{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T13:03:48Z","timestamp":1772283828392,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A quasi-symmetric design (QSD) is a 2-$(v,k,\\lambda)$ design with intersection numbers $x$ and $y$ with $x&lt; y$.\u00a0The 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 $3$.\u00a0We rule out the possibility of a QSD whose block graph is the\u00a0Latin 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 $\\lambda &gt; 2(x+2)$ or if it is a $3$-design is proven.\u00a0The 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.<\/jats:p>","DOI":"10.37236\/3954","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:32:12Z","timestamp":1578702732000},"source":"Crossref","is-referenced-by-count":5,"title":["Conditions for the Parameters of the Block Graph of Quasi-Symmetric Designs"],"prefix":"10.37236","volume":"22","author":[{"given":"Rajendra M.","family":"Pawale","sequence":"first","affiliation":[]},{"given":"Mohan S.","family":"Shrikhande","sequence":"additional","affiliation":[]},{"given":"Shubhada M.","family":"Nyayate","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,2,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p36\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p36\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:37:28Z","timestamp":1579257448000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i1p36"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2,16]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/3954","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,2,16]]},"article-number":"P1.36"}}