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A GEOMETRIC CHARACTERIZATION
OF THE LINE GRAPH OF A PROJECTIVE PLANE
by
T.A
DOWling and Renu Laskar
University of North Carolina
Institute of Statistics Mimeo Series No. 516
March 1967
Abstract
The line graph of a graph G is defined to be the graph whose vertices
are the edges of G, two vertices being adjacent if and only if the correspondingedges of G have a common vertex. If rr is a projective plane of
order n, then the bi~artite graph of rr, denoted by G(rr), is the graph whose
vertices are the 2(n +n+l) points and lines of rr, two vertices of G(rr) being
adjacent if and only if one is a point and the other is a line containing
the point. The line graph of G(rr) will be denoted by L(rr). A characterization of the class [L(rr)) in terms of the eigenvalues of the adjacency
matrix of L(rr) has been given by Hoffman. An alternative characterization
of the class [L(rr) )is presented here. If d(x,y) denotes the distance
between two vertices x and y and A(x,y) denotes the number of vertices
adjacent ,to both x and y, then G = L(rr~ has the following properties: (b )
The number of vertices in G is (n+l)(n +n+l). ,(b ) G is connected and l
2
regular of valence 2n. (b3) A(x,y) = n-l if d(x,y)
= 1. (b4) A(x,y) = 1
if d(x,y) = 2. (b5) If d(x,y) = 2, the number of vertices z, such that
d(x;z) = 1 and d(y,z) = 2, is at most n-l. In this paper we show that any
graph G with no loops and no multiple edges having the properties (bl)-(b5)
must be the line graph of a projective plane of order n.
This research was su~orted by the National Science
Foundation Grant No. GP-3792 and the Air Force Office of
Scientific Research Gra~ No. AF-AFOSR-76o-65.
~
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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A GEOMETRIC CHARACTERIZATION
OF THE LINE GRAPH OF A PROJECTIVE PLANE
by
T. A. Dowling and Renu Laskar
University of North Carolina
Chapel Hill, N. C.
Introduction.
L
We shall consider only finite, undirected graphs G with
no loops and no multiple edges.
The line graph of G is the graph whose
vertices are the edges of G, two vertices being adjacent if and only i f the
corresponding edges of G have a common vertex.
of order n, then the bipartite graph of
~,
~
is a projective plane
denoted by
whose vertices are the 2(n2 +n+l) points and
G(~)
If
li~es
of
G(~),
~,
is the graph
two vertices of
being adjacent if and only if one is a point and the other is a line
The line graph of G(~) will be denoted by L(~).
.
2
Hoffman [2] showed that if G is a regular connected graph on (n+l)(n +n+l)
containing the point.
vertices such that the adjacency matrix of G has the same distinct eigenvalues
as those of the adjacency matrix of
L(~),
projective plane of order n.
the eigenvalues of the adjacency matrix
of
L(~)
order n.
S~nce
then G =
L(~')
where
~'
is a
depend only on n, they are the same for every projective plane
~
of
Thus Hoffman's result states in effect that these eigenvalues
characterize the class (L(~)) consisting of the line graphs of all projective planes of order n.
In this paper we obtain an alternative charac-
terization of the class (L(~)J based on properties of L(~) other thaa the
This research was supported by the National SCience Foundation Grant
No. GP-3792 and the Air Force Office of SCientific Research Grant No.
AF-AFOSR-76o-65.
1
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eigenvalues of its adjacency matrix.
2.
Properties of L(1f).
In this section we shall deteI'llline several properties
of L(1f) which will subsequently be shown to characterize the class of graphs
(L(1f»).
It is convenient here to recall a few definitions.
Let G be a finite, undirected graph with no loops and no multiple
A path C "" (x ,x , ••• 'XII.) is a sequence of ra ~ 2 vertfces of G, not
l 2
necessarily all distinct, such that any two consecutive vertices in the
edges.
sequence are adjacent and the pairs (xl ,x2 ), (x2,~), ••• ,(xm_l,xm) are
distinct edges of G.
The number of edges m-l is the length of C, and C is
said to join Xl and xm•
an arc.
If no vertex appears twice in C, then C is called
If' Xl = xm' but no other vertex appears tWice, C is called a
circuit.
G is said to be connected if there is a path joining x and y for every
pair of distinct vertices x,y in G.
For a connected graph the distance
d(x,y) is defined to be the length of the shortest path joining x and y.
Clearly d(x,y)
define d(x,x)
=1
if and only if x and y are adjacent.
In addition, we
= o.
The valence v(x) of a vertex x in G is the number of vertices adjacent
to x.
If there exists an integer v such that v(x) = v for every vertex x
in G, then G is said to be regular of valence v.
For any two distinct vertices x and y, A(x,y) denotes the number of
vertices adjacent to both x and y.
It x and y are adjacent vertices,
ll.(x,y) is called the edge-degree of the edge (x,y).
A regular graph G for
which all edges have the same edge-degree A is said to be edge-regular with
-.
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edge-degree ll..
2
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If G
Lemma (2.1) •
= L(n),
where n is a projective plane of order n,
then
(b )
l
(b )
2
2
G contains (n+l)(n +n+l) vertices;
G is connected and regular of valence 2nj
G is edge-regular with edge-degree n-l,
(b )
3
i.e., A(x,y) = n-l if d(x,y) = 1;
A(x,y)
(b )
if d(x,y)
5
d(x,z)
=1
(b 4 )
=1
and d(y,z)
= 2,
if d(x,y)
= 2,
= 2j
the number of vertices z, such t:hat
is at most n-l.
Proof.
(b ): The vertices of L(n) can be represented by pairs (p,!),
l
where P is a point in nand! is a line in n containing P. Since n contains
n2 +n+l points and each point is incident with n+l lines, the number of such
2
pairs is (n+l)(n +n+l).
(b ):
2
Two distinct vertices
and only if P
= PI
two categories:
or
1= 11.
(p,l) and (Pl'!l) of L(n) are adjacent if
Thus the vertices adjacent to
(i) those of the form (P,m) where P
adjacent to
Let
I
m and m ~
I;
and
€
I
and Q 1= P.
and n points on
I
other than P, the number of vertices
(ii) those of the form (Q,l) where Q
through P other than
€
(p,l) fall into
Since there are n lines
(p,l) is clearly 2n.
(p,l) and (pl,ll) be two distinct vertices of L(n). We wish to
show that there is a path C joining
(p,l) and (Pl,ll).
the result is obvious, so we may assume P 1= Pl ,
unique line m in n containing P and Pl.
If m
I
= I,
1=
11 •
If they are adjacent
There exists a
we can take C
= ((p ,[),
m 1=
(Pl,l l ))· If m = 11 , we let C =((Px!), (p,ll ), (Pl,ll)). If m ~ I,
II' then C = ((p,I), (P,m), (pl,m), (pl,ll)) is a path joining (p,l)
and
(Pl,ll).
(pl,l),
It follows that L(n) is connected.
In addition, we note that
(p,l) and (Pl,ll) are at distance two if and only if P
3
1= PI'
I
1=
11 ,
but
either P
11
€
or P
€
l
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I·
If (p,l) and (Pl,/ ) are adjacent, then either P = P or K = 1 •
l
1
l
3
Consider first the case P = Pl. Clearly every vertex of the form (P,m)
(b ):
where P
€
I,
m and m ~
m~
11 ,
(p,l) and (p,l l ).
is adjacent to both
There are n-l lines m and hence n-l such vertices.
can be adjacent to both
Clearly no other vertex
1=11 .
(p,l) and (p,/ l ). Next suppose
vertex of the form (Q,/) where Q
€
1 and
Then every
Q ~ P, Q ~ P , is adjacent to both
l
(p,/) and (pl,/). There are n-l points Q and hence n-l such vertices, and
these are clearly the only vertices adjacent to both (p,!) and (pl,l).
Hence in either case we have A(x,y)
= n-l
when x
= (p,!)
and y
= (pl,l l )
are adjacent.
(b 4 ):
If x
= (p,/),
proof of (b 2 ), we have P ~
first that P
€
11 •
= (pl,/ l ),
P , 1 ~ /~,
l
y
This implies P ,
l
and d(x,y)
= 2,
but either P
I.
€
then, as noted in the
11
or ~
€
I.
Suppose
The only pairs (Q,m) which could
(p,/) and (pl,l l ) are (p,/ l ) and (P l ,().
since P € 1 ; the second, however, is not a
1
have one element in common with both
The first is a vertex of
vertex s,ince P1 ¢
I·
L(~)
A similar argument shows that (p1,1) is the only
(p,l) and (Pl'!l) in the case P l
either case we have A(x,y) = 1 if d(x,y) = 2.
vertex adjacent to both
€
/.
Hence in
Again if x = (p,l) , y = (Pl,/ l ), and d(x,y) = 2, then P ~ ~,
5
1 ~ 11 , but either P € /1 or Pl € I. Suppose first that Pl € I. Then
clearly the n-l vertices of the form z = (P,m) where P € m and m ~ K,
(b ):
m~
11 ,
are adjacent to (p,l) and at distance two from (Pl,/l).
vertex adjacent to
(p'/) is of the form (Q,I) where Q
€
I
Any other
and Q ~ P.
If
(Q,I) is at distance two from (Pl,l ), then either P € 1 or Q € 11 • But
l
l
P € 1 implies P , f and Q , 1 • Hence there are exactly n-l vertices
1
l
1
adjacent to
(p,/) and at distance two from (Pl,ll).
4
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Next suppose P
z
=
(Q,t) where Q
f.
€
l
f
€
and Q
tance two from (Pl,l ).
l
(P,m) where P
€
In this case the n-l vertices of the form
~
P, Q
~
P , are adjacent to
l
Any other vertex adjacent to
m and m~
t.
(p,t) and at dis-
(p,t) is of the form
If (P,m) is at distance two from (Pl,t ), then
l
€ m. But P € 1 implies P ¢ /1 and P ¢ m. Hence
l
l
l
again we have exactly n-l vertices z adjacent to x = (p,!) and at distance
either P
€
two from y
3.
11
=
or P
It follows a fortiori that (b ) holds for L(~).
(Pl,t l ).
5
Geometric characterization of {L(~)J.
We shall prove in this section
that properties (b )-(b ) of Lemma (2.1) characterize the class {L(~)J.
l
5
The proof reqUires a theorem due to Bose and Laskar on the grand cliques of
an edge-regular graph.
For completeness we shall quote the theorem here
along with some definitions.
Let G be a connected, edge-regular graph with valence r(k-l) and edgedegree k - 2 + a, and in addition let G satisfy A(x,y)
pair of non-adjacent vertices x and y, where r, k, a,
such that r
~
1, k
~
2, a
~
0,
~ ~
0, and r
~
~
~
- 2 a > 0.
1 +
~
for every
are fixed integers
A grand clique
K
in G is a set of mutually adjacent vertices satisfying IK I ~ k - (r-l)a,
and such that for any vertex y
not adjacent to y.
Theorem.
¢ K,
there exists at least one vertex in K
Bose and Laskar [lJ proved the following theorem for G:
If k > max[p(r,a,~), p(r,a,~)J, where p(r,a,~)
~(r+l)(r~-2a), p(r,a,~)
=1
=1
+
+ ~ + (2r-l)a, then each vertex of G is contained
in exactly r grand cliques and each pair of adjacent vertices is contained
in exactly one grand clique.
In the remainder of this paper we shall let G denote a graph possessing
properties (b )-(b ) of Lemma (2.1) for some positive integer n.
l
5
5
Lemma (3.1).
Each vertex of G is contained in exactly two grand cliques,
and each pair of adjacent vertices is contained in exactly one grand clique.
Froof.
It is easily verified that G satisfies the conditions of the
previous theorem when r
p(r,a,~)
= 1.
(r-l)a
= n+1.
k
= n+l, a = 0,
~
= 0.
In this case p(r,a,~)
=
Hence if k> 1, i.e., n > 0, the theorem holds for G.
Lemma (3.2).
Proof.
= 2,
Each grand clique in G contains n+l vertices.
If K is a grand clique in G, then by definition
If
I K 12: n+2,
IK 12:
k-
then given any two vertices x,y in K, there
would exist at least n vertices adjacent to both, which contradicts (b ).
3
Lemma. (3.3).
Proof.
2
The number of grand cliques in G is 2(n +n+l).
Let N be the number of grand cliques in G.
By counting pairs
(x,K) where x is a vertex in G and K is a grand clique containing x, we
have from (b l ) and Lemmas (3.1) and (3.2)
(n+l) N
= 2(n+l)(n2 +n+l),
from which the lemma follows immediately.
From Lemma (3.1) it is clear that two distinct grand cliques of G
have at most one vertex in common.
If K and L have a common vertex x,
we say that K and L intersect and write x
= K n L.
Since each vertex is
contained in exactly two grand cliques, the above representation of x as
the intersection of two grand cliques is unique.
Hence there is a one-one
correspondence between vertices of G and pairs of intersecting grand cliques
such that x <---> (K,L) if and only if x
A clique path C
= (Sl,S2, ••• ,Sm)
= K n L.
is a sequence of m 2: 2 grand cliques
of G, not necessarily a.ll distinct, such that any two consecutive grand
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cliques in the sequence intersect and the vertices Si
are all distinct.
n Si+l' i = 1,2, ••• ,m-l
The number m-l of such vertices will be called the length
of C and denoted by
I(C). C is said to join Sl and Sm'
appears twice in C, then C is a
clique~.
If Sl
If no grand clique
= Sm'
but no other grand
clique appears tWice, then C is called a clique circuit.
Lemma (3.4).
If C is a clique circuit in G, then [(C) ~ 6.
Proof.
= (Sl,S2"'.'S[(C)'
Let C
Sl)'
The cases [(C) = 1,2 are clearly
impossible, so it is sufficient to consider the cases 3
(i)
y
Suppose [(C) = 3.
= S2 n S3'
[(C)
= 3,
Consider the adjacent vertices x
then z
=2
= S3 n Sl
is adjacent to both x and y and
n S3
d(y,z)
~
Hence
n S2 and y = S3 n S4. Clearly
But if
I(C)
= 4, then zl
= 2,
= 2.
= 5.
Let x
= Sl n S2
and y
= S4 n Sl
and z2
=
= S3 n S4.
Since
by (b ) there can be at most n-l vertices z such that d(x,z)
5
But each of the n-l vertices in S2 other than x and x* = S2
is of this type, and in addition so is z' = S5
[(C)
¢ S2.
If
are adjacent to both x and y, which contradicts (b ).
4
(iii) Suppose [(C)
d(X,y)
Z
and
for otherwise there would exist a clique circuit of length 3,
contradicting (i) above.
S2
= Sl n S2
There are n-l vertices in S2 adjacent to both x and y.
we have A(x,y) ~ n, contradicting (b ).
3
(ii) Suppose [(C) = 4. Let x = Sl
d(x,y)
S [(C) S 5.
n Sl'
= 1,
n S3
It follows that
6.
Let x
=KnL
be an arbitrary vertex of G.
Denote the n grand cliques
intersecting K other than L by L , L , ••• ,L , and the n grand cliques interl
n
2
secting L other than K by K ,K , ••• ,K •
n
l 2
For i
= 1,2, ••• ,n,
let K ,K , •.. ,
il i2
Kin be the n grand cliques intersecting L other than K, and let Lil , Li2 , ••• ,
i
L.
~n
be the n grand cliques intersecting K. other than L.
~
7
Lemma
(3.5). The 2(n2 +n+l) grand cliques K,L,Ki,Li,Kij,Lij, i,j
I
I
=
-.
1,2, •.• ,n, defined above are all distinct, and hence constitute all the
grand cliques of G.
This lemma follows directly from Lemma
Proof.
(3.4), for if any two
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of these grand cliques are identical then the common clique will be contained in a clique circuit of length 5 or less.
then the sequence C
A
= (K,Li,Kij,Kil,L,K)
For example, if Kij
= Li
I
jI
,
is a clique circuit bf length 5.
similar result holds for any pair of grand cliques in the above collection
as can be easily verified.
2
The 2(n +n+l) grand cliques of G can be divided into two
sets 'P( and
S,
2
each containing n +n+l grand cliques, such that no two grand
cliques in the same set intersect.
Proof.
i,j
Let the grand cliques of G be denoted by K,L,Ki,Li,Kij,Lij'
= 1,2, ••• ,n,
el
as defined above, and let
1'<
= [K,
K. ,K..
~
~J,
s, = [L, L , L..
i
~J,
i, j
= 1,2, ••• ,nl,
i, j
= 1,2, ... ,nl.
These two sets satisfy the conditions of the lemma, for it is easily verified that
if two cliques in the same set intersect, then these cliques are con-
(3.4).
tained in a clique circuit of length five or less, contradicting Lemma
Lemma (3.7).
For any two distinct grand cliques S and T in G, there
exists a clique path C joining S and T such that
[(C)
~
3.
Proof. Without loss of generality we can take S = K where K, L, K., L.,
~
K.. , L ., i, j
iJ
~J
= 1,2, ••• ,n
are as defined above.
~
Then for each possible
choice of T, the sequence C defined below is seen to be a clique path satis-
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fying the conditions of the lemma.· If T
= L, c = (K, L); if T = L.,
~
C = (K,L ); if T = Ki , C = (K,L,Ki ); if T = Kij , C = (K,Li , Kij ); and
i
if T = L. ., C = (K, L, K., L. j ) .
~J
~
~
Consider now the graph H whose vertices are the grand cliques of G,
where two vertices S* and T* of H are adjacent if and only if the corresponding grand cliques S and T, respectively, of G intersect.
Lemma (3.8).
(i)
If H is the graph defined above, then
H is regular of valence n+l;
(ii)
(iii)
H is bipartite and each vertex set of H contains n2 +n+l vertices;
two distinct vertices of H are at distance two if and only if
they are in the same vertex set, and in this case there is exactly one
vertex adjacent to both.
Proof.
(i)
The fact that H is regular of valence n+l follows
immediately from Lemmas (3.1) and (3.2).
(ii)
A graph is bipartite if the vertices can be assigned to disjoint
sets U, V such that every edge (u, v) connects a vertex u
v
E:
V [3J.
E:
U to a vertex
Consider the two sets'l*, ~* of vertices in H corresponding to
the sets't, ~ of grand cliques in G described in Lemma (3.6).
If (S*,T*) is
an edge in H, then S and T are intersecting grand cliques in G, and thus
by Lemma (3.6) one of S and T must be in'K and the other in i:..
that one of S*, T* is in 'K * and the other is in
~*.
It follows
Hence H is bipartite
2
and each vertex set of H contains n +n+l vertices.
(iii)
If S*, T* are vertices in H such that d(S*, T*)
clearly they are both in the same vertex set.
= 2,
then
Conversely, if S* and T*
are in the same vertex set, then d(S*,T*) = 2t is even.
There exists a
path C joining S* and T* of length 2t, but no shorter path joins S* and T*.
9
This implies that in G there exists a corresponding clique path C of length
2t joining Sand T, but no shorter path joins S and T.
Hence by Lemma
(3.7)
we have [(C) = 2t ~ 3, which implies t=l and therefore d(S*,T*) = 2.
Suppose next that d(S*,T*) = 2 and that there exist two vertices
U*, V* adjacent to both S* and T*.
Then C* = (S*, U*, T*, V*, S*) is a
circuit of length four in H, and thus C = (S,U,T,V,S) is a clique circuit
of length four in G, which contradicts Lemma (3.4).
Hence A(S*, T*) = 1
if d(S*,T*) = 2.
Lemma
(3.9). H =
Proof.
G(~I)
for some projective plane
~I
of order n.
Let the vertices in one vertex set be called lines and the
vertices in the other vertex set be called points.
We define a point and
a line to be incident if and only if the corresponding vertices are adjacent
in H.
Then it follows from Lemma
(3.8) that there are n2+n+l points and
2
n +n+1 lines, that each point is incident with n+1 lines and each line is
incident with n+l points, and that any two points are incident with
exactly one line and any two lines are incident with exactly one point.
Thus the set of points and lines together with the incidence relation
defined above constitute a projective plane
graph of
~I
~I
of order n.
The bipartite
is clearly H.
Theorem.
If G is an undirected graph with no loops and no multiple edges
possessing the properties (b )-(b ) of Lemma (2.1), then G
l
5
~'
= L(~')
where
is a projective plane of order n.
Proof.
graph of H.
Since H
= G(~I)
it will suffice to show that
G is the line
We must show that there is a one-one correspondence between
edges of H and vertices of G such that two edges of H have a common vertex
if and only if the corresponding vertices of G are adjacent.
10
If we define
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a correspondence by saying that the vertex z in G corresponds to the edge
(5*,T*) in H if and only if z
is one-one.
If (51, T*) and
= 5 n T,
(5~,
then clearly this correspondence
T*) are two edges in H with a common
vertex T*, then the corresponding vertices zl = 51
nT
and z2
= 52 n T
in G are both contained in the grand clique T and are therefore adjacent.
Conversely, if zl and z2 are adjacent vertices in G, let T be the grand
clique containing both zl and z2 and let 5 , i=1,2,be the grand clique
i
containing zi other than T.
Then the vertices zl = 51
n T and z2 = 52 n T
in G correspond to the edges (5 * ,T* ), (5 * ,T* ), respectively, in H.
1
2
Hence
the edges in H corresponding to the adjacent vertices zl' z2 in G have a
*
common vertex T.
This completes the proof.
It follows from the above theorem that properties (b )-(b ) of
l
5
Lemma (2.1) characterize the class (L(1C)) consisting of the line graphs
of all projective planes of order n.
In the early stages of this research
attempts were made to show that property (b ) is redundant (inasmuch as
5
Bose and LaskarIs theorem and hence Lemma (3.1) hold without assuming (b ))
5
and that, consequently, the class (L(1C)) could be characterized by
properties (b )-(b 4 ).
l
The following conterexample shows that property
The graph G below satisfies (b )-(b ) for
4
l
n=2, but it contains many clique circuits of length 5. Hence H contains
(b ) is, indeed, necessary.
5
circuits of length 5 and cannot be bipartite.
represent grand cliques.
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The lines in the diagram
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Figure 1.
Counterexample showing that
condition
(b~)
;)
is not redundant (n=2).
-.
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References
[lJ
R. C. Bose and Renu Laskar, A characterization of the tetrahedral
graph, Institute of Statistics Mimeo Series No. 509, University
of North Carolina, March, 1967.
[2J
A. J. Hoffman, On the line graph of a projective plane, Proceedings of
the American Mathematical Society, Vol. 16, 1965.
[3J
o.
Ore, Theory of Graphs, American Mathematical Society ,Colloquium
Publications, Vol. XXXVIII, 1962.
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