• Journal of the Korean Mathematical Society
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• v.43 no.1
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• pp.215-224
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• 2006

We call the bipartite graph G is normal provided the reduced adjacency matrix A of G is normal. In this paper we give graph representations of normal matrices. In addition we shall have the characterization of signed bipartite normal graphs.

• Proceedings of the Korean Society of Computational and Applied Mathematics Conference
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• 2002.09a
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• pp.26-26
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• 2002

• Korean Journal of Mathematics
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• v.12 no.2
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• pp.201-209
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• 2004

We define the incidence matrices of oriented and nonoriented ${\kappa}$-hypergraphs, respectively. We discuss the ranks of some circulant matrices and show that the rank of the incidence matrices of oriented and nonoriented ${\kappa}$-hypergraphs H are $n$ under a certain condition on the ${\kappa}$-edge set or ${\kappa}$-arc set of H.

• Honam Mathematical Journal
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• v.29 no.3
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• pp.341-350
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• 2007

A dynamical system with a skew-commuting involution map is called a flip system. Every flip system on a subshift of finite type is represented by a pair of matrices, one of which is a permutation matrix. The transposition number of this permutation matrix is studied. We define an invariant, called the flip number, that measures the complexity of a flip system, and prove some results on it. More properties of flips on subshifts of finite type with symmetric adjacency matrices are investigated.

• Kyungpook Mathematical Journal
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• v.19 no.1
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• pp.127-134
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• 1979

A (0,1) matrix is acyclic if it does not have a permutation matrix of order 2 as a submatrix. A bipartite tournament is acyclic if and only if its adjacency matrix is acyclic. The concepts of (maximal) order of cyclicity of a matrix and a bipartite tournament are introduced and studied.

• Journal of the Korean Mathematical Society
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• v.42 no.4
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• pp.871-881
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• 2005

In 1977, Ganter and Teirlinck proved that any $2t\;\times\;2t$ matrix with 2t nonzero elements can be partitioned into four sub-matrices of order t of which at most two contain nonzero elements. In 1978, Kramer and Mesner conjectured that any $mt{\times}nt$ matrix with kt nonzero elements can be partitioned into mn submatrices of order t of which at most k contain nonzero elements. In 1995, Brualdi et al. showed that this conjecture is true if $m = 2,\;k\;\leq\;3\;or\;k\geq\;mn-2$. They also found a counterexample of this conjecture when m = 4, n = 4, k = 6 and t = 2. When t = 2, we show that this conjecture is true if $k{\leq}5$.

• The Pure and Applied Mathematics
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• v.18 no.2
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• pp.141-155
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• 2011

We compute zeta functions of 1-solenoids. When our 1-solenoid is nonorientable, we compute Artin-Mazur zeta function and Lefschetz zeta function of the 1-solenoid and its orientable double cover explicitly in terms of adjacency matrices and branch points. And we show that Artin-Mazur zeta function of orientable double cover is a rational function and a quotient of Artin-Mazur zeta function and Lefschetz zeta function of the 1-solenoid.

• Journal of Institute of Control, Robotics and Systems
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• v.20 no.12
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• pp.1225-1230
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• 2014

This paper investigates the group average-consensus and group formation-consensus problems for first-order multi-agent systems. The control protocol for group consensus is designed by considering the positive adjacency elements. Since each intra-group Laplacian matrix cannot be satisfied with the in-degree balance because of the positive adjacency elements between groups, we decompose the Laplacian matrix into an intra-group Laplacian matrix and an inter-group Laplacian matrix. Moreover, average matrices are used in the control protocol to analyze the stability of multi-agent systems with a fixed and undirected communication topology. Using the graph theory and the Lyapunov functional, stability analysis is performed for group average-consensus and group formation-consensus, respectively. Finally, some simulation results are presented to validate the effectiveness of the proposed control protocol for group consensus.

• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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• pp.125-135
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• 2004

One of the intriguing and fundamental algorithmic graph problems is the computation of the strongly connected components of a directed graph G. In this paper we first introduce a simple procedure for determining the location of the nonzero elements occurring in $B^{-1}$ without fully inverting B, where EB\;{\equiv}\;(b_{ij)\;and\;B^T$ are diagonally dominant matrices with $b_{ii}\;>\;0$ for all i and $b_{ij}\;{\leq}\;0$, for $i\;{\neq}\;j$, and then, as an application, show that all of the strongly connected components of a directed graph G can be recognized by the location of the nonzero elements occurring in the matrix $C(G)\;=\;(D\;-\;A(G))^{-1}$. Here A(G) is an adjacency matrix of G and D is an arbitrary scalar matrix such that (D - A(G)) becomes a diagonally dominant matrix.

• Kyungpook Mathematical Journal
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• v.59 no.2
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• pp.335-340
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• 2019

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. Some variants of energy can also be found in the literature, in which the energy is defined for the Laplacian matrix, Distance matrix, Commonneighbourhood matrix or Seidel matrix. The Seidel matrix of the graph G is the square matrix in which $ij^{th}$ entry is -1 or 1, if the vertices $v_i$ and $v_j$ are adjacent or non-adjacent respectively, and is 0, if $v_i=v_j$. The Seidel energy of G is the sum of the absolute values of the eigenvalues of its Seidel matrix. We present here some families of pairs of graphs whose Seidel matrices have different eigenvalues, but who have the same Seidel energies.