On the impossibility of certain (n2+ n + kn+1) configurations
Abstract
This paper investigates the impossibility of certain $({n^2+n+k}_{n+1})$ configurations. Firstly, for $k=2$, the result of \cite{gropp1992non} that $\frac{n^2+n}{2}$ is even and $n+1$ is a perfect square or $\frac{n^2+n}{2}$ is odd and $n-1$ is a perfect square is reproved using the incidence matrix $N$ and analysing the form of $N^TN$. Then, for all $k$, configurations where paralellism is a transitive property are considered. It is then analogously established that if $n\equiv0$ or $n\equiv k-1$ mod $k$ for $k$ even, then $\frac{n^2+n}{k}$ is even and $n+1$ is a perfect square or $\frac{n^2+n}{k}$ is odd and $n-(k-1)$ is a perfect square. Finally, the case $k=3$ is investigated in full generality.
Fields of Study & Subjects
graph theory and CDMA systems
Coding theory and cryptography
Polynomial and algebraic computation
Publication details
- Journal / source
- Involve a Journal of Mathematics
- Volume
- 19
- Issue
- 2
- Pages
- 297–310
- Publication date
- 8 March 2026
- Language
- EN