Distance-Regular Shilla Graphs with~$b_2=c_2$

A. A. Makhnev ; M. S. Nirova

Matematičeskie zametki, Tome 103 (2018) no. 5, pp. 730-744

Résumé

A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigenvalue $\theta_1$ equal to $a_3$. For a Shilla graph, let us put $a=a_3$ and $b=k/a$. It is proved in this paper that a Shilla graph with $b_2=c_2$ and noninteger eigenvalues has the following intersection array: $$ \biggl\{\frac{b^2(b-1)}2\mspace{2mu}, \frac{(b-1)(b^2-b+2)}2\mspace{2mu}, \frac{b(b-1)}4\mspace{2mu};1, \frac{b(b-1)}4\mspace{2mu}, \frac{b(b-1)^2}2\biggr\}. $$ If $\Gamma$ is a $Q$-polynomial Shilla graph with $b_2=c_2$ and $b=2r$, then the graph $\Gamma$ has intersection array $$ \{2rt(2r+1),(2r-1)(2rt+t+1),r(r+t);1,r(r+t),t(4r^2-1)\} $$ and, for any vertex $u$ in $\Gamma$, the subgraph $\Gamma_3(u)$ is an antipodal distance-regular graph with intersection array $$ \{t(2r+1),(2r-1)(t+1),1;1,t+1,t(2r+1)\}. $$ The Shilla graphs with $b_2=c_2$ and $b=4$ are also classified in the paper.

Détail

Keywords: distance-regular graph, Shilla graph
Mots-clés : graph automorphism.