{48, 35, 9; 1, 7, 40} #

(1)

ÓÄÊ519.17

DOI10.46698/n0833-6942-7469-t

ÀÂÒÎÌÎÔÈÇÌÛ ÄÈÑÒÀÍÖÈÎÍÍÎ ÅÓËßÍÎÎ ÀÔÀ

ÑÌÀÑÑÈÂÎÌ ÏÅÅÑÅ×ÅÍÈÉ

{48, 35, 9; 1, 7, 40} ^#

À. À. Ìàõíåâ

1

, Â. Â.Áèòêèíà

2

, À. Ê. óòíîâà

2

1

Èíñòèòóòìàòåìàòèêèèìåõàíèêèèì.Í.Í.Êðàñîâñêîãî,

îññèÿ,620990,Åêàòåðèíáóðã,óë.Ñ.Êîâàëåâñêîé,16;

2

Ñåâåðî-Îñåòèíñêèéãîñóäàðñòâåííûéóíèâåðñèòåòèì.Ê.Ë.Õåòàãóðîâà,

îññèÿ,362025,Âëàäèêàâêàç,óë.Âàòóòèíà,4446

E-mail:makhnevimm.uran.ru, bviktoriyavmail.ru,gutnovaalinagmail.om

Àííîòàöèÿ.Åñëèäèñòàíöèîííîðåãóëÿðíûéãðà

Γ

^{äèàìåòðà}³^{ñîäåðæèò}ìàêñèìàëüíûéëîêàëüíî ðåãóëÿðíûé1-êîä,ñîâåðøåííûéîòíîñèòåëüíîïîñëåäíåéîêðåñòíîñòè,òî

Γ

^èìååò^ìàññèâ^{ïåðåñå÷å-}

íèé

{a(p + 1), cp, a + 1; 1, c, ap}

^èëè

{a(p + 1), (a + 1)p, c; 1, c, ap}

^,^ãäå

a = a 3

^,

c = c 2

^,

p = p ³ 33

^{(Þðèøè÷}

èÂèäàëè).Âïåðâîìñëó÷àå

Γ

^èìååòñîáñòâåííîå çíà÷åíèå

θ 2 = −1

^è

Γ 3

^{ÿâëÿåòñÿ}ïñåâäîãåîìåòðè-

÷åñêèì ãðàîì äëÿ

GQ(p + 1, a)

^. ^Åñëè

c = a − 1 = q

^,

p = q − 2

^, ^òî

Γ

^èìååò ^ìàññèâ ïåðåñå÷åíèé

{q ² − 1, q(q − 2), q + 2; 1, q, (q + 1)(q − 2)}

^,

q > 6

^.^Â^ðàáîòå^{èçó÷åíû}^{ïîðÿäêè}^è^{ïîäãðàû}íåïîäâèæíûõ òî÷åê àâòîìîðèçìîâ ãèïîòåòè÷åñêîãî äèñòàíöèîííî ðåãóëÿðíîãîãðàà ñ ìàññèâîì ïåðåñå÷åíèé

{48, 35, 9; 1, 7, 40}

⁽

q = 7

^).^Ïóñòü

G = Aut(Γ)

íåðàçðåøèìàÿãðóïïà,äåéñòâóþùàÿòðàíçèòèâíîíà

ìíîæåñòâåâåðøèíãðàà

Γ

^,

K = O 7 (G)

^,

T ¯

^öîêîëü^ãðóïïû

G ¯ = G/K

^.^Òîãäà

T ¯

^{ñîäåðæèò}^{åäèíñòâåí-}

íóþêîìïîíåíòó

L ¯

^,^òî÷íîäåéñòâóþùóþíà

K

^,

L ¯ ∼ = L 2 (7)

^,

A 5

^,

A 6

^,

P Sp 4 (3)

^è^äëÿ^{ïîëíîãî}^{ïðîîáðàçà}

L

ãðóïïû

L ¯

^èìååì

L a = K a × O 7 ′ (L a )

^è

|K| = 7 ³

^â^ñëó÷àå

L ¯ ∼ = L 2 (7)

^,

|K| = 7 ⁴

^â^{ïðîòèâíîì}^{ñëó÷àå.}

Êëþ÷åâûåñëîâà:ñèëüíîðåãóëÿðíûéãðà,äèñòàíöèîííîðåãóëÿðíûéãðà,àâòîìîðèçìãðàà.

Mathematial Subjet Classiation (2010):05C25.

Îáðàçåööèòèðîâàíèÿ:ÌàõíåâÀ.À.,ÁèòêèíàÂ.Â.,óòíîâàÀ.Ê.Àâòîìîðèçìûäèñòàíöèîííî

ðåãóëÿðíîãîãðààñìàññèâîìïåðåñå÷åíèé

{48, 35, 9; 1, 7, 40}

^/^/^{Âëàäèêàâê.}^ìàò.æóðí.2020.Ò.22, âûï.2.Ñ. 2433.DOI:10.46698/n0833-6942-7469-t.

1. Ââåäåíèå

Ìûðàññìàòðèâàåìíåîðèåíòèðîâàííûå ãðàûáåç ïåòåëüè êðàòíûõðåáåð. Äëÿâåð-

øèíû

a

^ãðàà

Γ

^÷åðåç

Γ i (a)

^{îáîçíà÷èì}

i

-îêðåñòíîñòü âåðøèíû

a

^, ^ò^. ^å. ^{ïîäãðà,} ^èíäó-

öèðîâàííûé

Γ

^íà ^{ìíîæåñòâå} ^âñåõ ^{âåðøèí,} íàõîäÿùèõñÿ íà ðàññòîÿíèè

i

^îò

a

^. ^{Ïîëîæèì}

[a] = Γ ₁ (a)

^,

a ^⊥ = {a} ∪ [a]

^.

Ïóñòü

Γ

^ãðà,

a, b ∈ Γ

^, ^÷èñëî ^âåðøèí ^â

[a] ∩ [b]

îáîçíà÷àåòñÿ ÷åðåç

µ(a, b)

^(÷åðåç

λ(a, b))

^,^åñëè

a

^,

b

^{íàõîäÿòñÿ}^íà^{ðàññòîÿíèè}²^{(ñìåæíû)}^â

Γ

^.^Äàëåå,èíäóöèðîâàííûé

[a]∩[b]

ïîäãðà íàçûâàåòñÿ

µ

^{-ïîäãðàîì}⁽

λ

-ïîäãðàîì).

Ñèñòåìà èíöèäåíòíîñòèñìíîæåñòâîì òî÷åê

P

^è^{ìíîæåñòâîì} ^ïðÿìûõ

L

íàçûâàåòñÿ

α

^{-÷àñòè÷íîé} ^{ãåîìåòðèåé}^{ïîðÿäêà}

(s, t)

^,^åñëè^êàæäàÿ^ïðÿìàÿ^{ñîäåðæèò}^ðîâíî

s+ 1

^òî÷êó^,

#

àáîòàâûïîëíåíà ïðè ïîääåðæêåïðîãðàììû óíäàìåíòàëüíûõ íàó÷íûõèññëåäîâàíèé èÔÅÍ

Êèòàÿ,ïðîåêò20-51-53013.

²⁰²⁰ ^Ìàõíåâ^À.^À.,^{Áèòêèíà}^Â.^Â.,^óòíîâà^À.^Ê.

(2)

êàæäàÿòî÷êàëåæèòðîâíîíà

t+1

^{ïðÿìîé,}^ëþáûå^äâå^òî÷êè^ëåæàò^íå^áîëåå^÷åì^íà^îäíîé

ïðÿìîé, è äëÿ ëþáîãîàíòèëàãà

(a, l) ∈ (P, L )

^{íàéäåòñÿ} ^òî÷íî

α

^{ïðÿìûõ,} ^{ïðîõîäÿùèõ}

÷åðåç

a

^èïåðåñåêàþøèõ

l

(îáîçíà÷åíèå:

pG _α (s, t)

^).^Â ^ñëó÷àå

α = 1

^{ãåîìåòðèÿ} ^{íàçûâàåò-}

ñÿ îáîáùåííûì÷åòûðåõóãîëüíèêîìè îáîçíà÷àåòñÿ

GQ(s, t)

^. ^Ò^{î÷å÷íûé} ^ãðà ^{ãåîìåòðèè}

îïðåäåëÿåòñÿ íà ìíîæåñòâå òî÷åê

P

^è ^äâå ^òî÷êè ^{ñìåæíû,} ^åñëè ^îíè ^ëåæàò ^íà ^{ïðÿìîé.}

Òî÷å÷íûéãðàãåîìåòðèè

pG _α (s, t)

^ñèëüíî^{ðåãóëÿðåí}^ñ

v = (s + 1)(1 + st/α)

^,

k = s(t + 1)

^,

λ = s − 1 + t(α − 1)

^,

µ = α(t + 1)

^. ^Ñèëüíî ^{ðåãóëÿðíûé} ^ãðà ^ñ ^òàêèìè ïàðàìåòðàìè äëÿíåêîòîðûõíàòóðàëüíûõ÷èñåë

α

^,

s

^,

t

^{íàçûâàåòñÿ}ïñåâäîãåîìåòðè÷åñêèìãðàîìäëÿ

pG _α (s, t)

^.

Åñëè âåðøèíû

u

^,

w

^{íàõîäÿòñÿ} ^íà ^{ðàññòîÿíèè}

i

^â

Γ

^, ^òî ^÷åðåç

b _i (u, w)

^(÷åðåç

c _i (u, w)

⁾

îáîçíà÷èì ÷èñëî âåðøèí â ïåðåñå÷åíèè

Γ _i+1 (u)

⁽

Γ _i−1 (u)

⁾ ^ñ

[w]

^. ^ðà

Γ

^{äèàìåòðà}

d

^íà-

çûâàåòñÿ äèñòàíöèîííîðåãóëÿðíûì ñ ìàññèâîì ïåðåñå÷åíèé

{b ₀ , b ₁ , . . . , b _d−1 ; c ₁ , . . . , c _d }

^,

åñëè çíà÷åíèÿ

b _i (u, w)

^è

c _i (u, w)

^íå ^{çàâèñÿò} ^îò ^âûáîðà ^âåðøèí

u

^,

w

^íà ^{ðàññòîÿíèè}

i

^â

Γ

äëÿëþáîãî

i = 0, . . . , d

^. ^{Ïîëîæèì}

a _i = k − b _i − c _i

^.

Äëÿïîäìíîæåñòâà

X

àâòîìîðèçìîâ ãðàà

Γ

^÷åðåç

Fix(X)

îáîçíà÷àåòñÿìíîæåñòâî âñåõ âåðøèí ãðàà

Γ

^, íåïîäâèæíûõ îòíîñèòåëüíî ëþáîãî àâòîìîðèçìà èç

X

^. ^Äàëåå,

÷åðåç

p ^l _ij (x, y)

^{îáîçíà÷èì} ^÷èñëî^âåðøèí ^â^{ïîäãðàå}

Γ _i (x) ∩ Γ _j (y)

^äëÿ ^âåðøèí

x

^,

y

^, ^íàõî-

äÿùèõñÿ íà ðàññòîÿíèè

l

^â ^ãðàå

Γ

^. ^Â äèñòàíöèîííî ðåãóëÿðíîì ãðàå ÷èñëà

p ^l _ij (x, y)

íåçàâèñÿò îò âûáîðàâåðøèí

x

^,

y

^, îáîçíà÷àþòñÿ

p ^l _ij

^è ^{íàçûâàþòñÿ}^{÷èñëàìè}ïåðåñå÷åíèé ãðàà

Γ

^[1℄.

ðà íàçûâàåòñÿ âåðøèííî ñèììåòðè÷íûì, åñëè åãî ãðóïïà àâòîìîðèçìîâ äåé-

ñòâóåò òðàíçèòèâíî íàìíîæåñòâåâåðøèí.

Ïóñòü

Γ

^ãðà^{äèàìåòðà}

d

^è

e

íàòóðàëüíîå ÷èñëî. Ïîäìíîæåñòâî

C

^âåðøèí ^ãðà-

à

Γ

^{íàçûâàåòñÿ}

e

^-êîäîì,^åñëè ìèíèìàëüíîå ðàññòîÿíèå ìåæäó äâóìÿ âåðøèíàìè èç

C

íå ìåíüøå

2e + 1

^. ^Äëÿ

e

^-êîäà ^â äèñòàíöèîííî ðåãóëÿðíîì ãðàå äèàìåòðà

d = 2e + 1

âûïîëíÿåòñÿ ãðàíèöà

|C| 6 p ^d _dd + 2

^. ^Â^ñëó÷àå ^{ðàâåíñòâà} ^êîä ^{íàçûâàåòñÿ} ìàêñèìàëüíûì.

Äëÿ ìàêñèìàëüíîãî

e

d = 2e + 1

^âû-

ïîëíÿåòñÿãðàíèöà

c _d > a _d p ^d _dd

^. ^Â^ñëó÷àå^{ðàâåíñòâà}^êîä^{íàçûâàåòñÿ}^{ëîêàëüíî} ðåãóëÿðíûì.

Íàêîíåö, äëÿ

e

d = 2e + 1

^{âûïîëíÿ-}

åòñÿ ãðàíèöà

|C| 6 k _d / P _e

i=0 p ^d _id + 1

^. ^Â ^ñëó÷àå ^{ðàâåíñòâà} ^êîä ^{íàçûâàåòñÿ} ñîâåðøåííûì îòíîñèòåëüíî ïîñëåäíåé îêðåñòíîñòè [2℄.

Åñëèäèñòàíöèîííîðåãóëÿðíûéãðà

Γ

^{äèàìåòðà}³^{ñîäåðæèò}ìàêñèìàëüíûé1-êîä

C

^,

ÿâëÿþùèéñÿëîêàëüíîðåãóëÿðíûìèñîâåðøåííûìîòíîñèòåëüíîïîñëåäíåéîêðåñòíîñòè,

òî ïî ïðåäëîæåíèþ 5 èç [2℄

Γ

^èìååò ^ìàññèâ ïåðåñå÷åíèé

{a(p + 1), cp, a + 1; 1, c, ap}

^èëè

{a(p + 1), (a + 1)p, c; 1, c, ap}

^, ^ãäå

a = a 3

^,

c = c 2

^,

p = p ³ ₃₃

^. ^Â ^ïåðâîì ^ñëó÷àå

Γ

^èìååò

ñîáñòâåííîåçíà÷åíèå

θ ₂ = −1

^è^ãðà

Γ ₃

^{ÿâëÿåòñÿ}ïñåâäîãåîìåòðè÷åñêèìäëÿ

GQ(p+1, a)

^.

Âñëó÷àå

c = a − 1 = q

^,

p = q − 2

^ïî^[2℄^ãðà

Γ

^èìååò ^ìàññèâïåðåñå÷åíèé

{q ² − 1, q ² − 2q, q + 2; 1, q, (q + 1)(q − 2)}

^,

q > 6

^, ^ñïåêòð

(q ² − 1) ¹

^,

(2q − 1) ^q(q ² ^−1)/6

^,

−1 ^(q+1)(q ² ^+q−2)/2

^,

−(q + 1) q(q−1)(q−2)/3

è

Γ ¯ ₂

^{ÿâëÿåòñÿ} ïñåâäîãåîìåòðè÷åñêèì ãðàîì äëÿ

pG ₂ (q − 1, 2q + 2)

^.

Ïðè

q = 7

^{ïîëó÷èì}^ìàññèâ ïåðåñå÷åíèé

{48, 35, 9; 1, 7, 40}

^.

Â äàííîé ðàáîòå èçó÷àþòñÿ àâòîìîðèçìû ãèïîòåòè÷åñêîãî äèñòàíöèîííî ðåãóëÿð-

íîãî ãðàà

Γ

^ñ ^{ìàññèâîì} ïåðåñå÷åíèé

{48, 35, 9; 1, 7, 40}

^. ^Ýòîò ^ãðà ^èìååò

v = 1 + 48 + 240 + 54 = 343 = 7 ³

^âåðøèí ^è ^ñïåêòð

48 ¹

^,

13 ⁵⁶

^,

−1 ²¹⁶

^,

−8 ⁷⁰

^. ^Ââèäó ^{ãðàíèöû} ^{Äåëüñàðòà}

ìàêñèìàëüíûéïîðÿäîêêëèêèâ

Γ

^íå^áîëüøå

7

^,^àìàêñèìàëüíûéïîðÿäîêêîêëèêè â

Γ

^íå

áîëüøå

49

^. ^Äàëåå,^ãðà

Γ 3

^{ÿâëÿåòñÿ} ïñåâäîãåîìåòðè÷åñêèì äëÿ

GQ(6, 8)

^.

Òåîðåìà1.Ïóñòü

Γ

äèñòàíöèîííîðåãóëÿðíûéãðà,èìåþùèéìàññèâïåðåñå÷åíèé

{48, 35, 9; 1, 7, 40}

^,

G = Aut(Γ)

^,

g

^{ýëåìåíò} ^èç

G

^{ïðîñòîãî}^{ïîðÿäêà}

p

^è

Ω = Fix(g)

^.^Òîãäà

(3)

π(G) ⊆ {2, 3, 5, 7}

^èâûïîëíÿåòñÿ îäíî èç ñëåäóþùèõóòâåðæäåíèé:

(1) Ω

^ïóñòîé^ãðà,

p = 7

^,

α ₁ (g) = 49(l + 1 + 3s)

^,

α ₃ (g) = 49(2l + 1)

^, ^è

3l + 2 + 3s 6 7

^;

(2) Ω

^{ÿâëÿåòñÿ}

1

^{-êëèêîé,}

p = 3

^,

α ₁ (g) = 3(7l +16+21s)

^,

α ₃ (g) = 42l+54

^è

9l +9s+15 6 49

^;

(3) Ω

^{ÿâëÿåòñÿ}

7

^{-êîêëèêîé,} ^ëþáûå ^äâå ^{âåðøèíû} ^èç

Ω

^{íàõîäÿòñÿ} ^íà ^{ðàññòîÿíèè}

3

^,

p = 2

^,

α ₁ (g) = 14l − 18 + 42s

^è

α ₃ (g) = 28l − 36

^;

(4) Ω

^{ñîäåðæèò}^ðåáðî ^è^ëèáî

Ω

^{ñîäåðæèò}^{âåðøèíû,}íàõîäÿùèåñÿíà ðàññòîÿíèè

2

^â

Γ

è

p 6 11

^,^ëèáî

Ω

îáúåäèíåíèåäâóõèçîëèðîâàííûõ

4

^-êëèê,

p = 5

^,

α ₁ (g) = 35s+20+105t

è

α 3 (g) = 5 + 70s

^.

Ñëåäñòâèå 1. Ïóñòü

Γ

äèñòàíöèîííî ðåãóëÿðíûé ãðà, èìåþùèé ìàññèâ ïåðå- ñå÷åíèé

{48, 35, 9; 1, 7, 40}

^, ^è íåðàçðåøèìàÿ ãðóïïà

G = Aut(Γ)

^{äåéñòâóåò} òðàíçèòèâíî íà ìíîæåñòâå âåðøèí ãðàà. Åñëè

K = O ₇ (G)

^,

T ¯

^öîêîëü ^ãðóïïû

G ¯ = G/K

^, ^òî

T ¯

ñîäåðæèò åäèíñòâåííóþ êîìïîíåíòó

L ¯

^, ^òî÷íî äåéñòâóþùóþ íà

K

^,

L ¯ ∼ = L ₂ (7)

^,

A ₅

^,

A ₆

^,

P Sp 4 (3)

^è ^äëÿ ^{ïîëíîãî} ^{ïðîîáðàçà}

L

^ãðóïïû

L ¯

^èìååì

L a = K a × O ₇ ^′ (L a )

^è

|K| = 7 ³

^â

ñëó÷àå

L ¯ ∼ = L ₂ (7)

^,

|K| = 7 ⁴

^â ^{ïðîòèâíîì} ^{ñëó÷àå.}

Äëÿ äîêàçàòåëüñòâà ñëåäñòâèÿïîëåçíà

Òåîðåìà 2.Ïóñòü

Γ

^ñèëüíî ^{ðåãóëÿðíûé}^ãðà^ñïàðàìåòðàìè

(343, 54, 5, 9)

^è^ñïåê-

òðîì

54 ¹ , 5 ²¹⁶ , −9 ¹²⁶

^,

G = Aut(Γ)

^,

g

^{ýëåìåíò} ^{ïðîñòîãî} ^{ïîðÿäêà}

p

^èç

G

^,

α _i (g) = pw _i

^äëÿ

i > 0

^è

∆ = Fix(g)

^. ^Ò^îãäà âûïîëíÿåòñÿ îäíîèç ñëåäóþùèõ óòâåðæäåíèé:

(1) ∆

^ïóñòîé ^ãðà,

p = 7

^è

α ₁ (g) = 49(2s + 1)

^;

(2) ∆

^{ÿâëÿåòñÿ}

n

^{-êëèêîé,} ^ëèáî

p = 2

^,

n = 7

^è

α ₁ (g) = 28l

^, ^ëèáî

p = 3

^,

n = 1, 4, 7

è

α ₁ (g) = 5n + 7 + 42l

^;

(3) ∆

^{ÿâëÿåòñÿ}

m

^{-êîêëèêîé,}

m > 1

^,

p = 3

^,

m ∈ {4, 7, . . . , 49}

^è

α ₁ (g) = 5m + 7 + 42l

^;

(4) ∆

^{ñîäåðæèò}^ðåáðî^è^{ÿâëÿåòñÿ}îáúåäèíåíèåì èçîëèðîâàííûõêëèê,

p = 3

^è^{ïîðÿäîê}

ìàêñèìàëüíîé êëèêèèç

∆

^ðàâåí

1

^èëè

4

^;

(5) p 6 7

^.

2. Ïðåäâàðèòåëüíûå ðåçóëüòàòû

Ñíà÷àëà ïðèâåäåì îäèí âñïîìîãàòåëüíûé ðåçóëüòàò [3, òåîðåìà 2.3℄.

Ëåììà 1. Ïóñòü

Γ

^ñèëüíî ^{ðåãóëÿðíûé} ^ãðà ^ñ ïàðàìåòðàìè

(v, k, λ, µ)

^è ^âòî-

ðûì ñîáñòâåííûì çíà÷åíèåì

r

^. ^Åñëè

g

^{àâòîìîðèçì}

Γ

^è

∆ = Fix(g)

^, ^òî

|∆| 6 v · max{λ, µ}/(k − r)

^.

Ïî ëåììå1 äëÿãðàà ñïàðàìåòðàìè

(343, 54, 7, 9)

^{ïîëó÷èì}

|∆| 6 343 · 9/49 = 63

^.

Γ

äèñòàíöèîííî ðåãóëÿðíûé ãðà ñ ìàññèâîì ïåðåñå÷åíèé

{48, 35, 9; 1, 7, 40}

^. ^Ò^îãäà ^äëÿ^÷èñåë ïåðåñå÷åíèé ãðàà

Γ

^âåðíû^{ðàâåíñòâà}

(1) p ¹ ₁₁ = 12

^,

p ¹ ₁₂ = 35

^,

p ¹ ₂₂ = 160

^,

p ¹ ₂₃ = 45

^,

p ¹ ₃₃ = 9

^;

(2) p ² ₁₁ = 7

^,

p ² ₁₂ = 32

^,

p ² ₁₃ = 9

^,

p ² ₂₂ = 171

^,

p ² ₂₃ = 36

^,

p ² ₃₃ = 9

^;

(3) p ³ ₁₂ = 40

^,

p ³ ₁₃ = 8

^,

p ³ ₂₂ = 160

^,

p ³ ₂₃ = 40

^,

p ³ ₃₃ = 5

^.

⊳

Äîêàçàòåëüñòâî ñëåäóåòèç [1, ëåììà4.1.7℄.

⊲

Äîêàçàòåëüñòâî òåîðåì îïèðàåòñÿ íà ìåòîä Õèãìåíà ðàáîòû ñ àâòîìîðèçìàìè äè-

ñòàíöèîííî ðåãóëÿðíîãî ãðàà, ïðåäñòàâëåííûé â òðåòüåé ãëàâå ìîíîãðàèè Êàìåðî-

íà[4℄.Ïðèýòîìãðà

Γ

ðàññìàòðèâàåòñÿêàêñèììåòðè÷íàÿñõåìà îòíîøåíèé

(X, R )

^ñ

d

êëàññàìè,ãäå

X

^{ìíîæåñòâî}^âåðøèí^ãðàà,

R ₀

^{îòíîøåíèå}^{ðàâåíñòâà}^íà

X

^è^äëÿ

i 6 1

(4)

êëàññ

R _i

^{ñîñòîèò} ^èç ^ïàð

(u, w)

^òàêèõ, ^÷òî

d(u, w) = i

^. ^Äëÿ

u ∈ Γ

^{ïîëîæèì}

k _i = |Γ _i (u)|

^,

v = |Γ|

^. ^Êëàññó

R i

^{îòâå÷àåò} ^ãðà

Γ i

^íà ^{ìíîæåñòâå} ^âåðøèí

X

^, ^â ^{êîòîðîì} ^{âåðøèíû}

u

^,

w

ñìåæíû,åñëè

(u, w) ∈ R _i

^.^Ïóñòü

A _i

^{ìàòðèöà}^{ñìåæíîñòè} ^ãðàà

Γ _i

^äëÿ

i > 0

^è

A ₀ = I

åäèíè÷íàÿ ìàòðèöà.Òîãäà

A _i A _j = P

p ^l _ij A _l

^äëÿ ^÷èñåë ïåðåñå÷åíèé

p ^l _ij

^.

Ïóñòü

P _i

^{ìàòðèöà,} ^â ^{êîòîðîé} ^íà ^ìåñòå

(j, l)

^ñòîèò

p ^l _ij

^. ^Ò^îãäà ñîáñòâåííûå çíà÷å- íèÿ

p ₁ (0), . . . , p ₁ (d)

^{ìàòðèöû}

P ₁

^{ÿâëÿþòñÿ}ñîáñòâåííûìèçíà÷åíèÿìèãðàà

Γ

^{êðàòíîñòåé}

m ₀ = 1, . . . , m _d

^.^{Ìàòðèöû}

P

^è

Q

^, ^ó^{êîòîðûõ}^íà^ìåñòå

(i, j)

^ñòîÿò

p _j (i)

^è

q _j (i) = m _j p _i (j)/k _i

ñîîòâåòñòâåííî,íàçûâàþòñÿïåðâîé èâòîðîé ìàòðèöåéñîáñòâåííûõçíà÷åíèéñõåìû è

ñâÿçàíûðàâåíñòâîì

P Q = QP = vI

^.

Ïîäñòàíîâî÷íîå ïðåäñòàâëåíèå ãðóïïû

G = Aut(Γ)

^íà ^{âåðøèíàõ} ^ãðàà

Γ

^{îáû÷íûì}

îáðàçîì äàåò ìàòðè÷íîå ïðåäñòàâëåíèå

ψ

^ãðóïïû

G

^â

GL(v, C )

^. Ïðîñòðàíñòâî

C ^v

ÿâ- ëÿåòñÿîðòîãîíàëüíîé ïðÿìîéñóììîéñîáñòâåííûõïîäïðîñòðàíñòâ

W ₀ , . . . , W _d

^{ìàòðèöû}

ñìåæíîñòè

A ₁

^ãðàà

Γ

^.^Äëÿ^ëþáîãî

g ∈ G

^{ìàòðèöà}

ψ(g)

ïåðåñòàíîâî÷íàñ

A

^,^{ïîýòîìó}^ïîä-

ïðîñòðàíñòâî

W _i

^{ÿâëÿåòñÿ}

ψ(G)

-èíâàðèàíòíûì.Ïóñòü

χ _i

^{õàðàêòåð}ïðåäñòàâëåíèÿ

ψ _W i

^.

Òîãäà (ñì.[4, 3.7℄) äëÿ

g ∈ G

^{ïîëó÷èì}

χ _i (g) = v ⁻¹

d

X

j=0

Q _ij α _j (g),

ãäå

α _j (g)

^÷èñëî ^òî÷åê

x

^èç

X

^òàêèõ,^÷òî

d(x, x ^g ) = j

^.

Γ

{48, 35, 9; 1, 7, 40}

^,

G = Aut(Γ)

^. ^Åñëè

g ∈ G

^,

χ ₁

^{õàðàêòåð} ^{ïðîåêöèè} ïðåäñòàâëåíèÿ

ψ

íàïîäïðîñòðàíñòâîðàçìåðíîñòè

56

^,

χ ₂

^{õàðàêòåð}^{ïðîåêöèè}ïðåäñòàâëåíèÿ

ψ

^íà^{ïîäïðî-}

ñòðàíñòâîðàçìåðíîñòè

216

^, ^òî

α _i (g) = α _i (g ^l )

^äëÿ ^ëþáîãîíàòóðàëüíîãî ÷èñëà

l

^, ^{âçàèìíî}

ïðîñòîãîñ

|g|

^,

χ ₁ (g) = (7α ₀ (g)+2α ₁ (g)−α ₃ (g))/42−7/6

^è

χ ₂ (g) = (9α ₀ (g)+α ₃ (g))/14−9/2

^.

Åñëè

|g| = p

^{ïðîñòîå}^÷èñëî, ^òî

χ ₁ (g) − 56

^è

χ ₂ (g) − 216

^{äåëÿòñÿ} ^íà

p

^.

⊳

^Èìååì

Q =







1 1 1 1

56 91/6 −7/6 −28/3 216 −9/2 −9/2 20

70 −35/3 14/3 −35/3





 .

Çíà÷èò,

χ 1 (g) = (8α 0 (g) + 13α 1 (g)/6 − α 2 (g)/6 − 4α 3 (g)/3)/49

^. ^{Ïîäñòàâëÿÿ}

α 2 (g) = 343 − α ₀ (g) − α ₁ (g) − α ₃ (g)

^, ^{ïîëó÷èì}

χ ₁ (g) = (7α ₀ (g) + 2α ₁ (g) − α ₃ (g))/42 − 7/6

^.

Àíàëîãè÷íî

χ ₂ (g) = (432α ₀ (g) − 9α ₁ (g) − 9α ₂ (g) + 40α ₃ (g))/686

α ₁ (g) + α ₂ (g) = 343 − α ₀ (g) − α ₃ (g)

χ ₂ (g) = (9α ₀ (g) + α ₃ (g))/14 − 9/2

^.

Îñòàëüíûå óòâåðæäåíèÿëåììû ñëåäóþò èç [5, ëåììà2℄.

⊲

3.Àâòîìîðèçìû ñèëüíî ðåãóëÿðíîãî ãðàà ñ ïàðàìåòðàìè

(343, 54, 5, 9)

Â ëåììàõ 46 ïðåäïîëàãàåòñÿ, ÷òî

Γ

^ñèëüíî ^{ðåãóëÿðíûé} ^ãðà ^ñ ïàðàìåòðàìè

(343, 54, 5, 9)

^è ^{ñïåêòðîì}

54 ¹

^,

5 ²¹⁶

^,

−9 ¹²⁶

^,

G = Aut(Γ)

^,

g

p

èç

G

^,

α _i (g) = pw _i

^äëÿ

i > 0

^è

∆ = Fix(g)

^. ^Ââèäó ^{ãðàíèöû} ^Õîìàíà ìàêñèìàëüíûé ïî- ðÿäîê êëèêè

K

^èç

Γ

^íå ^áîëüøå

1 − k/θ _d = 7

^, ìàêñèìàëüíûé ïîðÿäîê êîêëèêè

C

^èç

Γ

íå áîëüøå

−vθ _d /(k − θ _d ) = 49

^. ^Â ^ñëó÷àå

|K| = 7

^ëþáàÿ ^{âåðøèíà} ^èç

Γ − K

^ñìåæíà ^ñ

åäèíñòâåííîéâåðøèíîéèç

K

^,^à^â^ñëó÷àå

|C| = 49

^ëþáàÿ^{âåðøèíà}^èç

Γ −C

^ñìåæíà^òî÷íî

ñäåâÿòüþ âåðøèíàìè èç

C

^.

Ëåììà 4.Âûïîëíÿþòñÿñëåäóþùèåóòâåðæäåíèÿ:

(5)

(1)

^åñëè

∆

^ïóñòîé ^ãðà,^òî

p = 7

^è

α ₁ (g) = 49(2s + 1)

^;

(2)

^åñëè

∆

^{ÿâëÿåòñÿ}

n

^{-êëèêîé,}^òî

p = 2

^,

n = 7

^è

α ₁ (g) = 28l

^;

(3)

^åñëè

∆

^{ÿâëÿåòñÿ}

m

m > 1

^, ^òî

p = 3

^,

m ∈ {4, 7, . . . , 49}

^è

α ₁ (g) = 5m + 7 + 42l

^;

(4)

^åñëè

∆

^{ñîäåðæèò}^ðåáðî ^è ^{ÿâëÿåòñÿ} îáúåäèíåíèåì èçîëèðîâàííûõ êëèê, òî

p = 3

è ïîðÿäîê ìàêñèìàëüíîé êëèêè èç

∆

^ðàâåí

1

^èëè

4

^.

⊳

^Èìååì

Q =





1 1 1

216 20 −9/2 126 −21 7/2



 .

Ïóñòü

ϕ ₂

^{ïðîåêöèÿ} ìîíîìèàëüíîãî ïðåäñòàâëåíèÿ

G

^íà ïîäïðîñòðàíñòâî ðàçìåðíî- ñòè 126. Òîãäà

ϕ ₂ (g) = (36α ₀ (g) − 6α ₁ (g) + α ₂ (g))/98

α ₂ (g) = 343 − α ₀ (g) − α 1 (g)

ϕ 2 (g) = (5α 0 (g) − α 1 (g) + 49)/14

^.

Ïóñòü

∆

^ïóñòîé ^ãðà. ^Ò^àê ^êàê

v = 7 ³

^, ^òî

p = 7

^. ^Äàëåå, ^÷èñëî

ϕ ₂ (g) = (−α ₁ (g) + 49)/14

^{äåëèòñÿ}^íà ^7,^{ïîýòîìó}

α ₁ (g) = 49(2s + 1)

^.

Ïóñòü

∆

^{ÿâëÿåòñÿ}

n

^{-êëèêîé.}^Åñëè

n = 1

^, ^òî

p

^äåëèò ⁵⁴ ^è ^288,^{ïîýòîìó}

p = 2

^. ^Â ^ýòîì

ñëó÷àå ÷èñëî

ϕ ₂ (g) = (54 − α ₁ (g))/14

^÷åòíî, ^{ïîýòîìó}

α ₁ (g) = 28l

^. ^Äàëåå, ^÷èñëà

λ

^è

µ

íå÷åòíû,ïîýòîìó ëþáàÿ âåðøèíàèç

Γ − ∆

^ñìåæíà^ñ ^{âåðøèíîé}^èç

∆

^,ïðîòèâîðå÷èå.

Åñëè

n > 1

^, ^òî^äëÿ^äâóõ^âåðøèí

a, b ∈ ∆

^{ýëåìåíò}

g

^{äåéñòâóåò}^áåç íåïîäâèæíûõ òî÷åê íà

[a] ∩ [b] − ∆

^,

[a] − b ^⊥

^è^íà

Γ − a ^⊥

^.^Îòñþäà

p

^äåëèò

7 − n

^,⁴⁸^è^288,^{ïîýòîìó}

p = 2

^.^Äàëåå,

÷èñëà

λ

^è

µ

^{íå÷åòíû,}^{ïîýòîìó}^ëþáàÿ^{âåðøèíà}^èç

Γ − ∆

^ñìåæíà^ñ^{âåðøèíîé}^èç

∆

^è

n = 7

^.

Äàëåå,÷èñëî

ϕ ₂ (g) = 6 − α ₁ (g)/14

^÷åòíî,^{ïîýòîìó}

α ₁ (g) = 28l

^.

Ïóñòü

∆

^{ÿâëÿåòñÿ}

m

m > 1

^.^Äëÿ^äâóõ^âåðøèí

a, b ∈ ∆

^{ýëåìåíò}

g

^{äåéñòâóåò}

áåç íåïîäâèæíûõ òî÷åê íà

[a] ∩ [b]

^è ^íà

Γ − (a ^⊥ ∪ b ^⊥ ∪ ∆)

^. ^Îòñþäà

p

^äåëèò ⁹ ^è

244 − m

^,

ïîýòîìó

p = 3

^,

m ∈ {4, 7, . . . , 49}

^,

ϕ 2 (g) = (5m − α 1 (g) + 49)/14

^è

α 1 (g) = 5m + 49 + 42l

^.

Åñëè

m = 49

^, ^òî ^êàæäàÿ ^{âåðøèíà}^èç

Γ − ∆

^ñìåæíà^ñ ⁹ ^{âåðøèíàìè}^èç

∆

^è

α ₁ (g) = 0

^.

Ïóñòü

∆

^{ñîäåðæèò} ^ðåáðî ^è ^{ÿâëÿåòñÿ} îáúåäèíåíèåì èçîëèðîâàííûõ êëèê. Òîãäà

p

äåëèò 9è

7 − t

^, ^ãäå

t

^{ïîðÿäîê} ìàêñèìàëüíîé êëèêè èç

∆

^, ^{ïîýòîìó}

t ∈ {1, 4}

^.

⊲

Ëåììà 5. Âûïîëíÿþòñÿñëåäóþùèåóòâåðæäåíèÿ:

(1) [a]

^íå ^{ñîäåðæèòñÿ} ^â

∆

^äëÿ^ëþáîé ^{âåðøèíû}

a ∈ Γ

^;

(2) Γ

^íå ^{ñîäåðæèò} ñîáñòâåííûõ ñèëüíî ðåãóëÿðíûõ ïîäãðàîâ ñ ïàðàìåòðàìè

(v ^′ , k ^′ , 5, 9)

^;

(3) p 6 7

^.

⊳

^Ïóñòü

[a] ⊂ ∆

^äëÿ^{íåêîòîðîé} ^{âåðøèíû}

a

^. ^Ò^îãäà^äëÿ^ëþáîé ^{âåðøèíû}

u ∈ Γ ₂ (a) − ∆

îðáèòà

u ^hgi

^íå ^{ñîäåðæèò}ãåîäåçè÷åñêèõ 2-ïóòåé è ÿâëÿåòñÿêîêëèêîé.

Åñëè

|∆| = 55

^, ^òî

p

^äåëèò ^288,

ϕ ₂ (g) = (324 − α ₁ (g))/14 = 324/14

^, ïðîòèâîðå÷èå. Åñëè æå

b ∈ ∆ − a ^⊥

^, ^òî

[b] ⊂ ∆

^, ïðîòèâîðå÷èå.

Äîïóñòèì,÷òî

Γ

^{ñîäåðæèò}ñîáñòâåííûéñèëüíîðåãóëÿðíûéïîäãðà

Σ

^ñïàðàìåòðàìè

(v ^′ , k ^′ , 5, 9)

^. ^Ò^îãäà

4(k ^′ − 9) + 16 = n ²

^,^{ïîýòîìó}

n = 2l

^,

k ^′ = l ² + 5

^,

l 6 6

^,

Σ

^èìååò ^{íåãëàâíûå}

ñîáñòâåííûå çíà÷åíèÿ

l − 2

^,

−2 − l

^è^{êðàòíîñòü}

l − 2

^ðàâíà

(l + 1)(l ² + 5)(l ² + l + 7)/18l

^.

Îòñþäà

l = 5

^,

v ^′ − k ^′ − 1 = 80

^è

Σ

^èìååò ^{ïàðàìåòðû}

(111, 30, 5, 9)

^. ^Ò^åïåðü ^÷èñëî ^ðåáåð

ìåæäó

Σ

^è

Γ − Σ

^ðàâíî

111 · 24

^, ïðîòèâîðå÷èå ñ òåì, ÷òî íåêîòîðàÿ âåðøèíà èç

Γ − Σ

ñìåæíàïî êðàéíåé ìåðå ñäâóìÿ âåðøèíàìè èç

Σ

^.

Åñëè

p > 11

^, ^òî

∆

^ñèëüíî ^{ðåãóëÿðíûé}^ïîäãðà ^ñ ïàðàìåòðàìè

(v ^′ , k ^′ , 5, 9)

^, ^ïðîòè-

âîðå÷èå. Çíà÷èò,

p 6 7

^.

⊲

Èç ëåìì45 ñëåäóåò òåîðåìà 2.

(6)

4. Àâòîìîðèçìû äèñòàíöèîííî ðåãóëÿðíîãî ãðàà

ñ ìàññèâîì ïåðåñå÷åíèé

{48, 35, 9; 1, 7, 40}

Â ëåììàõ 67 ïðåäïîëàãàåòñÿ, ÷òî

Γ

{48, 35, 9; 1, 7, 40}

^,

G = Aut(Γ)

^,

g

p

^èç

G

^è

Ω = Fix(g)

^.

(1)

^åñëè

Ω

^ïóñòîé ^ãðà, ^òî

p = 7

^,

α ₁ (g) = 49(l + 1 + 3s)

^è

α ₃ (g) = 49(2l + 1)

^,

3l + 2 + 3s 6 7

^;

(2)

^åñëè

Ω

^{ÿâëÿåòñÿ}

n

^{-êëèêîé,}^òî

n = 1

^,

p = 3

^,

α ₁ (g) = 3(7l + 16+ 21s)

^,

α ₃ (g) = 42l + 54

è

9l + 9s + 15 6 49

^;

(3)

^åñëè

Ω

^{ÿâëÿåòñÿ}

m

m > 1

^, ^òî ^ëþáûå ^äâå ^{âåðøèíû} ^èç

Ω

^{íàõîäÿòñÿ}

íàðàññòîÿíèè

3

^â

Γ

^,

p = 2

^,

m = 7

^,

α ₁ (g) = 14l − 18 + 42s

^è

α ₃ (g) = 28l − 36

^;

(4)

^åñëè

Ω

^{ñîäåðæèò}^ðåáðî,^òî^ëèáî

Ω

^{ñîäåðæèò}^{âåðøèíû,}íàõîäÿùèåñÿíàðàññòîÿíèè

2

^â

Γ

^, ^ëèáî

Ω

îáúåäèíåíèå äâóõèçîëèðîâàííûõ

4

^-êëèê,

p = 5

^,

α ₁ (g) = 35s + 20 + 105t

è

α ₃ (g) = 5 + 70s

^.

⊳

^Ïóñòü

Ω

^ïóñòîé ^ãðà.^Ò^àê ^êàê

v = 343

^, ^òî

p = 7

^, ^÷èñëî

χ ₂ (g) = (α ₃ (g) − 63)/14

ñðàâíèìî ñ6 ïî ìîäóëþ 7,ïîýòîìó

α ₃ (g) = 63 + 14(7l + 6) = 49 + 98l

^.

χ ₁ (g) = (α ₁ (g) − 49(l + 1))/21

^{äåëèòñÿ}^íà^7,^{ïîýòîìó}

α ₁ (g) = 49(l + 1) + 147s = 49(l + 1 + 3s)

^. Óòâåðæäåíèå (1)äîêàçàíî.

Ïóñòü

Ω

^{ÿâëÿåòñÿ}

n

^{-êëèêîé.} ^Åñëè

n = 1

^, ^òî

p

^äåëèò ⁴⁸^è ^54,^{ïîýòîìó}

p = 2, 3

^. ^Èìååì

p ¹ ₃₃ = 9

^,

p ² ₃₃ = 9

^è

p ³ ₃₃ = 5

^,^{ïîýòîìó}

p 6= 2

^.^Åñëè

p = 3

^, ^òî^÷èñëî

χ ₂ (g) = (9 + α ₃ (g) − 63)/14

äåëèòñÿíà 3,

α ₃ (g) = 42l + 54

^.

χ ₁ (g) = (7 + 2α ₁ (g) − 6(7l + 9) − 49)/42 = (α ₁ (g) − 21l − 48)/21

^{ñðàâíèìî}

ñ2 ïî ìîäóëþ3, ïîýòîìó

α 1 (g) = 21l + 48 + 63s

^.

Åñëè

n > 1

^, ^òî

p

^äåëèò

14 − n

^,

54

^è

35

^, ïðîòèâîðå÷èå. Óòâåðæäåíèå (2)äîêàçàíî.

Ïóñòü

Ω

^{ÿâëÿåòñÿ}

m

m > 1

^. ^Åñëè ^ëþáûå ^äâå ^{âåðøèíû} ^èç

Ω

^{íàõîäÿòñÿ} ^íà

ðàññòîÿíèè 3,òî èç ðàâåíñòâ

p ³ ₁₃ = 8

^,

p ³ ₃₃ = 5

^{ñëåäóåò,} ^÷òî

p = 2

^è

m ∈ {3, 5, 7}

^. ^Ò^àê ^êàê

ëþáàÿ âåðøèíàèç

Γ − Ω

^{íàõîäèòñÿ}^íà ^{ðàññòîÿíèè} ³ ^îò^{âåðøèíû} ^èç

Ω

^, ^òî

m = 7

^. ^Äàëåå,

÷èñëî

χ ₂ (g) = (63 + α ₃ (g) − 27)/14

^÷åòíî, ^{ïîýòîìó}

α ₃ (g) = 28l − 36

^. ^{Àíàëîãè÷íî} ^÷èñëî

χ 1 (g) = (α 1 (g) − 14l + 18)/21

^÷åòíî ^è

α 1 (g) = 14l − 18 + 42s

^.

Åñëè

Ω

^{ñîäåðæèò} ^äâå ^{âåðøèíû} ^íà ^{ðàññòîÿíèè} ^2, ^òî

p

^äåëèò ⁴⁸ ^è ^7, ïðîòèâîðå÷èå.

Óòâåðæäåíèå (3)äîêàçàíî.

Ïóñòü

Ω

^{ñîäåðæèò} ^ðåáðî ^è ^íå ^{ñîäåðæèò} ^{âåðøèí,} íàõîäÿùèõñÿ íà ðàññòîÿíèè 2 â

Γ

^.

Òîãäà

Ω

^{ÿâëÿåòñÿ}îáúåäèíåíèåì

l

èçîëèðîâàííûõêëèê,èëþáûåäâåâåðøèíûèçðàçíûõ êëèê íàõîäÿòñÿ íà ðàññòîÿíèè 3 â

Γ

^. ^Ò^àê ^êàê

p ¹ ₁₂ = 35

^è

p ³ ₁₂ = 40

^, ^òî

p = 5

^è ^{ïîðÿäêè}

ìàêñèìàëüíûõ êëèê èç

Ω

^ðàâíû ^4. ^Äàëåå,

p ³ ₃₃ = 5

l = 2

^, ^÷èñëî

χ ₂ (g) = (72 + α ₃ (g) − 63)/14

^{ñðàâíèìî} ^ñ ¹ ^ïî ^ìîäóëþ ^5,^{ïîýòîìó}

α ₃ (g) = 5 + 70s

^. ^×èñëî

χ ₁ (g) = (1 + α ₁ (g) − 35s)/21

^{ñðàâíèìî} ^ñ¹ ^ïî ^ìîäóëþ ⁵^è

α ₁ (g) = 35s + 20 + 105t

^.

⊲

Ëåììà 7.Åñëè

Ω

^{ñîäåðæèò}^{âåðøèíû}

a

^,

b

^íà ^{ðàññòîÿíèè}

2

^â

Γ

^, ^òî

p 6 11

^.

⊳

^Ïóñòü

Ω

^{ñîäåðæèò} ^{âåðøèíû}

a

^,

b

^íà ^{ðàññòîÿíèè} ² ^â

Γ

^è

Ω ₀

^{ñîäåðæàùàÿ}

a

^,

b

ñâÿçíàÿ êîìïîíåíòà ãðàà

Ω

^.

Åñëè

p > 13

^, ^òî

Ω ₀

^âïîëíå ^{ðåãóëÿðíûé} ^ãðà ^ñ ïàðàìåòðàìè

(v ^′ , k ^′ , 12, 7)

^. ^Åñëè

Ω ₀

^ñèëüíî ^{ðåãóëÿðíûé} ^ãðà, ^òî

4(k ^′ − 7) + 25 = n ²

^. ^Â ^ýòîì ^ñëó÷àå

n = 2w + 1

^è

k ^′ = w ² + w + 1

^. ^{Íåãëàâíûå} ñîáñòâåííûå çíà÷åíèÿ

Ω ₀

^ðàâíû

w + 3

^è

2 − w

^, ^ïðè÷åì

êðàòíîñòü

w + 3

^ðàâíà

(w − 3)(w ² + w + 1)(w ² + 2w − 1)/(7(2w + 1))

^.^Äàëåå,

(w − 3, 2w + 1)

äåëèò 7,

(w ² + w + 1, 2w + 1) = (2w ² + 2w + 2, 2w ² + w) = (w + 2, 2w + 1)

^äåëèò ³ ^è

(7)

(w ² + 2w − 1, 2w + 1) = (2w ² + 4w − 2, 2w ² + w) = (3w − 2, 2w + 1)

^äåëèò^7.^Îòñþäà

2w + 1

äåëèò

49 · 3

^è

2w + 1 = 7

Åñëè

Ω

^{íåñâÿçíûé} ^ãðà, ^òî

k ^′ 6 8

^, ïðîòèâîðå÷èå. Èòàê,

Ω

^{ñâÿçíûé} ^ãðà ^äèà-

ìåòðà, áîëüøåãî2. Âñëó÷àå

k ^′ > 25

^{ïîëó÷èì}

|Ω| > 1 + 25 + 25 · 12/7 + 1

^. Ïðîòèâîðå÷èå ñ òåì,÷òîïî ëåììå 1èìååì

|Ω| 6 63

^. ^Â^ñëó÷àå

k ^′ 6 17

^{ïîëó÷èì}

b 1 (Ω) 6 4

^,

k ^′ > 3b 1 (Ω)

^è

äèàìåòð

Ω

^íå ^áîëüøå^2,ïðîòèâîðå÷èå.

Åñëè

k ^′ = 18

^,^òî

p

^äåëèò^30.^Åñëè

k ^′ = 19

^,^òî

p = 29

^,

b ₁ (Ω) = 6

^,

k ^′ > 3b ₁ (Ω)

^è^{äèàìåòð}

Ω

íå áîëüøå2,ïðîòèâîðå÷èå.

Åñëè

k ^′ = 20

^, ^òî

p

^äåëèò ^28. ^Åñëè

k ^′ = 21

^, ^òî

p

k ^′ = 22

^, ^òî

p = 13

^,

|Ω 3 (a)| > 15

^è

|Ω| > 1 + 22 + 22 · 9/7 + 15

Åñëè

k ^′ = 23

^, ^òî

p

k ^′ = 24

^, ^òî

p

^äåëèò ^24. ^Â ^ëþáîì ^ñëó÷àå ^èìååì

ïðîòèâîðå÷èå.

⊲

Èç ëåìì67 ñëåäóåò òåîðåìà 1.

5. Âåðøèííî ñèììåòðè÷íûé äèñòàíöèîííî ðåãóëÿðíûé ãðà

ñ ìàññèâîì ïåðåñå÷åíèé

{48, 35, 9; 1, 7, 40}

Äîêîíöàðàáîòûáóäåìïðåäïîëàãàòü,÷òî

Γ

äèñòàíöèîííî ðåãóëÿðíûéãðàñìàñ- ñèâîìïåðåñå÷åíèé

{39, 36, 4; 1, 1, 36}

^èíåðàçðåøèìàÿãðóïïà

G = Aut(Γ)

^{äåéñòâóåò}^òðàí-

çèòèâíîíàìíîæåñòâåâåðøèíãðàà.Äëÿâåðøèíû

a ∈ Γ

^{ïîëó÷èì}

|G : G a | = 343

^. ^Ââèäó

òåîðåì 12 èìååì

π(G) ⊆ {2, 3, 5, 7}

^. ^Ïóñòü

K = O ₇ (G)

^,

T ¯

^öîêîëü ^ãðóïïû

G ¯ = G/K

^.

U

ýëåìåíòàðíàÿ àáåëåâà ïîäãðóïïà èç

G

^{ïîðÿäêà}

49

^,

g _i

^,

i ∈ {1, 2, . . . , 8}

^{ïîðîæäàþò} ^{ðàçëè÷íûå} ^{ïîäãðóïïû} ^{ïîðÿäêà}

7

^èç

U

^,

Ω ⁱ = Fix(g _i )

^è

Ω ⁰ = Fix(U )

^. ^Ò^îãäàâûïîëíÿþòñÿ ñëåäóþùèåóòâåðæäåíèÿ:

(1)

^åñëè

a ∈ Ω ⁰

^, ^òî

[a]

^è

Γ 3 (a)

^{ñîäåðæàòñÿ} ^â

∪ i Ω ⁱ

^, ^è ^íà

Γ

^íåò

U

^-îðáèò^äëèíû

49

^;

(2) Ω ⁰

^ïóñòîé ^ãðà.

⊳

^×èñëî

χ ₂ (g _i ) = (9|Ω ⁱ | + α ₃ (g _i ) − 63)/14

^{ñðàâíèìî} ^ñ⁶^ïî^ìîäó^ëþ

7

^,^{ïîýòîìó}

α ₃ (g _i ) = 98l _i + 49 − 9|Ω ⁱ |

^. ^Äàëåå,^÷èñëî

χ ₁ (g _i ) = (8|Ω ⁱ |+ α ₁ (g _i ) − 49l _i − 49)/21

^{äåëèòñÿ}^íà^7,^{ïîýòîìó}

α ₁ (g _i ) = 49l _i + 49 − 8|Ω ⁱ | + 147s _i

^. ^Åñëè

α ₀ (g _i ) = 35

^, ^òî

α ₃ (g _i ) = 98l _i − 294

^è

α ₁ (g _i ) = 49l _i − 280 + 147s _i

^.

Ïóñòü

a ∈ Ω ⁰

^. ^Èç^{äåéñòâèÿ}

U

^íà

[a]

^, ^íà

Γ ₂ (a)

^è^íà

Γ ₃ (a)

^{ñëåäóåò}^, ^÷òî

Ω ⁰

^{ñîäåðæèò} ^íå

ìåíåå 6 âåðøèí èç

[a]

^, ^íå ^ìåíåå ²^âåðøèí ^èç

Γ 2 (a)

^è^íå ^ìåíåå ⁵ ^âåðøèí ^èç

Γ 3 (a)

^.

Äëÿ

b ∈ [a] ∩ Ω ⁰

^{ñëåäóåò,}^÷òî

[a] ∩ [b]

^{ñîäåðæè} ⁵^èëè¹²^âåðøèí^èç

Ω ⁰

^. ^{Çàìåòèì,}^÷òî

[a]

è

Γ ₃ (a)

^{ñîäåðæàòñÿ} ^â

∪ _i Ω ⁱ

^. ^Â ^{ïðîòèâíîì} ^ñëó÷àå

Γ ₃ (a)

^{ñîäåðæèò}

U

^{-îðáèòó} ^äëèíû ^49.

Ïðîòèâîðå÷èå ñ äåéñòâèåì

U

^íà

[d] ∩ Γ 3 (a)

^äëÿ

d ∈ Γ 3 (a) ∩ Ω ⁰

^. ^Åñëè

Γ 2 (a)

^{ñîäåðæèò}

U

^-

îðáèòó

∆

^äëèíû^49,

u ∈ ∆

^,^òî

u

^ñìåæíà^ñ⁹^{âåðøèíàìè} ^èç

Γ ₃ (a)

^. ^{Âåðøèíà}

d ∈ Γ ₃ (a) ∩ [u]

ïîïàäàåòâ

Ω ⁱ

^äëÿ^{íåêîòîðîãî}

i

^è

d

^ñìåæíà^ñ⁷^{âåðøèíàìè}^èç

∆

^.Ïðîòèâîðå÷èå ñòåì,÷òî

÷èñëîðåáåð ìåæäó

∆

^è

Γ 3 (a)

^ðàâíî

9 · 49

^, ^íî^íå^áîëüøå

54 ·7

^. Óòâåðæäåíèå(1)äîêàçàíî.

Åñëè

|Ω ⁰ | > 28

^, ^òî ^ââèäó ^ëåììû ¹ ^èìååì

|Γ − Ω ⁰ | 6 8 · 35 = 280

Çíà÷èò,

|Ω ⁰ | 6 21

^. ^Äàëåå,

₃ = 40

^, ^{ïîýòîìó} ^{âåðøèíà} ^èç

Γ ₃ (a) ∩ Ω ⁰

^ñìåæíà ^ñ ⁵ ^âåðøè-

íàìè èç

Γ ₂ (a) ∩ Ω ⁰

^. ^{Ïîýòîìó}

Ω ⁰

^{ñîäåðæèò} ^òî÷íî ⁶ ^âåðøèí ^èç

[a]

^, ⁹ ^âåðøèí ^èç

Γ ₂ (a)

^è

5 âåðøèí èç

Γ ₃ (a)

^äëÿ ^ëþáîé ^{âåðøèíû}

a ∈ Ω ⁰

^. ^Òåïåðü ^÷èñëî ^ðåáåð ^ìåæäó

Γ ₃ (a) ∩ Ω ⁰

^è

Γ ₂ (a) ∩ Ω ⁰

^ðàâíî

25

^, ^à^÷èñëî^ðåáåð ^ìåæäó

Γ ₂ (a) ∩ Ω ⁰

^è

Γ ₃ (a) ∩ Ω ⁰

^ðàâíî ^18,ïðîòèâîðå÷èå.

⊲

Ââèäó ëåììû 8 ãðóïïà

G _a

^èìååò öèêëè÷åñêóþ ñèëîâñêóþ 7-ïîäãðóïïó.

(8)

(1)

^åñëè

7

^äåëèò ^{ïîðÿäîê} ^{êîìïîíåíòû}

L

^ãðóïïû

T ¯

^, ^òî

L

^{èêñèðóåò} ^{âåðøèíó} ^èç

Γ

^,

|K : K _a | = 7 ³

^, ^ãðóïïà

L

^{èçîìîðíà}

L ₂ (7)

^è ^òî÷íî ^{äåéñòâóåò} ^íà

K

^;

(2)

^åñëè

7

^íå^äåëèò^{ïîðÿäîê}^{êîìïîíåíòû}

M

^ãðóïïû

T ¯

^,^òî^ãðóïïà

M = M a

^{èçîìîðíà}

A ₅

^,

A ₆

^èëè

P Sp ₄ (3)

^,

M

^íå öåíòðàëèçóåò

K

^è

|K : K _a |

^äåëèò

7 ³

^.

⊳

^Ïî ^{òàáëèöå} ¹ ^èç ^[6℄ ^ãðóïïà

L

^{èçîìîðíà}

L ₂ (7)

^,

L ₂ (8)

^,

U ₃ (3)

^,

A ₇

^,

L ₂ (49)

^,

U ₃ (5)

^,

L 3 (4)

^,

A 8

^,

A 9

^,

A 10

^,

J 2

^,

U 4 (3)

^,

P Sp 5 (7)

^,

Sp 6 (2)

^èëè

Ω ⁺ ₈ (2)

^.

Òàê êàê

|L : L _a |

^äåëèò

7 ³

^, ^òî ^ãðóïïà

L

^{èçîìîðíà}

L ₂ (7)

^èëè

A ₇

^. ^Åñëè

|L : L _a | = 7

^,

òî

|K : K _a | = 7 ²

^,

L _a

öåíòðàëèçóåò

K

^è ^{ïîòî÷å÷íî} ^{èêñèðóåò}

a ^K

^. ^Åñëè

K

^íåàáå-

ëåâà ãðóïïà, òî êîììóòàíò

K ^′

^{ñîäåæèòñÿ} ^â

K _a

^, ïðîòèâîðå÷èå. Òåïåðü äëÿ ïîäãðóïïû

U = [K, L _a ]

^{ïîðÿäêà} ⁹ ^îðáèòà

a ^U

^{ñîäåðæèò} ⁴⁹ ^{âåðøèí,} ïðîòèâîðå÷èå ñ ëåììîé 8. Èòàê,

L = L _a

^,

|K : K _a | = 7 ³

^è

L

^òî÷íî ^{äåéñòâóåò} ^íà

K

^. ^Îòñþäà ^ãðóïïà

L

^{èçîìîðíà}

L ₂ (7)

^.

Åñëè

7

^íå ^äåëèò ^{ïîðÿäîê} ^{êîìïîíåíòû}

M

^ãðóïïû

T ¯

^, ^òî ^ãðóïïà

M = M _a

^{ÿâëÿåòñÿ}

{2, 3, 5}

^{-ãðóïïîé.} ^{Ïîýòîìó}

M

^{èçîìîðíà}

A ₅

^,

A ₆

^èëè

P Sp ₄ (3)

^. ^Äàëåå

|K : K _a |

^äåëèò

7 ³

^.

Åñëè

M

öåíòðàëèçóåò

K

^, ^òî ^{ïîëó÷èì} ïðîòèâîðå÷èå ñ òåì, ÷òî

M

^{ïîòî÷å÷íî} ^{èêñèðó-}

åò

a ^K

^.

⊲

Ëåììà 10.

T ¯

^{ñîäåðæèò} åäèíñòâåííóþ êîìïîíåíòó

L ¯

^, ^òî÷íî äåéñòâóþùóþ íà

K

^,

L ¯ ∼ = L 2 (7)

^,

A 5

^,

A 6

^,

P Sp 4 (3)

^è^äëÿ^{ïîëíîãî}^{ïðîîáðàçà}

L

^ãðóïïû

L ¯

^èìååì

L a = K a ×O ₇ ^′ (L a )

è

|K| = 7 ³

^â^ñëó÷àå

L ¯ ∼ = L ₂ (7)

^,

|K| = 7 ⁴

^â^{ïðîòèâíîì} ^{ñëó÷àå.}

⊳

^Ïî ^ëåììå ⁹ ^ëþáàÿ ^{êîìïîíåíòà}

L ¯

^ãðóïïû

T ¯

^{èêñèðóåò} ^{âåðøèíó}

a

^. ^Åñëè

| L| ¯

^íå

äåëèòñÿ íà 7, òî ïî ëåììå 9

L ¯ ∼ = L 2 (7)

^,

A 5

^,

A 6

^,

P Sp 4 (3)

^è

L ¯

^íå öåíòðàëèçóåò

K

^. ^Ïóñòü

L

^ïîëíûé^{ïðîîáðàç} ^{êîìïîíåíòû}

L ¯

^. ^Ò^îãäà

L _a = K _a × O ₇ ′ (L _a )

^.

Åñëè

| L| ¯

^{äåëèòñÿ}^íà^7,^òî ^ïî^ëåììå⁹^èìååì

|K : K _a | = 7 ³

K

ýëåìåíòàðíàÿ àáåëåâàïîäãðóïïàïîðÿäêà

7 ³

^,

L ¯

^{èçîìîðíà}

L ₂ (7)

^è

| G ¯ : ¯ L|

^äåëèò ^2.

Äîïóñòèì, ÷òî

|L _a |

^íå ^{äåëèòñÿ} ^íà ^7. ^Ò^àê ^êàê

GL ₃ (7)

^íå ^èìååò ^{ñåêöèé,} ^{èçîìîðíûõ}

A ₅

^,

A ₆

^èëè

P Sp ₄ (3)

^, ^òî

|K : (K )|

^{äåëèòñÿ} ^íà

7 ⁴

^. ^Äàëåå, ^ãðóïïà

G _a

^èìååò öèêëè÷åñêóþ

ñèëîâñêóþ 7-ïîäãðóïïó, ïîýòîìó

(K)

^{èêñèðóåò}

a

^,

(K ) = 1

^è

|K| = 7 ⁴

^.

⊲

Ñëåäñòâèå 1 äîêàçàíî.

Ëèòåðàòóðà

1. Brouwer A. E., Cohen À. M., Neumaier A. Distane-Regular Graphs.BerlinHeidelbergN.Y.:

Springer-Verlag.1989. DOI:10.1007/978-3-642-74341-2.

2. JurisiA.,VidaliJ.Extremal1-odesindistane-regulargraphsofdiameter3//Des.CodesCryptogr.

2012.Vol. 65.P.2947.

3. Behbahani M., Lam C. Strongly regular graphs with nontrivial automorphisms// Disrete Math.

2011.Vol. 311.P.132144.DOI:10.1016/j.dis.2010 .10 .00 5

4. CameronP.J.PermutationGroups.Cambridge:CambridgeUniv.Press, 1999.(LondonMath.So.

StudentTexts,45).DOI:10.1017/CBO97805116 236 77 .

5. àâðèëþêÀ.Ë.,ÌàõíåâÀ.À.Îáàâòîìîðèçìàõ äèñòàíöèîííîðåãóëÿðíûõãðàîâñìàññèâîì

ïåðåñå÷åíèé

{56, 45, 1; 1, 9, 56}

^/^/^Äîêë.^ÀÍ.^2010.Ò.^432,^5.Ñ. ^512515.

6. Zavarnitsine A. V. Finite simple groups with narrow prime spetrum // Siberian Eletr. Math.

Reports.2009.Vol.6.P.112.

Ñòàòüÿïîñòóïèëà 30ìàðòà2020ã.

ÌàõíåâÀëåêñàíäð Àëåêñååâè÷

Èíñòèòóòìàòåìàòèêèèìåõàíèêèèì.Í.Í.Êðàñîâñêîãî,

çàâ.îòäåëîìàëãåáðûèòîïîëîãèè

ÎÑÑÈß,620990,Åêàòåðèíáóðã,óë.Ñ.Êîâàëåâñêîé,16

E-mail:makhnevimm.uran.ru

(9)

ÁèòêèíàÂèêòîðèÿÂàñèëüåâíà

äîöåíòêàåäðûïðèêëàäíîéìàòåìàòèêè

ÎÑÑÈß,362025,Âëàäèêàâêàç,óë.Âàòóòèíà,4446

E-mail:bviktoriyavmail.ru

óòíîâàÀëèíàÊàçáåêîâíà

äîöåíòêàåäðûàëãåáðûèãåîìåòðèè

ÎÑÑÈß,362025,Âëàäèêàâêàç,óë.Âàòóòèíà,4446

E-mail:gutnovaalinagmail.om

Vladikavkaz MathematialJournal

2020,Volume 22,Issue 2,P. 2433

AUTOMORPHISMS OF ADISTANCE REGULARGRAPH

WITHINTERSECTIONARRAY

{48, 35, 9; 1, 7, 40}

Makhnev, A.A.

1

, Bitkina,V. V.

2

andGutnova,A.K.

2

1

N.N.KrasovskiiInstituteofMathematisandMehanis,

16S.KovalevskajaSt.,Ekaterinburg620990,Russia

2

NorthOssetianStateUniversity,

4446VatutinSt.,Vladikavkaz362025,Russia;

E-mail:makhnevimm.uran.ru, bviktoriyavmail.ru, gutnovaalinagmail.om

Abstrat. Ifadistane-regulargraph

Γ

ôf^diameter³ôntainsâ^maximal^loally^regular^1-ode^perfet

with respet to the last neighborhood, then

Γ

^has ^an intersetion array

{a(p + 1), cp, a + 1; 1, c, ap}

^or

{a(p + 1), (a + 1)p, c; 1, c, ap}

^, ^where

a = a 3

^,

c = c 2

^,

p = p ³ ₃₃

^(Jurisi ând ^Vidali).În ^the ^rst âse,

Γ

^has

aneigenvalue

θ 2 = −1

^and

Γ 3

^is^apseudo-geometrigraphfor

GQ(p + 1, a)

^.^If

c = a − 1 = q

^,

p = q − 2

^,^then

Γ

hasanintersetionarray

{q ² − 1, q(q − 2), q + 2; 1, q, (q + 1)(q − 2)}

^,

q > 6

^.^Theôrdersând^subgraphsôf^xed

pointsof automorphismsofa hypothetialdistane-regular graphwithintersetion array

{48, 35, 9; 1, 7, 40}

(

q = 7

⁾^are^studiedⁱⁿ^the^paper.^Let

G = Aut(Γ)

^beânînsoluble^groupâtingtransitivelyonthesetofverties ofthegraph

Γ

^,

K = O 7 (G)

^,

T ¯

^be^the^sole^of^the^group

G ¯ = G/K

^.^Then

T ¯

ôntains^theônlyômponent

L ¯

^,

L ¯

^that âtsêxatlyôn

K

^,

L ¯ ∼ = L 2 (7), A 5 , A 6 , P Sp 4 (3)

ând^for^the ^full^theînverseîmage ôf

L

^of^the^group

L ¯

wehave

L a = K a × O 7 ′ (L a )

^and

|K| = 7 ³

ⁱⁿ^the^ase^of

L ¯ ∼ = L 2 (7)

^,

|K| = 7 ⁴

^otherwise.

Keywords:stronglyregulargraph,distane-regulargraph,automorphismofgraph.

Mathematial Subjet Classiation (2000):05C25.

Foritation:Makhnev,A.A.,Bitkina,V.V.andGutnova,A.K.AutomorphismsofaDistaneRegular

Graph with Intersetion Array

{48, 35, 9; 1, 7, 40}

^, Vladikavkaz Math. J., 2020, vol. 22, no. 2, pp. 2433 (inRussian).DOI:10.46698/n0833-6942-7469-t.

Referenes

1. Brouwer, A. E., Cohen, À. M. and Neumaier, A. Distane-Regular Graphs, BerlinHeidelbergNew

York,Springer-Verlag,1989.DOI:10.1007/978-3-642-74341-2.

2. Jurisi,A.andVidali,J.Extremal1-CodesinDistane-RegularGraphsofDiameter3,DesignsCodes

andCryptography, 2012,vol.65,pp.2947.

3. Behbahani,M.andLam,C.StronglyRegularGraphswithNontrivialAutomorphisms,DisreteMath.,

2011,vol.311,pp.132144.DOI:10.1016/j.dis.2010.1 0.0 05 .

4. Cameron,P.J.PermutationGroups,LondonMath.So.StudentTexts,no.

45

^,^Cambridge,^Cambridge

Univ.Press, 1999.DOI:10.1017/CBO97805116 236 77.

(10)

5. Gavrilyuk, A. L. and Makhnev, A. A. On Automorphisms of Distane-Regular Graphs with

Intersetion Array

{56, 45, 1; 1, 9, 56}

^, ^Doklady Mathematis, 2010, vol. 81, no. 3,pp. 439442. DOI:

10.1134/S1064562410 030 28 2.

6. Zavarnitsine,A.V.FiniteSimpleGroupswithNarrowPrimeSpetrum,SiberianEletr.Math.Reports,

2009,vol.6,pp.112.

ReeivedMarh 30,2020

AlexanderA.Makhnev

N.N.KrasovskiiInstituteofMathematisandMehanis,

16S.KovalevskajaSt.,Ekaterinburg620990,Russia,

HeadofDepartamentofAlgebraandTopology

E-mail:makhnevimm.uran.ru

https://orid.org/0000-0003-2868-67 13

ViktoriyaV.Bitkina

4446VatutinSt.,Vladikavkaz362025,Russia,

AssoiateProfessoroftheDepartmentofAppliedMathematis

E-mail:bviktoriyavmail.ru

AlinaK.Gutnova

4446VatutinSt.,Vladikavkaz362025,Russia,

AssoiateProfessoroftheDepartmentofAlgebraandGeometry

E-mail:gutnovaalinagmail.om

https://orid.org/0000-0001-7467-72 4X

{48, 35, 9; 1, 7, 40} #

{48, 35, 9; 1, 7, 40} #

1

2

2

1

2

Γ

Γ

{a(p + 1), cp, a + 1; 1, c, ap}

{a(p + 1), (a + 1)p, c; 1, c, ap}

a = a 3

c = c 2

p = p 3 33

Γ

θ 2 = −1

Γ 3

GQ(p + 1, a)

c = a − 1 = q

p = q − 2

Γ

{q 2 − 1, q(q − 2), q + 2; 1, q, (q + 1)(q − 2)}

q > 6

{48, 35, 9; 1, 7, 40}

q = 7

G = Aut(Γ)

Γ

K = O 7 (G)

T ¯

G ¯ = G/K

T ¯

L ¯

K

L ¯ ∼ = L 2 (7)

A 5

A 6

P Sp 4 (3)

L

L ¯

L a = K a × O 7 ′ (L a )

|K| = 7 3

L ¯ ∼ = L 2 (7)

|K| = 7 4

{48, 35, 9; 1, 7, 40}

a

Γ

Γ i (a)

i

a

Γ

i

a

[a] = Γ 1 (a)

a ⊥ = {a} ∪ [a]

Γ

a, b ∈ Γ

[a] ∩ [b]

µ(a, b)

λ(a, b))

a

b

Γ

[a]∩[b]

µ

λ

P

L

α

(s, t)

s+ 1

#

t+1

(a, l) ∈ (P, L )

α

a

l

pG α (s, t)

α = 1

GQ(s, t)

P

{48, 35, 9; 1, 7, 40} ^#

p = p ³ 33

{q ² − 1, q(q − 2), q + 2; 1, q, (q + 1)(q − 2)}

|K| = 7 ³

|K| = 7 ⁴

[a] = Γ ₁ (a)

a ^⊥ = {a} ∪ [a]

pG _α (s, t)

pG _α (s, t)

pG _α (s, t)

b _i (u, w)

c _i (u, w)

Γ _i+1 (u)

Γ _i−1 (u)

{b ₀ , b ₁ , . . . , b _d−1 ; c ₁ , . . . , c _d }

b _i (u, w)

c _i (u, w)

a _i = k − b _i − c _i

p ^l _ij (x, y)

Γ _i (x) ∩ Γ _j (y)

p ^l _ij (x, y)

p ^l _ij

|C| 6 p ^d _dd + 2

c _d > a _d p ^d _dd

|C| 6 k _d / P _e

i=0 p ^d _id + 1