SOME GENERAL CLASSES OF COMATCHING GRAPHS

Some sufficient conditions are given for two graphs to have the same matching polynomial (comatching graphs). Several general classes of comatching graphs are given. Also, techniques are discussed for extending certain pairs of comatching graphs to larger pairs of comatching graphs.


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 G to be a spanning subgraph of G, whose components are nodes and edges only.Let us associate with each node in G, an indeterminate or weight Wl, and with each edge, a weight w 2. Also, let us associate k with each matching in G with k edges, the weight w 2 Then, the matching polynomial of G is M(G;) a k wlP-2k w2 k, where a k is the number of matchings in G with k edges, is a weight vector (Wl,W2) and the summation is taken over all values of k.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 polynomial.In this paper, we investigate non-lsomorphlc comatchlng graphs.As far as we know, no investigation has been made into this interesting property of graphs.We will give some general results on comatching graphs.Then we will introduce various classes of comatching graphs.In many cases, we will give results concerning the con- struction of pairs of larger comatching graphs from certain "basic" pairs of comatchlng graphs (comatching pairs).
Let us denote the node set of G by V(G) and the edge set by E(G).Let A V(G).Then G A will denote the graph obtained from G hy removing the nodes in A. When A is a singleton {a}, we will write G a. Let H be a subgraph of F..J. FARRELL AND S.A. WAHID of G. Then G H will denote the graph obtained from G by removing the nodes in H.For brevity, we will write m() for M(G;_w).

SOME GENERAL RESULTS.
The following lemma is taken from [I].It is called the un0n2z the0rem or matching polynomials.
LEMMA I. Let G be a graph and xy an edge in G, where x,y V(G).Let G' be the graph obtained from G by deleting xy and G" the graph G {x,y}.Then M(G;_w) M(G' ;w) + w 2 M(G";w).
We will refer to the algorithm implied by this lemma, as the reduction roce.
G' will be called the reduced graph and G", the incorporated graph.The following is called the Component Theorem.It was also proved in [I].
LFA 2. Let G be a graph consisting of r components GI, G2, Gr.Then r m(G) il m(Gi)" An immediate consequence of the definitions given in Section is the following theorem.
THEOREM i.Two graphs are comatching if and only if they have the same numbers of the same kinds of matchings.
Trivially, we have the following corollary in which G represents the complement of G. COROLLARY I.i.(G, G) is a comatching pair, for all graphs G. Also, if G is self- complementary, (G, G) is a comatching pair.
The following are the smallest non-trivial comatching pair (HI, H 2) and the smallest connected comatching pair (H 3, H4). is not self-complementary.H 3 is.the small- est (connected) non-selfcomplementary graph with the property that (H3, 3 is a comatching pair.It can be easily confirmed that m(H I) m(H 2) w + 3 w w 2 and that m(H 3) m(H 4) w + 5 w w 2 + 4WlW Let G be a graph.Two nodes u and v in G are called pAeudoAim if G u and G v are isomorphic, but no utomorphism of G maps u onto v.We will denote by % + H b the graph formed from two connected graphs G and H by identifying node a of G with node b of H. THEOREM 2. Let G be a graph with pseudosimilar nodes u and v. Let H be a graph containing a node x.Then m(G u + H x) m(G v + Hx).
PROOF.We will prove the result by induction on the valency d(x) of node x.Sup- pose that d(x) 0. Then H is a node and in this case we get and the result follows trivially.Let d(x) and let y be the node adjacent to node x.Apply the reduction process to G + H using edge xy.Then G' (see U X Theorem I) will consist of two components, G and H x. G" will consist of the com- ponents G u and H {x,y}. Hence by using the Component Theorem we get

2.1)
Apply the reduction process to G + H by deleting edge xy.Then G' will con- V X sist of the components G and H x. G" will consist of the components G v and H {x,y}. Therefore we get Hence from Equations (2.I) and (2.2), we get m(G u + H x) m(G v + Hx).
Let us assume that the result holds for all graphs H with d(x) < n, and let d(x) n in H. Apply the reduction process to G + H by deleting edge xy, where U X y is one of the nodes in H which is adjacent to node x.Then G' will be the graph G + H' where H' is the graph obtained from H by deleting edge xy.G" U X will consist of two components, G u and H {x,y}. Hence m(G u + H x) m(G u + H) + w 2 m(G u) m(H {x,y}).
(2.3) Similarly, by applying the reduction process to G + H we get (2.4) Therefore from Equations (2.3) and (2.4), we get m(G u + H x) m(G v + Hx).
Hence the result holds for d(x) n.It follows by the Principle of Induction, that the result is true for all n.
Theorem 2 is a useful result for constructing comatching pairs, since the graphs G + H and G + H will generally be non-isomorphic.
Let G and H be connected graphs.We ch H to G by identifying a node x of H with a node y of G.In the resulting graph, G and H will be subgraphs with exactly one node in common the node z formed in the identification process, z will be called the node of attachment.Let us denote G(H), the graph obtained from a given graph G by attaching in the same manner, a copy of the graph H to each of the nodes of G It is clear that if G contains Pl nodes and ql edges, and H contains P2 nodes and q2 edges, then G(H) will contain plP2 nodes and (ql + plq2 edges.If H is attached to only some of the nodes of G the resulting graph will be denoted by G*(H).
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. Let (A,B) be a comatchlng pair.Suppose that there exist nodes a and b in A and B respectively, such that (A a, B b) is also a comatchlng pair.
Then (P2(A), P2(B)) is a comatchlng pair, when a and b are used in the attachment process.
POOF.Apply the reduction process to P2(A) by deleting the edge belonging to P2" The reduced graph will consist of two components, each being the graph A. The incor- porated graph will consist of two copies of A a. We therefore get m(P 2(A)) m(A)2 + w2 m(A a)2.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.
Then (Pn(A) P (B)) is also a comatchlng pair when the attachments are made using n nodes a and b.
It is unnecessary to attach A and B to every node of P in order to obtain n a comatchlng pair.This is implied by the following theorem, which is a further generalization of the lemma.THEOREM 4. Let (A,B) be a comatchlng pair satisfying the conditions of Lemma 3.
Then (P(A), P*(B)) is also a comatchlng pair where the attachments are made to the n same nodes of P in forming the two graphs P(A) and P*(B) and nodes a and b n respectively are used in all the attachments.PROOF.We can apply the reduction process to P*(A) by always deleting edges incident n to nodes of attachment, until all the reduced and incorporated graphs are either graphs of the form P (A) or A a. The reduction process can then be applied to P*(B) in r n the identical manner.The result will then follow from Lemma I, the Component Theorem and Theorem 3.
Theorem 4 can be generalized even further.Consider the graph G*(A).We can apply the reduction process to this graph until the reduced and incorporated graphs are of the form P*(B) or B b.Then, by using Lemma 3 the Component Theorem and r Theorem 4, we obtain the following general result.
THEOREM 5. Let (A,B) be a comatchlng pair such that (A a, B b) is also a co- matching pair, where a and b are nodes in A and B respectively.Then (G*(A), G*(B)) is also a comatchlng pair, when the attachments are made using nodes a and b of A and B respectively.
Theorem 5 provides us wlth a general construction technique for large families of comatchlng graphs.The following is a comatching pair (A,B) satisfying the condi- tions of Lemma 3. A: 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 A and B (of Lemma 3) to other graphs, we could instead attach other graphs to A and B, in order to construct new comatching pairs.THEOREM 6.Let (A,B) be a comatching pair satisfying the conditions of Lemma 3. Let A*(G) and B*(G) be the graphs obtained by attaching a graph G to A and B re- spectively.Then (A*(G),B*(G)) is a comatching pair.PROOF.Apply the reduction process to A*(G) by deleting one of the edges of G which be the reduced graph.The is incident to the node of attachment node z.Let G incorporated graph will consist of two components G y and A a, where y is the and to subsenode used to attach G to A. We can apply the reduction process to G in the same manner until the reduced graph becomes disconnected with two com- quent G i ponents G y and A. Hence  We will denote by G r the graph formed by attaching the chain P (using an end- PROOF.Apply the reduction process to each graph in turn, by deleting an edge which is incident to the node of attachment.In both cases, the reduced graph will be Pa+b-I and the incorporated graph will consist of two components Pa-I and Pb-2" Hence the result follows from Lemma I. Let x and y be the nodes of G and Gb_l,a+ respectively, which are ad- a,b jacent to the node of attachment.Then it can be easily confirmed that G x will a,b be Pa+b-2 and that Gb_l,a+ y will also be Pa+b-2" Hence these graphs satisfy the conditions of Lemma 3. 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 P and r Cr+lhaS the same matching polynomial as the chain r+l It can be proved by appro- priate applications of the reduction process.THEOREM 8.For all.ointegers r, m(Pr) m(Cr+I) m( P2r + i 5. A GENERAL COMATCHING QUADRUPLE. We will give a technique for constructing four non-lsomorphic graphs with the same matching polynomial i.e., c0mtegOg qudaple.Let (A,B) be a comatching pair, satisfying the conditions of Lemma 3. Let G be a graph containing two nodes x and y.Construct four graphs as follows.H is the graph obtained by attaching a graph A (using node a) to G, at the nodes x and y.Similarly H 2, H 3 and H 4 are the graphs formed by attaching to G, at nodes x and y respectively, the graphs A and B, B and A and B and B respectively.Then we have the following theorem.
THEOREM 9.The graphs H H 2, H 3 and H 4 all have the same matching polynomial i.e., (H H 2, H 3, H 4) is a comatching quadruple.
PROOF.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.The reduced graphs will either be A or B. The incorporated graphs will be A a, 'B b and G {x,y} The result will then follow from Lemma 3.
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.
the valency of node y in G. Y By applying the reduction process to B*(G), in an identical manner, we get m(B*(G)) m(G y) m(B) dym(G y) m(B b).
(Ga,b, Gb_l,a+I) is a comatching pair, for all positive integers a and b.