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A Relationship between the Second Largest Eigenvalue and Local Valency of an Edge-regular Graph
Park, Jongyook Department of Mathematics 2021 Kyungpook mathematical journal Vol.61 No.3
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 = a<sub>1</sub>. 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.
An inequality involving the second largest and smallest eigenvalue of a distance-regular graph
Koolen, Jack H.,Park, Jongyook,Yu, Hyonju Elsevier 2011 Linear algebra and its applications Vol.434 No.12
<P><B>Abstract</B></P><P>For a distance-regular graph with second largest eigenvalue (resp., smallest eigenvalue) <SUB>θ1</SUB> (resp., <SUB>θD</SUB>) we show that (<SUB>θ1</SUB>+1)(<SUB>θD</SUB>+1)⩽-<SUB>b1</SUB> holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.</P>
Partially metric association schemes with a multiplicity three
van Dam, Edwin R.,Koolen, Jack H.,Park, Jongyook Elsevier 2018 Journal of combinatorial theory. Series B Vol.130 No.-
<P><B>Abstract</B></P> <P>An association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity three. Besides the association schemes related to regular complete 4-partite graphs, we obtain the association schemes related to the Platonic solids, the bipartite double scheme of the dodecahedron, and three association schemes that are related to well-known 2-arc-transitive covers of the cube: the Möbius–Kantor graph, the Nauru graph, and the Foster graph F048A. In order to obtain this result, we also determine the symmetric association schemes with a multiplicity three and a connected relation with valency three. Moreover, we construct an infinite family of cubic arc-transitive 2-walk-regular graphs with an eigenvalue with multiplicity three that give rise to non-commutative association schemes with a symmetric relation of valency three and an eigenvalue with multiplicity three.</P>
홈 네트워크 환경에 적합한 디바이스 인증 프로토콜 요구사항
이정환(JungHwan Lee),황유동(YouDong Hwang),박동규(DongGue Park),한종욱(JongYook Han) 한국정보보호학회 2005 情報保護學會誌 Vol.15 No.6
미래의 홈 네트워크 환경에서는 다양한 홈 디바이스들이 인터넷을 통하여 외부와 직접 또는 간접적으로 연결이 되기 때문에 기존의 인터넷에 존재하는 보안 위협 이외에도 다양한 보안 위협들이 존재한다. 이러한 홈 네트워크 환경에서 디바이스 사이에 안전한 데이터 통신을 위해서는 보안에 대한 필요성이 절실히 요구되며 그중에서도 디바이스 인증은 매우 중요하다. 따라서 본 논문에서는 홈 네트워크 환경에서 기존에 연구되던 방식들을 분석하고 신뢰할 수 있는 홈서비스를 제공하기 위하여 홈 디바이스 특성을 고려한 새로운 디바이스 인증 프로토콜 요구사항을 정리하고자 한다.