Distance-Regular Shilla Graphs with b2 = c2


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Resumo

A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ1 equal to a3. For a Shilla graph, let us put a = a3 and b = k/a. It is proved in this paper that a Shilla graph with b2 = c2 and noninteger eigenvalues has the following intersection array:

\(\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2},\frac{{\left( {b - 1} \right)\left( {{b^2} - b + 2} \right)}}{2},\frac{{b\left( {b - 1} \right)}}{4};1,\frac{{b\left( {b - 1} \right)}}{4},\frac{{b{{\left( {b - 1} \right)}^2}}}{2}} \right\}\)
If Γ is a Q-polynomial Shilla graph with b2 = c2 and b = 2r, then the graph Γ has intersection array
\(\left\{ {2tr\left( {2r + 1} \right),\left( {2r + 1} \right)\left( {2rt + t + 1} \right),r\left( {r + t} \right);1,r\left( {r + t} \right),t\left( {4{r^2} - 1} \right)} \right\}\)
and, for any vertex u in Γ, the subgraph Γ3(u) is an antipodal distance-regular graph with intersection array
\(\left\{ {t\left( {2r + 1} \right),\left( {2r - 1} \right)\left( {t + 1} \right),1;1,t + 1,t\left( {2r + 1} \right)} \right\}\)
The Shilla graphs with b2 = c2 and b = 4 are also classified in the paper.

Sobre autores

A. Makhnev

Krasovskii Institute of Mathematics and Mechanics; Yeltsin Ural Federal University

Autor responsável pela correspondência
Email: makhnev@imm.uran.ru
Rússia, Ekaterinburg; Ekaterinburg

M. Nirova

Berbekov State University of Kabardino-Balkaria

Email: makhnev@imm.uran.ru
Rússia, Nalchik

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