On cliques in isoregular graphs
A graph \(\Gamma\) is called an edge-regular graph with parameters \((v,k,\lambda)\) if \(\Gamma\) is a regular graph of degree \(k\) on \(v\) vertices in which each edge lies in exactly \(\lambda\) triangles. The main results in this work are: {\parindent=8mm \begin{itemize}\item[(1)]if an edge-regular graph with parameters \(\left(4r^4 + 10r^3 + 6r^2 - 2r - 2, 2r^4 + 3r^3, r^4 - 2r^2 + r\right)\) contains a \(2r\)-clique then \(r = 1\); \item[(2)]if an edge-regular graph with parameters \(\left(2r^4 + 3r^3, r^4 - 2r^2 + r, \frac{r^4 - 2r^3 - 3r^2 + 6r}{2}\right)\) contains a \((2r-1)\)-clique then \(r = 1\) and \item[(3)]if an edge-regular graph with parameters \(\left(2r^4 + 5r^3 + 3r^2 - r - 1, r^4 + r^3 - r^2,\frac{r^4 - r^3 - 3r^2 + 3r}{2}\right)\) contains a \(2r\)-clique then \(r = 1\). \end{itemize}}
strongly regular graph, edge-regular graph
scientific article; zbMATH DE number 5998444