Finiteness Results in Triangle‐Free Quasi‐Symmetric Designs

Pawale, Rajendra M.

doi:10.1002/jcd.21320

Cited by 1 publication

(3 citation statements)

References 14 publications

(16 reference statements)

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“…The classification of triangle-free QS 3-designs is given in [24]. Further investigation of triangle-free QSDs with non-zero intersection numbers are carried out in [21], [25], [26], [27], [29]. Considerable evidence is presented in support of the conjecture that the only triangle-free QSDs with x > 0 are the complements of QSDs with x = 0.…”

Section: Tmentioning

confidence: 99%

“…Result 41 ([21], [25], [26], [27], [29]). Let D be a triangle-free QSD with non-zero intersection numbers.…”

Section: Tmentioning

confidence: 99%

“…Such designs were first studied in [1] and [20]. Some later references are [21], [24], [25], [26], [27], [29], [33]. It was shown in [1], that for a QSD D with intersection numbers x = 0 and y, the block size k satisfies 2y k y(y + 1).…”

Section: Appendixmentioning

confidence: 99%

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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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