hazrul On (4,2) digraph Containing a Cycle of Length 2 2000
BULLETIN,Trr,.
Bull.Maky'Mi Malh.5c 5o(.lsecondS€nes)23 (2000)79 al
MALAYSIAN
MATHEMATICAL
ScIENcES
SocIETY
On (4,2)-digraphs
Containinga Cycleof Length2
rHeznullsweot e.wD2EDyTRIBAsKoRo
'Depaflment
JalanRayaKali Rungkut,
ofMadematicsandSciences,
UnivenityofSurabaya,
Surabaya
60292,Indonesia
-'Department
InstitutTeknologiBandung,
JalanGanesha
10,Bandung,Indonesra
ofMathematrcs,
'e mail : [email protected]
'ze-mail:
itbma0I @bds.centrin.net.id
Abstmct
A diregular digraph is a digraph with the in-degree and ouldegree of all venices is
conslant. The Moore botrnd for a diregular digraph of degree d and diameter * is
Ma,r=l+d+d2+..+
dk.
d>ltnd
d o n o t e x i s t . A ( d , l ) - d i g r a p hi s a d i r e g u l a rd i g r a p ho f d e g r e e d > 1 ,
dianeter t>l
diameter t>1,
I t i s w e l l k n o w n t h a t d i r e g u l a rd i g r a p h so f o r d e r M , / . t ,
and number of venices one less than the Moore bound. For degrees d=2
degree
nd3,it
has been shown tha! for diameter t > 3 there arc no such (4 l)-digmphs. However for diameter 2,
it is known that (d,2)-digraphsdo exist for an) degreed. The line digraph of (/+t
is one example
of such (42)-digaphs.
Furthermore, the recent study showed that there are three non,isomorphic
(2,2)-digraphs and exactly one non'isomorphic (3,2)-digraph. In this paper, we shall study
(4,2)-digraphs. We show that if(4,2)-digraph C contains a cycle oflength 2 then C; must be the line
digraph ofa complete digraph (i.
1. Introduction
A digraph G is a system consisting of a finite nonempty set y(G) of objects called
'fhe
vertices and a setE(G) of orderedpairs of distinct verticescalled arc.r.
order of G is
the cardinality of V(G). A subdigraph H of Gis a digraph having all verticesand arcs in
G. If (a,u) is an arc in a digraph G, tben r is said to be adjacent to y and y is said to be
adjacentfrom u. An in-neighbor of a vertex v in a digraph G is a ve ex lr such that
(u,v)e G. An out-neighbor of a vertex v in a digraph G is a vertex rr such that
(v,w)eG. The set of all out-neighborsof a vertex v is denotedby N*(v)and its
cardinality is called the out-deeree of v, d'(v) = | N- (v) | . Similarly, the set of all
in-neighborsof a vertex v is denotedby N (v) and its cardinality is called the ir-degree of
",
d- (")= N- (") A digraph
G is diregular of degree d if for any ve ex r in G,
|
l.
d*(v)=d (v)=d.
A walk of length , ftom a vertex ll to vertex v in G is a sequenceof vertices
( u = u o , u 1 , , . , , u =h v ) s u c ht h a t (u,-1,u,)e G for each i. A vertex u forms the trivial
H lswadi and E.T Baskoto
80
A path is a walk in which all points are
walk of length 0. A closed walk has 116= lo
distinct. A cycle C, of length lt > 0 is a closed walk with i distinct vertices (except 40
and |lr,). Ifthere is a path from & to v in 6 then we say thatv is teachable from u.
The distance from vertextl to vertex v in a digraph G, denotedby d(u,r''), is defined
as the length ofa shortestpath from u to v In generd, 6(u,v) is not necessarilyequal to
5(v,a). The diqmeter k of a digraph 6 is the maximum distancebetweenany two
verticesin G.
Let G be a diregular digraph of degree d and diameter t with ll vertices. Let one
vertex be distinguishedin G. Let n,,Vi=0,1," ,k, be the number of vertices at
distancei from the distinguishedve ex. Then,
.
Bull.Maky'Mi Malh.5c 5o(.lsecondS€nes)23 (2000)79 al
MALAYSIAN
MATHEMATICAL
ScIENcES
SocIETY
On (4,2)-digraphs
Containinga Cycleof Length2
rHeznullsweot e.wD2EDyTRIBAsKoRo
'Depaflment
JalanRayaKali Rungkut,
ofMadematicsandSciences,
UnivenityofSurabaya,
Surabaya
60292,Indonesia
-'Department
InstitutTeknologiBandung,
JalanGanesha
10,Bandung,Indonesra
ofMathematrcs,
'e mail : [email protected]
'ze-mail:
itbma0I @bds.centrin.net.id
Abstmct
A diregular digraph is a digraph with the in-degree and ouldegree of all venices is
conslant. The Moore botrnd for a diregular digraph of degree d and diameter * is
Ma,r=l+d+d2+..+
dk.
d>ltnd
d o n o t e x i s t . A ( d , l ) - d i g r a p hi s a d i r e g u l a rd i g r a p ho f d e g r e e d > 1 ,
dianeter t>l
diameter t>1,
I t i s w e l l k n o w n t h a t d i r e g u l a rd i g r a p h so f o r d e r M , / . t ,
and number of venices one less than the Moore bound. For degrees d=2
degree
nd3,it
has been shown tha! for diameter t > 3 there arc no such (4 l)-digmphs. However for diameter 2,
it is known that (d,2)-digraphsdo exist for an) degreed. The line digraph of (/+t
is one example
of such (42)-digaphs.
Furthermore, the recent study showed that there are three non,isomorphic
(2,2)-digraphs and exactly one non'isomorphic (3,2)-digraph. In this paper, we shall study
(4,2)-digraphs. We show that if(4,2)-digraph C contains a cycle oflength 2 then C; must be the line
digraph ofa complete digraph (i.
1. Introduction
A digraph G is a system consisting of a finite nonempty set y(G) of objects called
'fhe
vertices and a setE(G) of orderedpairs of distinct verticescalled arc.r.
order of G is
the cardinality of V(G). A subdigraph H of Gis a digraph having all verticesand arcs in
G. If (a,u) is an arc in a digraph G, tben r is said to be adjacent to y and y is said to be
adjacentfrom u. An in-neighbor of a vertex v in a digraph G is a ve ex lr such that
(u,v)e G. An out-neighbor of a vertex v in a digraph G is a vertex rr such that
(v,w)eG. The set of all out-neighborsof a vertex v is denotedby N*(v)and its
cardinality is called the out-deeree of v, d'(v) = | N- (v) | . Similarly, the set of all
in-neighborsof a vertex v is denotedby N (v) and its cardinality is called the ir-degree of
",
d- (")= N- (") A digraph
G is diregular of degree d if for any ve ex r in G,
|
l.
d*(v)=d (v)=d.
A walk of length , ftom a vertex ll to vertex v in G is a sequenceof vertices
( u = u o , u 1 , , . , , u =h v ) s u c ht h a t (u,-1,u,)e G for each i. A vertex u forms the trivial
H lswadi and E.T Baskoto
80
A path is a walk in which all points are
walk of length 0. A closed walk has 116= lo
distinct. A cycle C, of length lt > 0 is a closed walk with i distinct vertices (except 40
and |lr,). Ifthere is a path from & to v in 6 then we say thatv is teachable from u.
The distance from vertextl to vertex v in a digraph G, denotedby d(u,r''), is defined
as the length ofa shortestpath from u to v In generd, 6(u,v) is not necessarilyequal to
5(v,a). The diqmeter k of a digraph 6 is the maximum distancebetweenany two
verticesin G.
Let G be a diregular digraph of degree d and diameter t with ll vertices. Let one
vertex be distinguishedin G. Let n,,Vi=0,1," ,k, be the number of vertices at
distancei from the distinguishedve ex. Then,
.