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The Least Eigenvalue of Graphs whose Complements Are Uni- cyclic

100%
Wang Y., Fan Y.-Z., Li X.-X., Zhang F.-F.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
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nr 2
249-260
EN
A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n/2 ). In this paper we discuss the minimizing graphs of a special class of graphs of order n whose complements are connected and contains exactly one cycle (namely the class Ucn of graphs whose complements are unicyclic), and characterize the unique minimizing graph in Ucn when n ≥ 20.
2
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Characterization of knot complements in the n-sphere

100%
Liem V.-T., Venema G. A.
Fundamenta Mathematicae
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1995
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tom 147
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nr 2
189-196
EN
Knot complements in the n-sphere are characterized. A connected open subset W of $S^n$ is homeomorphic with the complement of a locally flat (n-2)-sphere in $S^n$, n ≥ 4, if and only if the first homology group of W is infinite cyclic, W has one end, and the homotopy groups of the end of W are isomorphic to those of $S^1$ in dimensions less than n/2. This result generalizes earlier theorems of Daverman, Liem, and Liem and Venema.
3
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On graphs G for which both g and G̅ are claw-free

100%
Fujita S.
Discussiones Mathematicae Graph Theory
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2005
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tom 25
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nr 3
267-272
EN
Let G be a graph with |V(G)| ≥ 10. We prove that if both G and G̅ are claw-free, then min{Δ(G), Δ(G̅)} ≤ 2. As a generalization of this result in the case where |V(G)| is sufficiently large, we also prove that if both G and G̅ are $K_{1,t}$-free, then min{Δ(G),Δ(G̅)} ≤ r(t- 1,t)-1 where r(t-1,t) is the Ramsey number.
4
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Domination Parameters of a Graph and its Complement

75%
Desormeaux W. J., Haynes T. W., Henning M. A.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
|
nr 1
203-215
EN
A dominating set in a graph G is a set S of vertices such that every vertex in V (G) \ S is adjacent to at least one vertex in S, and the domination number of G is the minimum cardinality of a dominating set of G. Placing constraints on a dominating set yields different domination parameters, including total, connected, restrained, and clique domination numbers. In this paper, we study relationships among domination parameters of a graph and its complement.
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