• Kyungpook Mathematical Journal
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• v.61 no.3
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• pp.671-677
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• 2021

For a distance-regular graph with valency k, second largest eigenvalue r and diameter D, it is known that r ≥ $min\{\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2},\;a_3\}$ if D = 3 and r ≥ $\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2}$ if D ≥ 4, where λ = a₁. This result can be generalized to the class of edge-regular graphs. For an edge-regular graph with parameters (v, k, λ) and diameter D ≥ 4, we compare $\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2}$ with the local valency λ to find a relationship between the second largest eigenvalue and the local valency. For an edge-regular graph with diameter 3, we look at the number $\frac{{\lambda}-\bar{\mu}+\sqrt{({\lambda}-\bar{\mu})^2+4(k-\bar{\mu})}}{2}$, where $\bar{\mu}=\frac{k(k-1-{\lambda})}{v-k-1}$, and compare this number with the local valency λ to give a relationship between the second largest eigenvalue and the local valency. Also, we apply these relationships to distance-regular graphs.

• Journal of the Korean Mathematical Society
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• v.56 no.4
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• pp.857-867
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• 2019

Suppose that G = (V, E) is a connected locally finite graph with the vertex set V and the edge set E. Let ${\Omega}{\subset}V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph G $$\{-{\Delta}_{pu}={\lambda}K(x){\mid}u{\mid}^{p-2}u+f(x,u),\;x{\in}{\Omega}^{\circ},\\u=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}^{\circ}$ and ${\partial}{\Omega}$ denote the interior and the boundary of ${\Omega}$, respectively, ${\Delta}_p$ is the discrete p-Laplacian, K(x) is a given function which may change sign, ${\lambda}$ is the eigenvalue parameter and f(x, u) has exponential growth. We prove the existence and monotonicity of the principal eigenvalue of the corresponding eigenvalue problem. Furthermore, we also obtain the existence of a positive solution by using variational methods.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.1
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• pp.7-11
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• 2008

The energy of a graph is the sum of the absolute values of its eigen values. We obtain some bounds for the energy of planar graphs in terms of its vertices, edges and faces.

• Kyungpook Mathematical Journal
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• v.57 no.2
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• pp.211-222
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• 2017

In this paper, we consider three inverse eigenvalue problems for a special type of acyclic matrices. The acyclic matrices considered in this paper are described by a graph called a broom on n + m vertices, which is obtained by joining m pendant edges to one of the terminal vertices of a path on n vertices. The problems require the reconstruction of such a matrix from given partial eigen data. The eigen data for the first problem consists of the largest eigenvalue of each of the leading principal submatrices of the required matrix, while for the second problem it consists of an eigenvalue of each of its trailing principal submatrices. The third problem has an eigenvalue and a corresponding eigenvector of the required matrix as the eigen data. The method of solution involves the use of recurrence relations among the leading/trailing principal minors of ${\lambda}I-A$, where A is the required matrix. We derive the necessary and sufficient conditions for the solutions of these problems. The constructive nature of the proofs also provides the algorithms for computing the required entries of the matrix. We also provide some numerical examples to show the applicability of our results.

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.75-86
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• 1996

Let G be a finite simple connected graph with vertex set V(G) and edge set E(G). Let A(G) be the adjacency matrix of G. The characteristic polynomial of G is the characteristic polynomial $\Phi(G;\lambda) = det(\lambda I - A(G))$ of A(G). A zero of $\Phi(G;\lambda)$ is called an eigenvalue of G.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.61-74
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• 2006

Let G be a simple graph and let G denote its complement. We say that $\bar{G}$ is integral if its spectrum consists of integral values. In this work we establish a characterization of integral graphs which belong to the class $\bar{{\alpha}K_{a,a}\cup{\beta}K_{b,b}}$, where mG denotes the m-fold union of the graph G.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.8 no.6
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• pp.1218-1227
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• 2004

Multi-domain engineering systems include electrical, mechanical, hydraulic, pneumatic, and thermal components, making it difficult to design a system because of their complexity and inter domain nature. In order to obtain an optimal design, a unified design approach for each domain and an automated search method are required. This paper suggests a method for automatically synthesizing designs for multi-domain systems using the combination of bond graph that is domain independent and genetic programming that is well recognized as a powerful tool for open-ended search. To investigate the effect of proposed approach, an eigenvalue design problem is tested for some sample target sets of eigenvalues with different embryos.

• Journal of applied mathematics & informatics
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• v.33 no.5_6
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• pp.627-633
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• 2015

In this paper, we generalize the concept of the energy of Seidel matrix S(G) which denoted by S^α(G) and obtain some results related to this matrix. Also, we obtain an upper and lower bound for S^α(G) related to all of graphs with |detS(G)| ≥ (n - 1); n ≥ 3.

• Bulletin of the Korean Mathematical Society
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• v.45 no.1
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• pp.95-99
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• 2008

It is known that each eigenvalue of a real symmetric, irreducible, tridiagonal matrix has multiplicity 1. The graph of such a matrix is a path. In this paper, we extend the result by classifying those trees for which each of the associated acyclic matrices has distinct eigenvalues whenever the diagonal entries are distinct.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2005.04a
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• pp.185-188
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• 2005

This paper suggests a control method for efficient topology/parameter evolution in a bond-graph-based GP design framework that automatically synthesizes designs for multi-domain, lumped parameter dynamic systems, We adopt a hierarchical breeding control mechanism with fitness-level-dependent differences to obtain better balancing of topology/parameter search - biased toward topological changes at low fitness levels, and toward parameter changes at high fitness levels. As a testbed for this approach, an eigenvalue assignment problem, which is to find bond graph models exhibiting minimal distance errors from target sets of eigenvalues, was tested and showed improved performance for various sets of eigenvalues.