A note on triangle-free quasi-symmetric designs

Pawale, Rajendra M.

doi:10.1002/jcd.20285

Cited by 4 publications

(10 citation statements)

References 12 publications

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“…Equivalent to the above results are the following observations given in : (1)

2 y - 1 \leq λ \leq y^{2} + y - 1

. (2)

λ = 2 y - 1

if and only if

k = 2 y

if and only if D is a Hadamard 3-design. (3)

λ = y^{2} + y - 1

λ = y^{2}

if and only if

k = y (y + 1)

. …”

Section: Introductionmentioning

confidence: 58%

Finiteness Results in Triangle‐Free Quasi‐Symmetric Designs

Pawale¹

2012

J of Combinatorial Designs

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Triangle‐free quasi‐symmetric 2‐(v,k,λ) designs with intersection numbers x,y; 0≤x≤y≤k and λ>1 are investigated. Possibility of triangle‐free quasi‐symmetric designs with λ=2y or k−x=3(y−x) is ruled out. It is also shown that, for a fixed x and a fixed ratio (k−x)/(y−x)≥3, there are only finitely many triangle‐free quasi‐symmetric designs. © 2012 Wiley Periodicals, Inc. J Combin Designs 00: 1‐6, 2012

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“…Equivalent to the above results are the following observations given in : (1)

2 y - 1 \leq λ \leq y^{2} + y - 1

. (2)

λ = 2 y - 1

if and only if

k = 2 y

if and only if D is a Hadamard 3-design. (3)

λ = y^{2} + y - 1

λ = y^{2}

if and only if

k = y (y + 1)

. …”

Section: Introductionmentioning

confidence: 58%

Finiteness Results in Triangle‐Free Quasi‐Symmetric Designs

Pawale¹

2012

J of Combinatorial Designs

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“…So quasi-symmetric designs whose block graph and its complement are connected give rise to primitive strongly regular graphs. So earlier papers, such as [1,11,[18][19][20][21][22]25], and [23] may be viewed under this wider umbrella.…”

Section: Triangle-free Quasi-symmetric Designsmentioning

confidence: 99%

On quasi-symmetric designs with intersection difference three

Mavron

McDonough

Shrikhande

2011

Des. Codes Cryptogr.

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In a recent paper, Pawale (Des Codes Cryptogr, 2010) investigated quasisymmetric 2-(v, k, λ) designs with intersection numbers x > 0 and y = x + 2 with λ > 1 and showed that under these conditions either λ = x + 1 or λ = x + 2, or D is a design with parameters given in the form of an explicit table, or the complement of one of these designs. In this paper, quasi-symmetric designs with y − x = 3 are investigated. It is shown that such a design or its complement has parameter set which is one of finitely many which are listed explicitly or λ ≤ x + 4 or 0 ≤ x ≤ 1 or the pair (λ, x) is one of

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“…In remaining cases from the fact that discriminant of the quadratic (5) is non-negative we get z = 1. If block graph of D is the Shrikhande graph, with parameters (16,6,2,2) or the Schläfli graph, with parameters (27,16,10,8) or the Clebsch graph, with parameters (16,10,6,6), then the discriminant of the same quadratic (5) is not a perfect square. If the block graph of D is one of the three Chang graphs, with parameters (28,12,6,4), then using the same equation 5 [28]) and the third is ruled out using [8].…”

Section: Block Graphs Of Qsdsmentioning

confidence: 99%

Conditions for the Parameters of the Block Graph of Quasi-Symmetric Designs

Pawale

Shrikhande

Nyayate³

2015

Electron. J. Combin.

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A quasi-symmetric design (QSD) is a 2-$(v,k,\lambda)$ design with intersection numbers $x$ and $y$ with $x< y$. The block graph of such a design is formed on its blocks with two distinct blocks being adjacent if they intersect in $y$ points. It is well known that the block graph of a QSD is a strongly regular graph (SRG) with parameters $(b,a,c,d)$ with smallest eigenvalue $ -m =-\frac{k-x}{y-x}$.The classification result of SRGs with smallest eigenvalue $-m$, is used to prove that for a fixed pair $(\lambda\ge 2,m\ge 2)$, there are only finitely many QSDs. This gives partial support towards Marshall Hall Jr.'s conjecture, that for a fixed $\lambda\ge 2$, there exist finitely many symmetric $(v, k, \lambda)$-designs.We classify QSDs with $m=2$ and characterize QSDs whose block graph is the complete multipartite graph with $s$ classes of size $3$. We rule out the possibility of a QSD whose block graph is the Latin square graph $LS_m (n)$ or complement of $LS_m (n)$, for $m=3,4$.SRGs with no triangles have long been studied and are of current research interest. The characterization of QSDs with triangle-free block graph for $x=1$ and $y=x+1$ is obtained and the non-existence of such designs with $x=0$ or $\lambda > 2(x+2)$ or if it is a $3$-design is proven. The computer algebra system Mathematica is used to find parameters of QSDs with triangle-free block graph for $2\le m \le 100$. We also give the parameters of QSDs whose block graph parameters are $(b,a,c,d)$ listed in Brouwer's table of SRGs.

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A note on triangle-free quasi-symmetric designs

Cited by 4 publications

References 12 publications

Finiteness Results in Triangle‐Free Quasi‐Symmetric Designs

Finiteness Results in Triangle‐Free Quasi‐Symmetric Designs

On quasi-symmetric designs with intersection difference three

Conditions for the Parameters of the Block Graph of Quasi-Symmetric Designs

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