ON THE SPECTRUM OF OCTAGON QUADRANGLE SYSTEMS OF ANY INDEX
DOI:
https://doi.org/10.11575/cdm.v12i1.62559Keywords:
Octagon quadrangle system, designs, decomposition.Abstract
An \emph{octagon quadrangle} is the graph consisting of a length $8$ cycle $(x_{1},x_{2},\dots,x_{8})$ and two chords, $\{x_{1},x_{4}\}$ and $\{x_{5},x_{8}\}$. An \emph{octagon quadrangle system} of order $v$ and index $\lambda$ is a pair $(X,\mathcal B)$, where $X$ is a finite set of $v$ vertices and $\mathcal B$ is a collection of octagon quadrangles (called blocks) which partition the edge set of $\lambda K_{v}$, with $X$ as vertex set. In this paper we determine completely the spectrum of octagon quadrangle systems for any index $\lambda$, with the only possible exception of $v=20$ for $\lambda=1$.
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