contrary to many graph-related matrices. In particular, we determined the minimum degree Wiener index of the bipartite Kneser graph, bipartite Kneser type-k graphs, Johnson graph and the set inclusion graphs. The terminal Wiener index of a graph G is the sum of distances between all unordered pairs of pendant vertices of G. Also, we determined Wiener index, hyper Wiener index and corresponding polynomials of the bipartite Kneser type-k graphs for k = 2, 3.

• Management Science and Financial Engineering
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• v.8 no.1
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• pp.1-19
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• 2002

Ordering is used to reduce the amount of fill-ins in the Cholesky factor of a symmetric positive definite matrix. One of the most efficient ordering methods is the minimum degree ordering algorithm(MDO). In this paper, we provide a few techniques that improve the performance of MDO implemented with the clique storage scheme. First, the absorption of nodes in the cliques is developed which reduces the number of cliques and the amount of storage space required for MDO. Second, we present a modified minimum degree ordering algorithm of which the number of degree updates can be reduced by introducing the lower bounds of degrees. Third, using both the lower and upper bounds of degrees, we develop an approximate minimum degree ordering algorithm. Experimental results show that the proposed algorithm is competitive with the minimum degree ordering algorithm that uses quotient graphs from the points of the ordering time and the nonzeros in the Cholesky factor.

• Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography
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• v.26 no.1
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• pp.85-91
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• 2008

When a large sparse matrix is calculated for a horizontal geodetic network adjustment, it needs to go through the process of matrix reordering for the efficiency of time and space. In this study, several reordering methods for sparse matrix were tested, using Sparse Matrix Manipulation System(SMMS) program, total processing time and Fill-in number produced in factorization process were measured and compared. As a result, Minimum Degree(MD) and Mutiple Minimum Degree(MMD), which are based on Minimum Degree are better than Gibbs-Poole-Stockmeyer(GPS) and Reverse Cuthill-Mckee(RCM), which are based on Minimum Bandwidth. However, the method of the best efficiency can be changed dependent on distribution of non-zero elements in a matrix. This finding could be applied to heighten the efficiency of time and storage space for national datum readjustment and other large geodetic network adjustment.

• Korean Journal of Mathematics
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• v.19 no.4
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• pp.367-379
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• 2011

In this paper, we find the maximum exponent of primitive simple graphs G under the restriction $deg(v){\geq}3$ for all vertex $v$ of G. Our result is also an answer of a Klee and Quaife type problem on exponent to find minimum number of vertices of graphs which have fixed even exponent and the degree of whose vertices are always at least 3.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.511-517
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• 2014

In this paper, we prove that the 3-cycle is light in the family of 1-planar graphs with minimum vertex degree at least 5 and minimum edge degree at least 12. This generates a known result of Fabrici and Madaras [8].

• Journal of the Korean Mathematical Society
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• v.46 no.4
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• pp.759-773
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• 2009

A dominating set for a graph G is a set D of vertices of G such that every vertex of G not in D is adjacent to a vertex of D. Reed [11] considered the domination problem for graphs with minimum degree at least three. He showed that any graph G of minimum degree at least three contains a dominating set D of size at most $\frac{3}{8}$ |V (G)| by introducing a covering by vertex disjoint paths. In this paper, by using this technique, we show that every graph on n vertices of minimum degree at least four contains a dominating set D of size at most $\frac{4}{11}$ |V (G)|.

• Journal of the Korea Society of Computer and Information
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• v.21 no.7
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• pp.47-52
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• 2016

This paper introduces an algorithm to construct an Eulerian cycle for Chinese postman problem. The Eulerian cycle is formed only when all vertices in the graph have an even degree. Among available algorithms to the Eulerian cycle problem, Edmonds-Johnson's stands out as the most efficient of its kind. This algorithm constructs a complete graph composed of shortest path between odd-degree vertices and derives the Eulerian cycle through minimum-weight complete matching method, thus running in $O({\mid}V{\mid}^3)$. On the contrary, the algorithm proposed in this paper selects minimum weight edge from edges incidental to each vertex and derives the minimum spanning tree (MST) so as to finally obtain the shortest-path edge of odd-degree vertices. The algorithm not only runs in simple linear time complexity $O({\mid}V{\mid}log{\mid}V{\mid})$ but also obtains the optimal Eulerian cycle, as the implementation results on 4 different graphs concur.

• Journal of the Korean Operations Research and Management Science Society
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• v.23 no.4
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• pp.21-31
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• 1998

Ordering is used to reduce the amount of fill-ins in the Cholesky factor of an symmetric definite matrix. One of most efficient ordering methods is the minimum degree ordering method. In this paper. we propose the two techniques to improve the performance of the minimum degree ordering which are implemented using clique storage structure. One is node absorption which is a generalized version of clique absorption. The other technique is using the lower bounds of degree to suspend the degree updates of nodes. finally, we provide computational results on the problems on NETLIB. These results show that the proposed techniques reduce the number of degree updates and the computational time considerably.

• Kyungpook Mathematical Journal
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• v.43 no.4
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• pp.567-577
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• 2003

In this paper, we classify the Steinhaus graphs with minimum degree two.

• Journal of the Korea Society of Computer and Information
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• v.20 no.5
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• pp.31-39
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• 2015

The degree-constrained minimum spanning tree (d-MST) problem is considered NP-complete for no exact solution-yielding polynomial algorithm has been proposed to. One thus has to resort to an heuristic approximate algorithm to obtain an optimal solution to this problem. This paper therefore presents a polynomial time algorithm which obtains an intial solution to the d-MST with the help of Kruskal's algorithm and performs k-opt on the initial solution obtained so as to derive the final optimal solution. When tested on 4 graphs, the algorithm has successfully obtained the optimal solutions.