• Korean Journal of Mathematics
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• 제29권3호
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• pp.577-580
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• 2021

For a finite simplicial graph 𝚪, let G(𝚪) denote the right-angled Artin group on the complement graph of 𝚪. For path graphs P_k, cycle graphs C_ℓ and complete bipartite graphs K_n,m, this article characterizes the embeddability of G(K_n,m) in G(P_k) and in G(C_ℓ).

• 호남수학학술지
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• 제36권2호
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• pp.399-415
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• 2014

We study the Seifert surfaces of a link by relating the embeddings of graphs with induced graphs. As applications, we prove that every link L is the boundary of an oriented surface which is obtained from a graph embedding of a complete bipartite graph $K_{2,n}$, where all voltage assignments on the edges of $K_{2,n}$ are 0. We also provide an algorithm to construct such a graph diagram of a given link and demonstrate the algorithm by dealing with the links $4^2_1$ and $5_2$.

• Kyungpook Mathematical Journal
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• 제61권3호
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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.

• East Asian mathematical journal
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• 제25권4호
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• pp.533-541
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• 2009

In this paper, we classify reflexible edge-transitive embeddings of complete bipartite graphs $K_{m,n}$ for any odd positive integers m and n. As a result, for any odd m, n, it will be shown that there exists only one reflexible edge-transitive embedding of $K_{m,n}$ up to isomorphism.

• 대한수학회논문집
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• 제30권3호
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• pp.313-325
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• 2015

An H-magic labeling in a H-decomposable graph G is a bijection $f:V(G){\cup}E(G){\rightarrow}\{1,2,{\cdots},p+q\}$ such that for every copy H in the decomposition, $\sum{_{{\upsilon}{\in}V(H)}}\;f(v)+\sum{_{e{\in}E(H)}}\;f(e)$ is constant. f is said to be H-V -super magic if f(V(G))={1,2,...,p}. In this paper, we prove that complete bipartite graphs $K_{n,n}$ are H-V -super magic decomposable where $$H{\sim_=}K_{1,n}$$ with $n{\geq}1$.

• 대한수학회보
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• 제52권2호
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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.

• Korean Journal of Mathematics
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• 제20권4호
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• pp.403-414
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• 2012

A voltage assignment on a graph was used to enumerate all possible 2-cell embeddings of a graph onto surfaces. The boundary of the surface which is obtained from 0 voltage on every edges of a very special diagram of a complete bipartite graph $K_{m,n}$ is surprisingly the ($m,n$) torus link. In the present article, we prove that every link is the boundary of a complete bipartite multi-graph $K_{m,n}$ for which voltage assignments are either -1 or 1 and that every link is the boundary of a complete bipartite graph $K_{2,n}$ for which voltage assignments are either -1, 0 or 1 where edges in the diagram of graphs may be linked but not knotted.

• Journal of applied mathematics & informatics
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• 제33권3_4호
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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.

• Kyungpook Mathematical Journal
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• 제56권3호
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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.

• 대한수학회지
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• 제54권5호
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• pp.1595-1604
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• 2017

We give a complete formula for the characteristic polynomial of hyperplane arrangements ${\mathcal{J}}_n$ consisting of the hyperplanes $x_i+x_j=1$, $x_k=0$, $x_l=1$, $1{\leq}i$, j, k, $l{\leq}n$. The formula is obtained by associating hyperplane arrangements with graphs, and then enumerating central graphs via generating functions for the number of bipartite graphs of given order, size and number of connected components.