• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.647-655
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• 2016

Let H be a graph, and $k{\geq}{\chi}(H)$ an integer. We say that H has a cyclic Gray code of k-colourings if and only if it is possible to list all its k-colourings in such a way that consecutive colourings, including the last and the first, agree on all vertices of H except one. The Gray code number of H is the least integer $k_0(H)$ such that H has a cyclic Gray code of its k-colourings for all $k{\geq}k_0(H)$. For complete bipartite graphs, we prove that $k_0(K_{\ell},r)=3$ when both ${\ell}$ and r are odd, and $k_0(K_{\ell},r)=4$ otherwise.

• Kyungpook Mathematical Journal
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• v.61 no.3
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• pp.661-669
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• 2021

For a simple, undirected graph G = (V, E), a perfect Roman dominating function (PRDF) f : V → {0, 1, 2} has the property that, every vertex u with f(u) = 0 is adjacent to exactly one vertex v for which f(v) = 2. The weight of a PRDF is the sum f(V) = ∑_v∈V f(v). The minimum weight of a PRDF is called the perfect Roman domination number, denoted by γ_R^P(G). Given a graph G and a positive integer k, the PRDF problem is to check whether G has a perfect Roman dominating function of weight at most k. In this paper, we first investigate the complexity of PRDF problem for some subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. Then we show that PRDF problem is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.467-481
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• 2015

Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. We introduce a signed a-polynomial which is a generalization of the signed a-number and gives a-, b-, and c-numbers. The signed a-polynomial of a graph G is related to the $Poincar\acute{e}$ polynomial $P_{M(G)}(z)$, which is the generating function for the Betti numbers of the real toric manifold M(G). We give the generating functions for the signed a-polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. As a consequence, we find the Euler characteristic number and the Betti numbers of the real toric manifold M(G) for complete multipartite graphs G.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.201-215
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• 2003

A map is a connected topological graph $\Gamma$ cellularly embedded in a surface. For any connected graph $\Gamma$, by introducing the concertion of semi-arc automorphism group Aut$\_$$\frac{1}{2}$/$\Gamma$ and classifying all embedding of $\Gamma$ undo. the action of this group, the numbers r$\^$O/ ($\Gamma$) and r$\^$N/($\Gamma$) of rooted maps on orientable and non-orientable surfaces with underlying graph $\Gamma$ are found. Many closed formulas without sum ∑ for the number of rooted maps on surfaces (orientable or non-orientable) with given underlying graphs, such as, complete graph K$\_$n/, complete bipartite graph K$\_$m, n/ bouquets B$\_$n/, dipole Dp$\_$n/ and generalized dipole (equation omitted) are refound in this paper.

• Annals of Fuzzy Mathematics and Informatics
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• v.16 no.3
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• pp.301-316
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• 2018

In this paper, the concept of connected geodesic number, $gn_c(G)$, of a fuzzy graph G is introduced and its limiting bounds are identified. It is proved that all extreme nodes of G and all cut-nodes of the underlying crisp graph $G^*$ belong to every connected geodesic cover of G. The connected geodesic number of complete fuzzy graphs, fuzzy cycles, fuzzy trees and of complete bipartite fuzzy graphs are obtained. It is proved that for any pair k, n of integers with $3{\leq}k{\leq}n$, there exists a connected fuzzy graph G : (V, ${\sigma}$, ${\mu}$) on n nodes such that $gn_c(G)=k$. Also, for any positive integers $2{\leq}a<b{\leq}c$, it is proved that there exists a connected fuzzy graph G : (V, ${\sigma}$, ${\mu}$) such that the geodesic number gn(G) = a and the connected geodesic number $gn_c(G)=b$.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.345-350
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• 2022

In this work, we confirm a weak version of a conjecture proposed by Hong Wang. The ideal of the work comes from the tree packing conjecture made by Gyárfás and Lehel. Bollobás confirms the tree packing conjecture for many small tree, who showed that one can pack T₁, T₂, …, $T_{n/\sqrt{2}}$ into Kn and that a better bound would follow from a famous conjecture of Erdős. In a similar direction, Hobbs, Bourgeois and Kasiraj made the following conjecture: Any sequence of trees T₁, T₂, …, T_n, with T_i having order i, can be packed into K_n-1,[n/2]. Further Hobbs, Bourgeois and Kasiraj [3] proved that any two trees can be packed into a complete bipartite graph K_n-1,[n/2]. Motivated by the result, Hong Wang propose the conjecture: For each k-partite tree T(𝕏) of order n, there is a restrained packing of two copies of T(𝕏) into a complete k-partite graph B_n+m(𝕐), where $m={\lfloor}{\frac{k}{2}}{\rfloor}$. Hong Wong [4] confirmed this conjecture for k = 2. In this paper, we prove a weak version of this conjecture.

• Journal of the Korean Statistical Society
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• v.23 no.2
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• pp.513-522
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• 1994

A multi-dimensional design is most easily constructed via the amalgamation of one-dimensional component block designs. However, not all sets of component designs are compatible to be amalgamated. The conditions for compatibility are related to the concept of a complete matching in a graph. In this paper, we give sufficient conditions for unequal-replicate designs. Two types of conditions are proposed; one is based on the number of verices adjacent to at least one vertex and the other is ona a degree of vertex, in a bipartite graph. The former is an extension of the sufficient conditions of equal-replicate designs given by Dean an Lewis (1988).

• Journal of the Chungcheong Mathematical Society
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• v.35 no.3
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• pp.205-225
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• 2022

In this paper, we derive analytical closed results for the first (a, b)-KA index, the Sombor index, the modified Sombor index, the first reduced (a, b)-KA index, the reduced Sombor index, the reduced modified Sombor index, the second reduced (a, b)-KA index and the mean Sombor index mSO_α for the OTIS biswapped networks by considering basis graphs as path, wheel graph, complete bipartite graph and r-regular graphs. Network theory plays a significant role in electronic and electrical engineering, such as signal processing, networking, communication theory, and so on. A topological index (TI) is a real number associated with graph networks that correlates chemical networks with a variety of physical and chemical properties as well as chemical reactivity. The Optical Transpose Interconnection System (OTIS) network has recently received increased interest due to its potential uses in parallel and distributed systems.

• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.87-98
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• 2014

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say ${\Gamma}(M)$, such that when M = R, ${\Gamma}(M)$ is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F. Anderson and S. B. Mulay, in [6], have been generalized for ${\Gamma}(M)$ in the present article. We show that ${\Gamma}(M)$ is connected with $diam({\Gamma}(M)){\leq}3$. We also show that for a reduced module M with $Z(M)^*{\neq}M{\backslash}\{0\}$, $gr({\Gamma}(M))={\infty}$ if and only if ${\Gamma}(M)$ is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, $x,y{\in}M{\backslash}\{0\}$ are adjacent if and only if $xR{\cap}yR=(0)$. Among other things, it is also observed that ${\Gamma}(M)={\emptyset}$ if and only if M is uniform, ann(M) is a radical ideal, and $Z(M)^*{\neq}M{\backslash}\{0\}$, if and only if ann(M) is prime and $Z(M)^*{\neq}M{\backslash}\{0\}$.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.685-695
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• 2022

The minimum number of complete bipartite subgraphs needed to partition the edges of a graph G is denoted by b(G). A known lower bound on b(G) states that b(G) ≥ max{p(G), q(G)}, where p(G) and q(G) are the numbers of positive and negative eigenvalues of the adjacency matrix of G, respectively. When equality is attained, G is said to be eigensharp and when b(G) = max{p(G), q(G)} + 1, G is called an almost eigensharp graph. In this paper, we investigate the eigensharpness and almost eigensharpness of lexicographic products of some graphs.