TY - JOUR

T1 - Дистанционно регулярные графы Шилла с b 2 = c 2

AU - Махнев, А. А.

AU - Нирова, М. С.

PY - 2018

Y1 - 2018

N2 - 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: {b2(b−1)2,(b−1)(b2−b+2)2,b(b−1)4;1,b(b−1)4,b(b−1)22} If Γ is a Q-polynomial Shilla graph with b2 = c2 and b = 2r, then the graph Γ has intersection array { 2 tr(2 r+ 1) , (2 r+ 1) (2 rt+ t+ 1) , r(r+ t) ; 1 , r(r+ t) , t(4 r2− 1) } and, for any vertex u in Γ, the subgraph Γ3(u) is an antipodal distance-regular graph with intersection array { t(2 r+ 1) , (2 r− 1) (t+ 1) , 1 ; 1 , t+ 1 , t(2 r+ 1) } The Shilla graphs with b2 = c2 and b = 4 are also classified in the paper.

AB - 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: {b2(b−1)2,(b−1)(b2−b+2)2,b(b−1)4;1,b(b−1)4,b(b−1)22} If Γ is a Q-polynomial Shilla graph with b2 = c2 and b = 2r, then the graph Γ has intersection array { 2 tr(2 r+ 1) , (2 r+ 1) (2 rt+ t+ 1) , r(r+ t) ; 1 , r(r+ t) , t(4 r2− 1) } and, for any vertex u in Γ, the subgraph Γ3(u) is an antipodal distance-regular graph with intersection array { t(2 r+ 1) , (2 r− 1) (t+ 1) , 1 ; 1 , t+ 1 , t(2 r+ 1) } The Shilla graphs with b2 = c2 and b = 4 are also classified in the paper.

UR - https://elibrary.ru/item.asp?id=32823048

U2 - 10.4213/mzm11503

DO - 10.4213/mzm11503

M3 - Статья

VL - 103

SP - 730

EP - 744

JO - Математические заметки

JF - Математические заметки

SN - 0025-567X

IS - 5

ER -