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We give a tight bound for the triple intersection numbers of Paley graphs. In particular, we show that any three vertices have a common neighbor in Paley graphs of order larger than 25. 相似文献
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Rajendra M. Pawale 《组合设计杂志》2007,15(1):49-60
The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.
- (1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z3 then λ < x + 1 + z + z3.
- (2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
- (3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z2.
- (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
- (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.
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Leonard H. Soicher 《Journal of Combinatorial Theory, Series A》2010,117(7):799-1141
The concept of intersection numbers of order r for t-designs is generalized to graphs and to block designs which are not necessarily t-designs. These intersection numbers satisfy certain integer linear equations involving binomial coefficients, and information on the non-negative integer solutions to these equations can be obtained using the block intersection polynomials introduced by P.J. Cameron and the present author. The theory of block intersection polynomials is extended, and new applications of these polynomials to the studies of graphs and block designs are obtained. In particular, we obtain a new method of bounding the size of a clique in an edge-regular graph with given parameters, which can improve on the Hoffman bound when applicable, and a new method for studying the possibility of a graph with given vertex-degree sequence being an induced subgraph of a strongly regular graph with given parameters. 相似文献
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Rajendra M. Pawale 《Designs, Codes and Cryptography》2011,58(2):111-121
Quasi-symmetric designs with intersection numbers x > 0 and y = x + 2 under the condition λ > 1 are investigated. If D(v, b, r, k, λ; x, y) is a quasi-symmetric design with above conditions then it is shown that either λ = x + 1 or x + 2 or D is a design with the parameters given in the Table 6 or complement of one of these designs. 相似文献
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In this article we study the n‐existential closure property of the block intersection graphs of infinite t‐(v, k, λ) designs for which the block size k and the index λ are both finite. We show that such block intersection graphs are 2‐e.c. when 2?t?k ? 1. When λ = 1 and 2?t?k, then a necessary and sufficient condition on n for the block intersection graph to be n‐e.c. is that n?min{t, ?(k ? 1)/(t ? 1)? + 1}. If λ?2 then we show that the block intersection graph is not n‐e.c. for any n?min{t + 1, ?k/t? + 1}, and that for 3?n?min{t, ?k/t?} the block intersection graph is potentially but not necessarily n‐e.c. The cases t = 1 and t = k are also discussed. © 2011 Wiley Periodicals, Inc. J Combin Designs 19: 85–94, 2011 相似文献
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Abstact: An α‐resolvable BIBD is a BIBD with the property that the blocks can be partitioned into disjoint classes such that every class contains each point of the design exactly α times. In this paper, we show that the necessary conditions for the existence of α‐resolvable designs with block size four are sufficient, with the exception of (α, ν, λ) = (2, 10, 2). © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 1–16, 2001 相似文献
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Geir Ellingsrud Stein Arild Strø mme 《Transactions of the American Mathematical Society》1998,350(6):2547-2552
We compute the intersection number between two cycles and of complementary dimensions in the Hilbert scheme parameterizing subschemes of given finite length of a smooth projective surface . The -cycle corresponds to the set of finite closed subschemes the support of which has cardinality 1. The -cycle consists of the closed subschemes the support of which is one given point of the surface. Since is contained in , indirect methods are needed. The intersection number is , answering a question by H. Nakajima.
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《组合设计杂志》2018,26(7):344-355
We derive a previously unknown lower bound of 41 for the frequency of of an E(s2)‐optimal and minimax‐optimal supersaturated design (SSD) with 20 rows and 76 columns. This is accomplished by an exhaustive computer search that uses the combinatorial properties of resolvable 2 − (20, 10, 36) designs and the parallel class intersection pattern method. We also classify all nonisomorphic E(s2)‐optimal 4‐circulant SSDs with 20 rows and . 相似文献
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Sharad Sane 《Designs, Codes and Cryptography》2001,23(3):291-296
A proof of the following conjecture of Jungnickel and Tonchev on quasi-multiple quasi-symmetric designs is given: Let D be a design whose parameter set (v,b,r,k,) equals (v,sv,sk,k, s) for some positive integer s and for some integers v,k, that satisfy (v-1) = k(k-1) (that is, these integers satisfy the parametric feasibility conditions for a symmetric (v,k,)-design). Further assume that D is a quasi-symmetric design, that is D has at most two block intersection numbers. If (k, (s-1)) = 1, then the only way D can be constructed is by taking multiple copies of a symmetric (v,k, )-design. 相似文献
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Martin Grüttmüller 《组合设计杂志》2000,8(2):122-131
A mandatory representation design MR[ν,K] is a pairwise balanced design on ν points with block sizes from K in which for each k ∈ K there is a block in the design of size k. Mendelsohn and Rees [4] investigated the existence of MR[ν,K]s, where 3 ∈ K. In this report we consider additional necessary conditions, where K = {3,k}. These conditions are proved to be sufficient for 4 ≤ k ≤ 50 with one genuine exception. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 122–131, 2000 相似文献
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Let B denote the set of values of b for which there exists a block design with b blocks and for k3, let Bk denote the subset of B determined by the designs with block size k. We present some information about B and the sets Bk. In particular, we discuss, for certain integers h, the question as to whether there exist integers k and k′ such that the equation b′=b+h has infinitely many solutions b,b′ satisfying bBk and b′Bk′. The study is restricted to the case λ=1. 相似文献
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将二项式系数的性质应用到Lucas数列的研究中,并结合Fibonacci数列与Lucas数列的恒等式得到几个有趣的Lucas数列的同余式. 相似文献
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Uli Walther 《Proceedings of the American Mathematical Society》2001,129(6):1631-1634
We collect some information about the invariants of a commutative local ring containing a field introduced by G. Lyubeznik in 1993 (Finiteness properties of local cohomology modules, Invent. Math. 113, 41-55). We treat the cases equal to zero, one and two, thereby answering in the negative a question raised in Lyubeznik's paper. In fact, we will show that has in the two-dimensional case a topological interpretation.
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Kevin G. Hare 《Journal of Number Theory》2004,105(2):262-274
For q∈(1,2), Erdös, Joó and Komornik studied the set: