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A recent result of Schmidt has brought Williamson matrices back into the spotlight. In this article, a new algorithm is introduced
to search for hard to find Williamson matrices. We find all nonequivalent Williamson matrices of odd order n up to n = 59. It turns out that there are none for n = 35, 47, 53, 59 and it seems that the Turyn class may be the only infinite class of these matrices.
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In this paper, we investigate connected nonregular graphs with four distinct Laplacian eigenvalues. We characterize all such
graphs which are bipartite or have exactly one multiple Laplacian eigenvalue. Other examples of interest are also presented. 相似文献
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The rank of a graph is that of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced trees as well as bipartite graphs with a given rank and characterize those graphs achieving the maximum order. 相似文献
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For a given graph , the -saturation number of a graph is the minimum number of edges in an edge-maximal -free subgraph of . Recently, the -saturation number of the Erd?s–Rényi random graph has been determined asymptotically for any complete graph . In this paper, we give an asymptotic formula for the -saturation number of when is a star graph. 相似文献
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W.H. Haemers A. Mohammadian B. Tayfeh-Rezaie 《Linear algebra and its applications》2010,432(9):2214-399
Let k be a natural number and let G be a graph with at least k vertices. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most , where e(G) is the number of edges of G. We prove this conjecture for k=2. We also show that if G is a tree, then the sum of the k largest Laplacian eigenvalues of G is at most e(G)+2k-1. 相似文献
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Journal of Algebraic Combinatorics - The notion of disjoint weighing matrices is introduced as a generalization of orthogonal designs. A recursive construction along with a computer search leads to... 相似文献
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The notion of type of quadruples of rows is proven to be useful in the classification of Hadamard matrices. In this paper, we investigate Hadamard matrices with few distinct types. Among other results, the Sylvester Hadamard matrices are shown to be characterized by their spectrum of types. 相似文献
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The existence of large sets of 5-(14,6,3) designs is in doubt. There are five simple 5-(14,6,6) designs known in the literature. In this note, by the use of a computer program, we show that all of these designs are indecomposable and therefore they do not lead to large sets of 5-(14,6,3) designs. Moreover, they provide the first counterexamples for a conjecture on disjoint t-designs which states that if there exists a t-(v, k, λ) design (X, D) with minimum possible value of λ, then there must be a t-(v, k, λ) design (X, D′) such that D∩ D′ = Ø. 相似文献