519
Internat. J. Math. & Math. Sci.
Vol. i0, No.3 (1987) 519-524
SOME GENERAL CLASSES OF COMATCHING GRAPHS
E.J. FARRELL and S.A. WAHID
Department of Mathematics
The University of the West Indies
St. Augustine, Trinidad
(Received May 6, 1985)
ABSTRACT.
Some sufficient conditions are given for two graphs to have the same match-
ing polynomial (comatching graphs).
Several general classes of comatching graphs are
Also, techniques are discussed for extending certain pairs of comatching graphs
given.
to larger pairs of comatching graphs.
KEY WORDS AND PHRASES. Matching polynomial, comatching graphs, weight, chains, cycles,
attaching graphs to other graphs.
1880 AMS SUBJECT CLASSIFICATION CODE. 05A88, 0C88.
I.
INTRODUCTION.
The graphs considered here will be finite and without loops or multiple edges.
Let
G
be such a graph.
We define a mtching in
indeterminate or
weight Wl, and with each edge,
with each matching in
polynomial of
G
G
with
k
a
k
(Wl,W2)
a weight
w
Also, let us associate
2.
w
edges, the weight
k
2
Then, the matching
is
M(G;)
where
G,
G, an
to be a spanning subgraph of
G
Let us associate with each node in
whose components are nodes and edges only.
a
k
is the number of matchings in
G
wlP-2k w2k,
with
edges,
k
and the summation is taken over all values of
k.
is a weight vector
The basic results on match-
ing polynomials are given in Farrell [I].
We define wo graphs to be comatching if and only if they have the same matching
In this paper, we investigate non-lsomorphlc comatchlng graphs.
polynomial.
As far as
we know, no investigation has been made into this interesting property of graphs.
We
Then we will introduce various
will give some general results on comatching graphs.
In many cases, we will give results concerning the construction of pairs of larger comatching graphs from certain "basic" pairs of comatchlng
classes of comatching graphs.
graphs (comatching pairs).
Let us denote the node set of
A
V(G).
in
A.
Then
When
A
G
G
by
V(G)
and the edge set by
A will denote the graph obtained from
is a singleton
{a},
we will write
G
a.
G
Let
E(G).
Let
hy removing the nodes
H
be a subgraph of
520
F..J. FARRELL AND S.A. WAHID
H
of
G.
H.
For brevity, we will write
2.
SOME GENERAL RESULTS.
or
matching polynomials.
G
Then
will denote the graph obtained from
m() for
G
Let
xy
be a graph and
the graph obtained from
G
by removing the nodes in
M(G;_w).
The following lemma is taken from [I].
LEMMA I.
G
an
It is called the
edge in
un0n2z the0rem
G, where
V(G).
x,y
M(G;_w)
xy
and
called the Component Theorem.
LFA 2.
Let
G
G",
reduced graph and
will be called the
the incorporated
It was also proved in [I].
be a graph consisting of
r components
graph.
be
Then
We will refer to the algorithm implied by this lemma, as the reduction
G’
G’
Let
{x,y}.
G" the graph G
M(G’ ;w) + w2 M(G";w).
by deleting
roce.
The following is
Gr.
GI, G2,
Then
r
m(G)
il m(Gi)"
is the following
An immediate consequence of the definitions given in Section
theorem.
Two graphs are comatching if and only if they have the same numbers of the
THEOREM i.
same kinds of matchings.
Trivially, we have the following corollary in which
G
represents the complement
G.
of
(G, G)
COROLLARY I.i.
is a comatching pair, for all graphs
complementary, (G, G)
G.
Also, if
G is self-
is a comatching pair.
The following are the smallest non-trivial comatching pair
(H 3,
smallest connected comatching pair
(HI, H 2)
and the
H4).
Figure
H
Notice that
est
H
4
3
and therefore
H
(connected) non-selfcomplementary graph with the property that
m(H I)
m(H 3)
Let
and
G
be a graph.
G
will denote by
3
is.
(H3,
3
the smallis a
w w
w + w 4WlW
m(H4)
Two nodes
+
m(H2)
and that
u
H
It can be easily confirmed that
comatching pair.
G
is not self-complementary.
3
5
u
and
3
w
v
in
w
2
2
+
G
are called pAeudoAim if
v
are isomorphic, but no utomorphism of
%
+ Hb
G
maps
the graph formed from two connected graphs
u
G
onto
v.
and
H
We
SOME GENERAL CLASSES OF COMATCHING GRAPHS
by identifying node
m(Gv +
m(Gu + Hx)
d(x)
0.
H
Then
+H
=G
X
U
Apply the reduction process to
x.
G
m(G
{x,y}.
H
and
u
u
+ Hx)
G
{x,y}.
Sup-
x.
G
+ HX
U
G
be the node adjacent to
y
using edge
H
and
Then
xy.
G" will
x.
G’
(see
consist of the com-
Hence by using the Component Theorem we get
x) + w 2 m(G
m(G) m(H
G
Apply the reduction process to
sist of the components
of node
=G
X
and let
Theorem I) will consist of two components,
ponents
+H
V
d(x)
Let
and the result follows trivially.
H
d(x)
is a node and in this case we get
G
node
be a graph
Hx).
We will prove the result by induction on the valency
PROOF.
pose that
H
Let
v.
and
u
Then
x.
containing a node
G
of
a
be a graph with pseudosimilar nodes
G
Let
THEOREM 2.
521
H.
of
b
with node
V
H
and
+ HX by deleting edge
G"
x.
2.1)
{x,y}).
u) m(H
Then
xy.
will consist of the components
G’
will con-
G
v
and
Therefore we get
m(G
+ Hx)
v) m(H- {x,y}).
m(G) m(H- x) + w 2 m(G
(2.2)
But from the definition of pseudosimilar nodes,
G-uG
m(G
-v
m(G
u)
v).
Hence from Equations (2. I) and (2.2), we get
m(G
+ Hx)
u
m(G
v
+
Hx).
H
Let us assume that the result holds for all graphs
d(x)
y
n
H.
in
is one of the nodes in
graph
G
U
+ H’X
G
Apply the reduction process to
H
H’
where
U
+ HX by
which is adjacent to node
is the graph obtained from
G
will consist of two components,
m(Gu + Hx)
Similarly, by applying the reduction
m(Gv + Hx)
m(Gv
deleting edge
Then
x.
H
d(x) < n, and let
G’
xy, where
will be the
by deleting edge
xy.
G"
H
{x,y}. Hence
w
{x,y}).
u) m(H
m(G
+
H) 2
process to GV + HX we get
(2.3)
Hx) + w m(G- v) m(H- {x,y}).
2
(2.4)
u
m(Gu +
with
and
But from the induction hypothesis,
m(G
u
+
H)
+
H’).x
m(Gv +
Hx).
m(G
v
Therefore from Equations (2.3) and (2.4), we get
m(Gu + Hx)
Hence the result holds for
d(x)
the result is true for all
n.
n.
It follows by the Principle of Induction, that
Theorem 2 is a useful result for constructing comatching pairs, since the graphs
G
U
3.
x
+ HX
and
G
V
+ HX
will generally be non-isomorphic.
A GENERAL TECHNIQUE FOR CONSTRUCTING COMATCHING GRAPHS.
Let G and H be connected graphs. We ch H to
of
H
with a node
y
of
G.
In the resulting graph,
G
G
by identifying a node
and
H
will be subgraphs
522
E.J. FARRELL AND S.A. WAHID
with exactly one node in common
will be called the
G
given graph
contains
G(H),
Let us denote
It is clear that if
nodes and
P2
If
edges.
plq2
node of attachment.
formed in the identification process,
z
H
q2 edges,
G
then
contains
G(H)
Pl
z
the graph obtained from a
H
by attaching in the same manner, a copy of the graph
G
the nodes of
the node
nodes
and
ql
plP2 nodes
will contain
G
is attached to only some of the nodes of
to each of
H
edges, and
(ql
and
+
the resulting graph
G*(H).
will be denoted by
By a chain, we will mean a tree with nodes of valencies
and 2 only. The chain
with n nodes will be denoted by P
The followlnglemma gives a simple technique for
n
constructing a larger comatching pair, starting with a given comatchlng pair, which
satisfies certain conditions.
LEMMA 3.
in
A
(A,B)
Let
and
B
be a comatchlng pair.
(P2(A), P2(B))
Then
(A
respectively, such that
Suppose that there exist nodes
b)
a, B
is a comatchlng pair, when a
and
a
b
is also a comatchlng pair.
b
and
are used in the attachment
process.
POOF.
P2(A)
Apply the reduction process to
by deleting the edge belonging to
The reduced graph will consist of two components, each being the graph
porated graph will consist of two copies of
m(P 2(A))
A
m(A)2 +
a.
w2
A.
P2"
The incor-
We therefore get
a)2.
m(A
(3.1)
Similarly, we get
m(P2(B))
(A,B)
Since
(A
and
a, B
b)
re(B) 2 + w 2 m(B
b) 2.
(3.2)
are comatching pairs
m(A)
m(B) and m(A
a)
m(B
Hence the result follows from Equations (3.1) amd (3.2).
b).
(3.3)
The following theorem gives a generalization of the lemma.
It can be easily
proved by induction on the number of nodes in the chain.
THEOREM 3. Let (A,B) be a comatchlng pair satisfying the conditions of Lemma 3.
(Pn(A)
Then
nodes
a
P (B))
is also a comatchlng pair
n
and
when the attachments are made using
b.
It is unnecessary to attach
a comatchlng pair.
A
and
B
to every node of
P
n
in order to obtain
This is implied by the following theorem, which is a further gener-
alization of the lemma.
THEOREM 4.
Then
Let
(P(A),
(A,B)
P*(B))
n
P
same nodes of
n
be a comatchlng pair satisfying the conditions of Lemma 3.
is also a comatchlng pair
in forming the two graphs
where the attachments are made to the
P(A)
and
P*(B)
and nodes
a
and
b
respectively are used in all the attachments.
PROOF.
We can apply the reduction process to
P*(A)
n
by always deleting edges incident
to nodes of attachment, until all the reduced and incorporated graphs are either graphs
of the form
the identical
P (A)
r
or
manner.
A
a.
The reduction process can then be applied to
P*(B)
n
in
The result will then follow from Lemma I, the Component Theorem
523
SOME GENERAL CLASSES OF COMATCHING GRAPHS
and Theorem 3.
Consider the graph
Theorem 4 can be generalized even further.
G*(A).
We can
apply the reduction process to this graph until the reduced and incorporated graphs
P*(B)
r
are of the form
or
B
Then, by using Lemma 3
b.
the Component Theorem and
Theorem 4, we obtain the following general result.
(A,B)
Let
THEOREM 5.
matching pair, where
G*(B))
b
and
b
A
are nodes in
B
and
b)
a, B
(A
be a comatchlng pair such that
a
is also a co-
(G*(A),
Then
respectively.
is also a comatchlng pair, when the attachments are made using nodes
A
of
B
and
a
and
respectively.
Theorem 5 provides us wlth a general construction technique for large families of
The following is a comatching pair
comatchlng graphs.
satisfying the condi-
(A,B)
tions of Lemma 3.
a
A:
B: b
Figure 2
It can be eaily confirmed that
re(A)
m(B)
w
+
6w w2 + 5WlW
and that
The following theorem shows that instead of attaching the graphs
Lemma 3) to other graphs, we could instead attach other graphs to
A
A
(of
B
and
B, in order
and
to construct new comatching pairs.
(A,B)
THEOREM 6.
Let
A*(G)
B*(G)
and
spectively.
be a comatching pair satisfying the conditions of Lemma 3.
Then
be the graphs obtained by attaching a graph
(A*(G),B*(G))
A*(G)
node
is incident to the node of attachment
G
G
quent
ponents
to
A.
G
y
A.
and
Let
re-
Let
G
G
y
G
and
A
a, where
We can apply the reduction process to
G
y
which
The
be the reduced graph.
is the
and to subse-
Hence we will get
m(G
y) m(A) +
d is the valency of node y in G.
Y
By applying the reduction process to
m(B*(G))
Since
B
and
in the same manner until the reduced graph becomes disconnected with two com-
i
m(A*(G))
where
A
by deleting one of the edges of
z.
incorporated graph will consist of two components
node used to attach
to
is a comatching pair.
Apply the reduction process to
PROOF.
G
m(A)
m(B)
m(A
and
m(G
a)
dym(G
B*(G),
y) m(A -a),
(3.3)
in an identical manner, we get
(3.4)
y) m(B)
dym(G y) m(B b).
m(B
b), the result follows from Equations (3.3)
and (3.4).
4.
COMATCHING COMPLEXES OF CHAINS AND CYCLES.
We will denote by
node) to the cycle
C
s
G
r ,s
with
the graph formed by attaching the chain
s nodes.
P r (using an end-
524
E.J. FARRELL AND S.A. WAHID
(Ga, b, Gb_l,a+ I)
THEOREM 7.
is a comatching pair, for all positive integers
a
and
b.
PROOF.
Apply the reduction process to each graph in turn, by deleting an edge which is
In both cases, the reduced graph will be
incident to the node of attachment.
Pa-I
and the incorporated graph will consist of two components
and
Pa+b-I
Hence the
Pb-2"
result follows from Lemma I.
Let
and
x
jacent to the node of attachment.
be
Pa+b-2
G
be the nodes of
y
and that
the conditions of Lemma 3.
and
respectively, which are ad-
Gb_l,a+
Then it can be easily confirmed that
y
Gb_l,a+
a,b
will also be
G
x
a,b
will
Hence these graphs satisfy
Pa+b-2"
Theorem 7 therefore gives us a general technique, not only
for constructing comatching pairs, but for constructing graphs which satisfy Lemma 3.
Hence we can construct comatching complexes of chains and cycles, by using Theorem 5.
The following theorem shows that the graph consisting of the components
Cr+lhaS
the same matching polynomial as the chain
P
and
r
It can be proved by appro-
r+l
priate applications of the reduction process.
THEOREM 8.
For all.ointegers r,
m(Pr) m(Cr+I)
5.
m(
P2r
+
i
A GENERAL COMATCHING QUADRUPLE.
We will give a technique for constructing four non-lsomorphic graphs with
c0mtegOg qudaple.
same matching polynomial i.e.,
graph
A
(using
node a)
G, at the nodes
to
A
and
B,
THEOREM 9.
(H
B
and
B
and
B respectively.
and
H 2, H
H
The graphs
H 2, H 3, H 4)
PROOF.
A
x
G, at nodes
are the graphs formed by attaching to
and
3
H
4
(A,B)
Let
be a comatching
and
y.
x
and
Similarly
y
H 2, H
3
H
and
4
respectively, the graphs
Then we have the following theorem.
all have the same matching polynomial i.e.,
is a comatching quadruple.
The result can be established by applying the reduction process to the graphs
until the attached subgraphs
A
and
B
become disconnected.
This is accomplished by
deleting the edges of
G
which are incident to the nodes of attachment.
graphs will either be
A
or
G
x
be a graph containing two nodes
is the graph obtained by attaching a
H
Construct four graphs as follows.
y.
and
G
Let
pair, satisfying the conditions of Lemma 3.
the
{x,y}
B.
The incorporated graphs will be
A
a,
The reduced
’B
b
and
The result will then follow from Lemma 3.
6. DISCUSSION.
We have given several classes of comatching graphs.
Also, we have given some gen-
eral results which can be used to construct larger and larger classes from any given
class.
We have found other kinds of comatching graphs, but have not mentioned them her
since they do not appear to be very interesting.
REFERENCES
I.
FARRELL, E. J. An Introduction to Matching Polynomials, J. Comb. Theory B, 27(1979)
75-86.